Optimal boundary control of quasilinear elliptic partial differential equations: theory and numerical analysis


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1 Optimal boundary control of quasilinear elliptic partial differential equations: theory and numerical analysis vorgelegt von Dipl.Math. Vili Dhamo von der Fakultät II  Mathematik und Naturwissenschaften der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften  Dr. rer. nat.  genehmigte Dissertation Promotionsausschuss: Vorsitzender: Berichter: Berichter: Prof. Dr. Yuri B. Suris Prof. Dr. Fredi Tröltzsch Prof. Dr. Eduardo Casas Tag der wissenschaftlichen Aussprache: Berlin 2012 D 83
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3 Preface This thesis is concerned with a class of optimal control problems governed by quasilinear elliptic partial differential equations with inhomogeneous Neumann boundary conditions. The boundary datum will be considered as the control variable which must satisfy given inequality constraints. Control of equations of this type is interesting, because in many practical applications of optimal control theory to problems in engineering and medical science the underlying PDE s are quasilinear; for instance, in models of heat conduction, where the heat conductivity coefficient depends on the spatial coordinate and on the temperature of the system. The heat conductivity of carbon steel depends on the temperature and also on the alloying additions contained; see Bejan [7]. If the different alloys of steel are distributed smoothly in the domain, then the conductivity coefficient should depend in a sufficiently smooth way on both, space variable and temperature. Similarly, this dependence is observed in the growth of silicon carbide bulk single crystals; see Klein et al. [73]. The quasilinear equation under consideration is not monotone, because the leading coefficient of the differential operator is dependent on the solution of the equation. Control problems with quasilinear elliptic equations of nonmonotone type were recently considered by Casas and Tröltzsch [36, 37, 38]. The authors have treated the case of distributed controls and their contributions include not only the derivation of necessary and sufficient optimality conditions but also the analysis of the numerical approximation of those control problems. It is known that in the case of boundary controls the analysis is more difficult, since the regularity of the states is lower than that of distributed controls. The goal of the present work is to extend the results obtained by Casas and Tröltzsch to the case of Neumann boundary controls problems in polygonal domains of dimension two. Most of the material of this thesis can be found in [20, 18, 19]. In order to tackle different aspects of the theoretical and numerical analysis of the control problem, it is necessary to perform a comprehensive study of the wellposedness of the state equation and further to analyze the differentiability of the controltostate mapping. The first chapter deals with these topics as well as contains some useful regularity results concerning the adjoint state equation. iii
4 iv PREFACE Chapter 2 is dedicated to the finiteelement based approximation of the state and adjoint state equation. Our main focus is the error analysis for these approximations. A serious difficulty in this analysis is that the uniqueness of a solution of the discrete quasilinear equation is an open problem. To overcome this difficulty a local uniqueness result can be provided which is sufficient for further investigations. In Chapter 3, the optimal control problem associated with the quasilinear equation is formulated and the question of existence of solutions is answered positively. Furthermore, first and secondorder optimality conditions are established and some higher regularity results for optimal controls are derived. Chapter 4 contains the numerical analysis of the control problem. Approximating the state and adjoint state by finite elements of degree one and the control by step functions, the strong convergence of discrete local optimal controls to a strict local optimal control of the continuous problem can be shown. Finally, the error analysis for the optimal controls is carried out and confirmed by numerical experiments. In the last chapter, some extensions of the results concerning the numerical approximation of the quasilinear equation to threedimensional polyhedral domains are presented. Acknowledgments. At first, I would like to express my gratitude to my supervisor Fredi Tröltzsch for always being open for any question, for introducing optimal control theory to me and also for the help in gaining experience in research and teaching. My special thanks are addressed to Eduardo Casas for suggesting this interesting subject. Without his enduring support and the countless fruitful discussions I would not have been able to realize this thesis. Besides, I want to thank him for the productive and successful collaboration in the last four years. I wish to thank the whole Optimal Control Group at the Technical University of Berlin for the exchange of ideas and for all the enjoyable time we spent together. Last but by no means least, I like to thank my parents, my brother and my girlfriend for their neverending encouragement and ongoing support in all its forms.
5 General notation Given a Banach space V, we shall denote by V the standard norm of V and by, V,V the duality product between the dual space V and V. When no ambiguity arises, we will abbreviate, V,V by,. In the sequel, we recall some Banach spaces which are frequently used in this thesis. Let R n, n 1, be bounded and measurable. We say that a property holds for almost all (for a.a.) x (or a.e. in ) if it is valid in except for a measurable set of Lebesgue measure zero. The space of all continuous functions in the closure of is denoted by C( ). We shall write C 0,α ( ) for the space of Hölder functions with Hölder exponent 0 < α 1 and we shall call f C 0,1 ( ) Lipschitz continuous. Moreover, for 1 q we define where L q () = { f : R f is measurable and f L q () is finite }, ( 1/q f(x) dx) q if 1 q <, f L q () := ess sup x f(x) if q =. In the last formula dx is the Lebesgue measure in. It is wellknown that the dual space to L q (), 1 < q <, can be identified by L q (), where q is the conjugate exponent of q satisfying (1/q) + (1/q ) = 1. For 1 q we set W 0,q () := L q () and, for m N, the Sobolev space W m,q () stands for the space of all measurable functions f : R, whose weak derivatives D α f of order α, with α m, belong to L q (), cf. Adams [1]. We equip the space W m,q () with the norm 1/q f W m,q () := D α f(x) q dx α m if 1 q < and f W m, () := α m D α f L () if q =. In particular, the space H m () := W m,2 () is a Hilbert space with the scalar product (f, g) H m () = (D α f, D α g) L 2 () = ( 1/2 D α fd α g dx) 2. α m α m For a real exponent s > 0 with s N we write s = [s] + σ, where [s] is an integer and 0 < σ < 1. Then the SobolevSlobodetskij space W s,q (), 1 q <, consists v
6 vi GENERAL NOTATION of all functions f W [s],q () such that D α f(x) D α f(y) q dx dy <. x y n+σq α =[s] We equip the space W s,q () with the norm f W s,q () := f q + W [s],q () α =[s] D α f(x) D α f(y) q dx dy x y n+σq More details on Sobolev and SobolevSlobodetskij spaces can be found, e.g., in Adams [1], Nečas [82], Wloka [92]. Let now have a Lipschitz boundary =, cf. Nečas [82] or Gajewski et al. [54] for a precise definition; σ denotes the usual (n 1)dimensional measure over induced by its parametrization. The spaces C(), C 0,α () and L q () are defined analogously. However, the introduction of Sobolev spaces on is more delicate. Let us denote W 1 /q,q () := { z z W 1,q () } endowed with the norm g W 1/q := inf{ z,q () W 1,q () z = g}, where z is the trace of z on, cf. Lions [77]. To avoid heavy formulas we will mostly write z instead of z. The dual space of W 1 /q,q () is denoted by W 1/q,q (). Since 1/q = 1 1/q (0, 1), we may equip W 1 /q,q () with the intrinsic norm ( g W 1/q = g q,q () L q () + g(x) g(y) q dσ(x) dσ(y) x y n 2+q which is equivalent to the previous one; see Grisvard [60, page 20]. For the corresponding trace theorems and embedding theorems for bounded Lipschitz domains the reader is referred to, e.g., Adams [1] and Nečas [82]; see also Ding [48]. We write S (v)h for the Fréchet derivative of an operator S : V Y, Y being another Banach space, at v V in the direction h V ; for the second Fréchet derivative in the directions h 1, h 2 V, we write S (v)[h 1, h 2 ] or simply S (v)h 2 if h 1 = h 2 = h. Given two sets A and B with A B, χ B A : B {0, 1} denotes the indicator function which is one in A and zero in B \ A. If there is no risk of notational confusion we will write χ A. Throughout the thesis, B V (x, ρ) is the open ball in V with radius ρ centered at x, and B V (x, ρ) stands for its closure. By C (without index) generic constants are denoted and in some formulas, the partial derivative / x j is abbreviated by j. ) 1/q 1/q
7 Preface General notation Contents Chapter 1. Analysis of quasilinear elliptic PDEs Introduction Main assumptions on the nonlinearities Wellposedness of the quasilinear equation Regularity of solutions of elliptic PDEs in domains with corners Higher regularity of solutions of the quasilinear equation Differentiability of the solution operator The adjoint problem 29 Chapter 2. Finite element approximation Introduction Assumptions and preliminary results Numerical analysis of the quasilinear equation Uniqueness of solutions of the discrete equation Numerical experiments The discrete solution operator Numerical analysis of the adjoint equation 66 Chapter 3. The optimal control problem Introduction Problem formulation and main assumptions Firstorder optimality conditions Pontryagin s principle Secondorder optimality conditions 88 Chapter 4. Numerical approximation of the control problem Introduction Approximation of the controls by piecewise constant functions Error estimates for optimal controls Numerical experiments 115 Chapter 5. Extensions to polyhedral domains of dimension three Wellposedness of the quasilinear equation 121 vii iii v
8 viii CONTENTS 5.2. Numerical analysis of the quasilinear equation 125 Conclusions and outlook 131 Zusammenfassung 133 Bibliography 135
9 CHAPTER 1 Analysis of quasilinear elliptic PDEs 1.1. Introduction This chapter is concerned with a particular class of quasilinear elliptic equations of the form { div [a(x, y(x)) y(x)] + f(x, y(x)) = 0 in, (1.1.1) a(x, y(x)) ν y(x) = u(x) on. Our main goal is to develop a comprehensive analysis of the above problem which will be widely used in the subsequent chapters. To make the setting more flexible for applications we will impose regularity assumptions on the nonlinear terms a and f which are as weak as necessary to study the above equation in the framework of PDE constrained optimal control problems; see Chapter 3 and 4. Another aim of this chapter is to provide useful results for the numerical analysis of the finite element based approximation of (1.1.1); see Chapter 2. One problem which arises in this study is the approximation of the domain. Typically, is approximated by a new domain h with piecewise polygonal (in 2D) or polyhedral (in 3D) boundary. In many cases, the simplest and the most convenient choice consists of replacing by a polygonal or polyhedral domain h. This requires the comparison of the boundary datum u of (1.1.1) and that of the discrete equation which is defined on h. Let us mention here that the effect of the domain change on the solution of problems associated with semilinear elliptic equations is investigated by Casas and Sokolowski [33]. To simplify our analysis we will restrict our consideration to polygonal domains of dimension two. The case when the domain is a polyhedral set of dimension three is discussed in Chapter 5. However, assuming that is a plane polygonal domain not necessary convex, introduces a new difficulty: the regularity of elliptic equations in corner domains needs special care. In the analysis of the quasilinear equation (1.1.1), we will be faced with several difficulties. In spite that f is considered monotone nondecreasing with respect to y, the above equation is not monotone, because the coefficient a of the divergence term depends on y. The first difficulty caused by the nonmonotone character of the equation (1.1.1) is found when deducing regularity properties for its solution. Other difficulties appear when analyzing the linearized equation and the associated adjoint equation which are both nonmonotone; see Section 1.6 and
10 2 1. ANALYSIS OF QUASILINEAR ELLIPTIC PDES In this chapter, we will also focus our attention to the differentiability properties of the solution operator u y. These properties are of utmost importance since they allow us to establish necessary and sufficient optimality conditions for control problems governed by PDEs of the type (1.1.1). The discussion of these differentiability properties will require a careful analysis of the corresponding linearized equation. Another issue we address in this chapter is the study of the adjoint problem. There are at least two reasons why it is of high interest to investigate the adjoint equation. First, it has become an essential element in optimal control theory; see Chapter 3 and 4. Secondly, it plays a key role in the proof of error estimates for the finite element approximation of (1.1.1); see Chapter 2. Up to now, there exist only a few contributions where quasilinear equations have been studied in the context of optimal control problems. We mention Lions [77], Casas and Fernández [21, 22], Casas and Yong [41] and Casas et al. [23], for problems with nonlinearity of gradient type. Recently, Casas and Tröltzsch considered in [36, 37, 38] the equation (1.1.1) with homogeneous Dirichlet boundary conditions. Here we extend the theory developed in [36] to the more delicate case of inhomogeneous Neumann boundary conditions. This chapter provides the analysis of all partial differential equations which occur when considering optimal control problems governed by (1.1.1). It is organized as follows: In the upcoming section, we state the main assumptions on the data involved in (1.1.1). In Sections , we discuss the wellposedness of equation (1.1.1) in different spaces. Section 1.4 contains some preliminary results which are used to obtain higher regularity of the solution of the quasilinear equation. Section 1.6 is devoted to the study of the linearized equation and to the derivation of some differentiability properties of the solution operator u y. The analysis of the adjoint problem is the scope of the last section Main assumptions on the nonlinearities We assume the following hypotheses about the data involved in (1.1.1). Assumption 1.1. R 2 is an open bounded polygonal domain with boundary. We denote the unit outward normal vector to at x by ν(x). Assumption 1.2. The function a : R R is a Carathéodory function, i.e. measurable with respect to the first variable and continuous with respect to the second one. Moreover, (1.2.1) α a > 0 such that a(x, y) α a for a.a. x and all y R, a(, 0) L () and for any M > 0 there exists a constant C a,m > 0 such that a(x, y 2 ) a(x, y 1 ) C a,m y 2 y 1 for a.a. x and all y i M, i = 1, 2.
11 1.3. WELLPOSEDNESS OF THE QUASILINEAR EQUATION 3 Assumption 1.3. The function f : R R is a Carathéodory function and there exists p > 4/3 such that f(, 0) L p (). Moreover, f is monotone nondecreasing with respect to the second variable for a.a. x and there exist a positive constant α f > 0 and a measurable set E with Lebesgue measure E > 0 such that (1.2.2) f(x, y 2 ) f(x, y 1 ) y 2 y 1 α f x E and y 1, y 2 R, with y 1 y 2. Finally, we assume that for any M > 0 there exists a function φ M L p () such that (1.2.3) f(x, y 2 ) f(x, y 1 ) φ M (x) y 2 y 1 for a.a. x and all y 1, y 2 M. Example 1.4. Taking a(x, y) = φ 0 (x)+y 2m and f(x, y) = y+y 3 ζ(x), with m N, φ 0 L (), φ 0 ( ) α a > 0 a.e. in, and ζ L p (), p > 4/3, we see that the equation { div [(φ0 (x) + y 2m (x)) y(x)] + y + y 3 (x) = ζ(x) in, [φ 0 (x) + y 2m (x)] ν y(x) = u(x) on, satisfies the above assumptions. Other possible choices are a(x, y) = φ 0 (x) + e y or f(x, y) = e y ζ(x). Throughout the thesis, p and E, introduced in Assumption 1.3, will be fixed and the solutions of PDEs are understood in the weak sense: We say that y H 1 () L () is a solution of (1.1.1) if the following identity holds (1.2.4) {a(x, y(x)) y(x) φ(x) + f(x, y(x))φ(x)} dx = u(x)φ(x) dσ(x) for any test function φ H 1 () Wellposedness of the quasilinear equation To prove the existence of a weak solution y of (1.1.1) in H 1 () we cannot apply the MintyBrowder theorem for monotone operators; see, for instance, Gajewski et al. [54, Theorem III.2.1]. This is due to the fact that the equation (1.1.1) does not lead to a monotone operator in general. For an example showing the nonmonotone character of (1.1.1) in a particular case the reader is referred to Hlaváček et al. [67, Remark 2.2]. In the following theorem, we establish the existence and uniqueness of a solution to (1.1.1). Theorem 1.5. Under the Assumptions , for any u L s () with s > 1 equation (1.1.1) has a unique solution y u H 1 () L (). Moreover, there exists µ (0, 1) independent of u such that y u C 0,µ ( ) and, for any set U bounded in L s (), (1.3.1) y u H 1 () + y u C 0,µ ( ) C U u U,
12 4 1. ANALYSIS OF QUASILINEAR ELLIPTIC PDES with some constant C U > 0. Proof. Existence of a solution. To show the existence of a solution of (1.1.1) we introduce the truncated functions a M and f M as follows. Depending on M > 0, the function a M is given by a M (x, y) = a(x, y) if y M, a(x, +M) if y > +M, a(x, M) if y < M. In the same way, we define the truncation f M of f. Now let u L s () and ε > 0 be arbitrary but fixed. Consider the mapping F : L 2 () L 2 () defined by F (w) = z ε, where z ε satisfies the linear equation { div [am (x, w) z ε ] + εχ E z ε = f M (x, w) in, (1.3.2) a M (x, w) ν z ε = u on. Thanks to Assumption 1.3, (1.3.2) is monotone, hence by applying the LaxMilgram theorem, we get the existence of a unique solution z ε H 1 () of (1.3.2), therefore F is welldefined. Next we will use the Poincaré inequality (see, for instance, Gajewski et al. [54, Lemma II.1.36]) ( ) (1.3.3) z 2 H 1 () C E z 2 L 2 () + z 2 L 2 (E) z H 1 (), with C E > 0 being independent of z. Taking z ε as test function in the weak formulation of (1.3.2), along with the positivity of a M, there holds { } min{α a, ε} z ε 2 H 1 () C E am (x, w) z ε 2 + εχ E zε 2 dx ( ) = C E uz ε dσ(s) f M (x, w)z ε dx C ( u L s () + f M (, w) L p ()) zε H 1 (). In the last inequality, we have used the continuity of the trace operator γ : H 1 () H 1 /2 (), γ(z)(x) = z (x), for z H 1 () C( ) and a.a. x, as well as the continuous embeddings H 1 () L p () and H 1 /2 () L s (), where p and s are the conjugate exponents to p and s, respectively. Thus, z ε H 1 () C a,ε ( u L s () + f M (, w) L p ()), where C a,ε depends only on, α a and ε, but neither on a M nor on f M. Because of the compact embedding of H 1 () in L 2 () we can apply Schauder s theorem to obtain the existence of a fixed point y ε H 1 () of F which is a solution of { div [am (x, y ε ) y ε ] + εχ E y ε = f M (x, y ε ) in, (1.3.4) a M (x, y ε ) ν y ε = u on.
13 1.3. WELLPOSEDNESS OF THE QUASILINEAR EQUATION 5 Next we show the boundedness of {y ε } ε>0 in H 1 (). Thanks to (1.2.2) and the definition of f M, we have (f M (, y ε ) f M (, 0)) y ε = (f M (,y ε) f M (,0))My ε α M f M y ε a.e. in if y ε > M, thus we arrive at α f yε 2 if y ε M, (f M (x, y ε ) f(x, 0)) y ε α f M y ε if y ε > M, for a.a. x. This leads to α f M y ε dx = α f M E E { y ε >M} E { y ε >M} y ε dx + α f M E { y ε M} y ε dx (f M (x, y ε ) f M (x, 0)) y ε dx + α f M 2 E (f M (x, y ε ) f M (x, 0)) y ε dx + α f M 2 E. Therefore, we have from this inequality and the weak formulation of (1.3.4) α a y ε 2 L 2 () + α f M y ε dx E a M (x, y ε ) y ε 2 dx + (f M (x, y ε ) f M (x, 0)) y ε dx + α f M 2 E C ( f M (, 0) L p () + u L ()) s yε H 1 () εyε 2 dx + α f M 2 E (1.3.5) C ( f M (, 0) L p () + u L ()) s yε H 1 () + α f M 2 E C ( f M (, 0) L p () + u L ()) ( ) s y ε L 2 () + y ε dx + α f M 2 E, where we have used that z := z L 2 () + is a norm equivalent to the H 1 () norm. Taking we deduce from (1.3.5) that (1.3.6) α a y ε 2 L 2 () α fm E z dx M > 2C ( f M (, 0) L p () + u L s ()) /αf, E y ε dx C ( f M (, 0) L p () + u L s ()) yε L 2 () + α f M 2 E. E E
14 6 1. ANALYSIS OF QUASILINEAR ELLIPTIC PDES Now for the term ( f M (, 0) L p () + u L ()) s yε L 2 () we use Young s inequality with q = q = 2 and get ( ) fm (, 0) L p () + u L s () yε L 2 () 1 ( ) 2 α a fm (, 0) L 2α p () + u L s () + a 2 y ε 2 L 2 (). Inserting the last inequality in (1.3.6), it follows y ε 2 L 2 () + y ε dx C ε > 0, E where C depends on u L s () but not on ε. This proves the boundedness of {y ε } ε>0 in H 1 (). Moreover, applying Stampacchia s truncation method (see, for instance, Stampacchia [89] or the exposition for semilinear elliptic equations in the textbook by Tröltzsch [91]), we deduce the uniform boundedness of y ε independently of ε, i.e. (1.3.7) y ε L () C ( u L s () + f(, 0) L p ()), where the constant C depends only on α a and α f but neither on a M (, y ε ) nor on f M (, y ε ) or on ε. By choosing M C ( u L s () + f(, 0) L p ()), (1.3.7) implies that a M (x, y ε (x)) = a(x, y ε (x)) and f M (x, y ε (x)) = f(x, y ε (x)) for a.a. x, therefore y ε H 1 () L () is a solution of (1.3.8) {a(x, y ε ) y ε φ +(εχ E y ε + f(x, y ε )) φ} dx = uφ dσ(x) φ H 1 (). By taking subsequences, {y ε } ε>0 converges weakly in L () and weakly in H 1 () to some y H 1 () L () when ε 0. Thanks to the compactness of the embedding H 1 () L q () for every q [1, ), the convergence y ε y is strong in every L q (). Passing to the limit in (1.3.8), we see that y satisfies (1.2.4), hence it is a solution of (1.1.1). Moreover, (1.3.7) and the weak convergence y ε y in L () lead to the inequality (1.3.9) y L () C ( u L s () + f(, 0) L p ()). The Hölder regularity is well known; see Murthy and Stampacchia [81], Stampacchia [88] or Griepentrog and Recke [59], Gröger [61]. Uniqueness of the solution. Here we follow a comparison principle proposed by Hlaváček et al. [67]. Let y i H 1 () L (), i = 1, 2, be two solutions of (1.1.1) and ε > 0. Since the regularity result proved above implies that y i C( ), we can define the open sets 0 = {x y 2 (x) y 1 (x) > 0} and ε = {x y 2 (x) y 1 (x) > ε}.
15 1.3. WELLPOSEDNESS OF THE QUASILINEAR EQUATION 7 Now we introduce the function z ε (x) = min{ε, (y 2 (x) y 1 (x)) + } which belongs to H 1 () and has the following properties: 0 z ε ε in, z ε (x) = 0 for a.a. x / 0 \ ε and z ε = (y 2 y 1 ) + = (y 2 y 1 ) a.e. in 0 \ ε. Choosing z ε as test function in the weak formulations of the equations corresponding to y i, along with (1.2.1) and the monotonicity of f, we get { } α a z ε 2 L 2 () a(x, y2 ) z ε 2 + (f(x, y 2 ) f(x, y 1 )) z ε dx = {a(x, y 2 ) (y 2 y 1 ) z ε + (f(x, y 2 ) f(x, y 1 )) z ε } dx = (a(x, y 1 ) a(x, y 2 )) y 1 z ε dx = (a(x, y 1 ) a(x, y 2 )) y 1 z ε dx 0 \ ε (1.3.10) C a,m y 2 y 1 L ( 0 \ ε) y 1 L 2 ( 0 \ ε) z ε L 2 ( 0 \ ε) C a,m ε y 1 L 2 ( 0 \ ε) z ε L 2 () with M max{ y 1 C( ), y 2 C( ) }. From Assumption 1.3 it follows 0 z ε (y 2 y 1 ) + = y 2 y 1 1 α f (f(, y 2 ) f(, y 1 )) in E 0, hence z 2 ε 1 α f (f(, y 2 ) f(, y 1 )) z ε in E, since z ε = 0 in E \ 0. By the Poi ncaré inequality (1.3.3), we then obtain as in (1.3.10) ( ) z ε 2 L 2 () C E zε 2 L 2 () + z ε 2 L 2 (E) C { } E a(x, y2 ) z ε 2 + (f(x, y 2 ) f(x, y 1 )) z ε dx min{α a, α f } (1.3.11) Cε y 1 L 2 ( 0 \ ε) z ε L 2 (). Combining (1.3.10) and (1.3.11), we infer z ε 2 L 2 () C ε 2 y 1 2 L 2 ( 0 \ ε). Since lim 0 \ ε = 0, we deduce from the last inequality ε 0 ε = ε 2 ε 2 dx = ε 2 z ε 2 dx C y 1 2 L 2 ( 0 \ ε) 0 as ε 0 ε ε which implies that 0 = lim ɛ 0 ε = 0 and hence y 2 y 1. In the same way, we prove that y 2 y 1.
16 8 1. ANALYSIS OF QUASILINEAR ELLIPTIC PDES Proof of (1.3.1). Following the arguments on page 4 and taking φ = y u in the equation (1.2.4) with y u substituted for y, along with the positivity of a and the monotonicity of f, there holds { } min{α a, α f } y u 2 H 1 () a(x, yu ) y u 2 + (f(x, y u ) f(x, 0)) y u dx Hence, C ( u L s () + f(, 0) L p ()) yu H 1 (). (1.3.12) y u H 1 () C a,f ( u L s () + f(, 0) L p ()). Finally, inequality (1.3.1) follows from (1.3.12), (1.3.9), with y replaced by y u, and the estimates in [88]. Remark 1.6. The Lipschitz property of a w.r.t. the second variable is necessary only for the uniqueness of a solution of (1.1.1), not for its existence. If this property is violated Hlaváček et al. have presented in [67] an onedimensional example of nonuniqueness of solutions. Remark 1.7. If f is differentiable w.r.t. the second component then Assumption 1.3 implies that ( f/ y) α f in E R and it is dominated by an L p () function. The proof of the existence of a solution in H 1 () C( ) of (1.1.1) is then easier; see [20, Theorem 2.4]. Let us briefly sketch the main steps of it. Utilizing the identity f(x, y(x)) = f(x, 0) + f 0 (x, y(x))y(x) with f 0 (x, y) := 1 0 f (x, θy) dθ, y it is enough to consider the mapping F : L 2 () L 2 () defined by F (w) = z, where z is the unique solution in H 1 () of { div [am (x, w) z] + f 0M (x, w)z = f(x, 0) in, a M (x, w) ν z = u on. Here, f 0M denotes the truncation of f 0 as described in the proof of the previous theorem. Once again, by Schauder s theorem, there exists a fixed point y M H 1 () of F. The rest of the proof is along the lines of Theorem 1.5. Remark 1.8. It is important to remark that in view of the previous theorem, the set {y u u U} is bounded in C( ) for bounded U L s (), s > 1. If a is continuous in R then we are going to prove that y u W 1,r () for some r > 2. For equations of the type (1.1.1) with homogeneous Dirichlet boundary conditions such a regularity result is standard; see Morrey [80, pp ] or Giaquinta [55, 18, p. 73]. To our best knowledge, in the case of quasilinear equations with inhomogeneous Neumann boundary conditions, W 1,r () regularity results have not yet appeared in the literature. To overcome this obstacle we will
17 1.3. WELLPOSEDNESS OF THE QUASILINEAR EQUATION 9 apply a result by Dauge [46] that holds true for equations with constant coefficients. For this reason, we follow the classical approach of freezing the coefficient a around certain points of the domain to perform a reduction from variable coefficient to constant coefficient. Theorem 1.9. Suppose that the Assumptions hold and that a : R R is continuous. Then there exists r > 3 such that, for any { } ( ) 2p 4 min r, if p (1.3.13) 2 < r 2 p 3, 2, r if p 2, and any u L r /2 (), the solution y u of (1.1.1) belongs to W 1,r (). Moreover, for any bounded set U L r /2 () there exists a constant C U > 0 such that (1.3.14) y u W 1,r () C U u U. In addition, if is convex then the above conclusions remain valid for some r Proof. Let us first assume that 2 < r (2p)/(2 p) if p < 2 and r (2, ) otherwise. The restriction r r for some r > 3 will be imposed later. By virtue of Theorem 1.5, (1.1.1) admits a unique solution y u in H 1 () C( ). We have to prove its W 1,r () regularity. Note that, thanks to our assumptions and the continuity of y u, ã( ) := a(, y u ( )) is continuous in. Let ρ > 0 then there exists a finite number of boundary points {x j } m j=1 such that m j=1 B R 2(x j, ρ). Further, let D be an open set with regular boundary such that D D and m j=1 B R 2(x j, ρ) D. We also take a partition of unity {ψ j } m j=0 C (R 2 ) with m j=0 ψ j (x) = 1 and 0 ψ j (x) 1 x and j = 0,..., m, supp ψ 0 D and supp ψ l B R 2(x l, ρ) for l = 1,..., m. Then y u = m j=0 y j, where y j := ψ j y u. We prove that y j W 1,r () for every j = 0,..., m. For j = 0 we have y 0 = ψ 0 y u, hence div [ã(x) y 0 ] + y 0 = div [ã(x)ψ 0 y u ] div [ã(x)y u ψ 0 ] + ψ 0 y u = ψ 0 div[ã(x) y u ] ã(x) y u ψ 0 div[ã(x)y u ψ 0 ] + ψ 0 y u (1.3.15) = ψ 0 f(x, y u ) ã(x) y u ψ 0 div [ã(x)y u ψ 0 ] + ψ 0 y u = G in and y 0 = 0 on. It suffices to show that G W 1,r () := W 1,r 0 (), where W 1,r 0 () denotes the closure in W 1,r () of the space Cc () consisting of all infinitely differentiable functions with compact support in. Then the W 1,r () regularity of y 0 follows from Morrey [80, pp ]. To prove that ψ 0 f W 1,r () we make use of the Sobolev embedding W 1,r 0 () L p () which holds true when 1/p 1/r 1/2 or equivalently 1/r 1/p 1/2. The last inequality is valid due to our assumption
18 10 1. ANALYSIS OF QUASILINEAR ELLIPTIC PDES on r. Moreover, ã( ) y u ψ 0 L 2 (), div [ã( )y u ψ 0 ] H 1 () and the inclusions L 2 () H 1 () W 1,r (), hence we conclude that G W 1,r (). We fix now j = 1,..., m and x j. Employing (1.2.4) with y replaced by y u, we have for arbitrary z H 1 () ã(x) (z y u ψ j + ψ j y u z) dx = ã(x) y u (ψ j z) dx = f(x, y u )ψ j z dx + uψ j z dσ(x), therefore {ã(x) y j z + y j z}dx = {ã(x) (y u ψ j z + ψ j y u z) + y u ψ j z} dx = {ã(x) (y u ψ j z z y u ψ j ) + (y u f(x, y u )) ψ j z} dx (1.3.16) + uψ j z dσ(x) =: F (z). F is a linear continuous functional on W 1,r (). To verify this consider, for instance, the terms z y u and uz with z W 1,r (): z y u L 1 () C z L 2r r 2 () y u L 2r r+2 () C z W 1,r () y u L 2 () is a consequence of the embedding W 1,r () L 2r r 2 () and the fact that the conjugate number of is 2r < 2. Moreover, z r 2 r+2 W 1 1/r,r () L r r 2 (), cf. 2r Grisvard [60, Theorem ], hence Hölder s inequality yields uz L 1 () u L r/2 () z L r r 2 () C u L r/2 () z W 1 1/r,r () C u L r/2 () z W 1,r (). From (1.3.16) we get for z W 1,r () {ã(x j ) y j z + y j z} dx = (ã(x j ) ã(x)) y j z dx + F (z). B R 2 (x j,ρ) Consider now the mapping F : W 1,r () W 1,r (), F(w) = y w, where y w is the solution of the problem (1.3.17) {ã(x j ) y w z + y w z} dx = (ã(x j ) ã(x)) w z dx + F (z), B R 2 (x j,ρ) for every z W 1,r (). According to Dauge [46, Corollary 3.10], there exists r > 3 such that, for any r satisfying (1.3.13), the equation (1.3.17) admits a unique solution
19 1.3. WELLPOSEDNESS OF THE QUASILINEAR EQUATION 11 y w W 1,r (). Consequently, F is welldefined for these values of r. If is convex then r 6 3, cf. [46, Corollary 3.12]. 5 Next we prove that F is a contraction so that Banach s fixedpoint theorem is applicable and yields y j W 1,r (). For w i W 1,r (), i = 1, 2, we find F(w 2 ) F(w 1 ) W 1,r () C ã(x j ) ã( ) L ( B R 2 (x j,ρ)) w 2 w 1 W 1,r (), where C depends only on α a and not on x j. Let us show this estimate. Consider first the mapping T : W 1,r () W 1,r (), T w, z = { w z + wz} dx. Then T is bijective, cf. [46, p. 233], and continuous, hence it is an isomorphism. The inverse of T is also an isomorphism, thus, given ξ W 1,r (), the solution w of the equation T w = ξ satisfies w W 1,r () C ξ W 1,r (). This fact can be used to deduce from the identity {a(x j ) (y w2 y w1 ) z + (y w2 y w1 )z} dx = (ã(x j ) ã(x)) (w 2 w 1 ) z dx =: L(z) z W 1,r (), B ρ(x j ) the following inequality F(w 2 ) F(w 1 ) W 1,r () C L W 1,r () C sup sup z W 1,r 1 x B () R 2 (x j,ρ) ã(x j ) ã(x) w 2 w 1 W 1,r () z W 1,r () C ã(x j ) ã( ) L ( B R 2 (x j,ρ)) w 2 w 1 W 1,r (). Now we can choose ρ sufficiently small such that C ã(x j ) ã( ) L ( B R 2 (x j,ρ)) < 1, hence F is a contraction. We finish the proof by verifying (1.3.14) which follows from the same estimate for y j, j = 0,..., m. For y 0 we get the desired estimate from known results for linear elliptic equations, cf. Morrey [80, pp ]. To estimate y j W 1,r (), with j 1, we are going to use an estimate for the linear case and the fact that F is a contraction. Let us take w 0 = 0 and w k+1 = F(w k ), k = 0, 1, 2,..., then w k+1 W 1,r () Λ F 1 Λ F w 1 W 1,r (), where Λ F is the contractivity constant of F. Since {w k } k=1 converges to the unique fixed point y j, y j W 1,r () Λ F 1 Λ F w 1 W 1,r (). Moreover, w 1 is the solution of the linear equation {ã(xj ) w 1 z + w 1 z } dx = (ã(x j ) ã(x)) w 0 z dx +F (z) B R 2 (x j,ρ) } {{ } =0 z W 1,r (), hence as above w 1 W 1,r () C F W 1,r () and the proof is complete.
20 12 1. ANALYSIS OF QUASILINEAR ELLIPTIC PDES Remark The paper by Dauge [46], cited in the proof of the previous theorem, deals only with problems in 3D but the result is also valid for dimension two. This can be seen as follows. Given the solution y H 1 () of the problem { y + cy = f in, ν y = u on, with a positive constant c, we introduce the prism = (0, 1) and consider the problem ỹ + cỹ = f in, ν ỹ = ũ on (0, 1), ν ỹ = 0 on {0, 1}, where f : R R and ũ : (0, 1) R are defined by f( x) = f(x 1, x 2 ) and ũ( x) = u(x 1, x 2 ), with x = (x 1, x 2, x 3 ). Then there holds ỹũ W 1,r ( ) and ỹũ( x) = y u (x 1, x 2 ), therefore y u W 1,r (). Remark Theorems 1.5 and 1.9 are still valid if we require in Assumption 1.3 that φ M and f(, 0) belong to L p () with p 2q/(2 + q) > 1 and q > 2. The reason for assuming p > 4/3 will become clear in Theorem Regularity of solutions of elliptic PDEs in domains with corners This section contains some auxiliary results concerning the regularity of solutions of the following Neumann problem { y = f in, (1.4.1) ν y = g on, where R 2 satisfies Assumption 1.1. Further, we assume that f L p () with p > 4/3, g L 2 () satisfying the compatibility condition (1.4.2) f(x) dx + g(x) dσ(x) = 0. It is a wellknown consequence of the LaxMilgram theorem that (1.4.1) has a solution in H 1 () that is unique up to an additive constant; see Grisvard [60, Theorem ] when g = 0. Theorem Suppose that Assumption 1.1 holds. There exists s > 3/2 depending on such that for every 3/2 < s < s the problem { z = ζ in, ν z = g on, with ζ H s 2 (), g L 2 () and (1.4.3) ζ, 1 H s 2 (),H 2 s () + g(x) dσ(x) = 0,
21 1.4. REGULARITY OF SOLUTIONS OF ELLIPTIC PDES IN DOMAINS WITH CORNERS 13 has a unique solution z H 3 /2 () up to an additive constant. Moreover, there exists a constant C s > 0 independent of ζ and g such that ( (1.4.4) z H 3/2 () C ) s ζ H s 2 () + g L 2 () + z(x) dx. The last term in (1.4.4) is a consequence of the uniqueness of z up to an additive constant. We should remark that H s 2 () := H0 2 s () if s < 2; the latter space is defined analogous to W 1,r 0 () on page 9. However, because 0 < 2 s < 1/2, there holds H0 2 s () = H 2 s (), hence H s 2 () = H 2 s () ; see, for instance, Grisvard [60, Theorem (c)]. Let us also mention that ζ, 1 H s 2 (),H 2 s () = ζ dx if ζ L p () with p 4/3; see Corollary 1.13 below. Proof. To simplify the notation we will write, instead of, H s 2 (),H 2 s (). Let us consider the problems (1.4.5) (1.4.6) (1.4.7) z 1 = 1 ζ, 1 in, z 1 = 0 on, z 2 = ζ 1 ζ, 1 in, ν z 2 = 0 on, { z3 = 0 in, ν z 3 = g ν z 1 on. According to Dauge [44, Theorem 23.3] or [45, Theorem 3], there exists s > 3/2, depending on the angles of and the minimum positive eigenvalue of the Laplace operator in, such that the problem (1.4.5) has a unique solution z 1 in H s () if ζ H s 2 () and 3/2 < s < s. Moreover, the following estimate holds (1.4.8) z 1 H s () C s,1 ζ H s 2 (). The regularity of z 1 implies that ν z 1 L 2 () (see also page 14) and by integrating (1.4.5), (1.4.9) ν z 1 dσ(x) = z 1 dx = ζ, 1. The existence of a unique solution z 2 H s () of (1.4.6), up to an additive constant, follows from Dauge [44, Corollary 23.5] for 3/2 < s < s and ζ H s 2 (). Now we have the estimate z 2 H s () C s,2 ζ 1 ζ, 1 + z 2 (x) dx H s 2 () ( ) C s,2 ζ H s 2 () + z 2 (x) dx.
22 14 1. ANALYSIS OF QUASILINEAR ELLIPTIC PDES Finally, taking into account (1.4.3) and (1.4.9), we get (g ν z 1 ) dσ(x) = g dσ(x) + ζ, 1 = 0. Thus, from Kenig [72, p. 121] we deduce the existence of a solution z 3 in H 3 /2 () of (1.4.7) that is unique up to the addition of a constant. Similarly, we obtain with the help of the continuity of the trace operator and (1.4.8) z 3 H 3/2 () ( g C ) ν z 1 L 2 () + z 3 (x) dx ( ) C g L 2 () + z 1 H s () + z 3 (x) dx ) C s,3 ( g L 2 () + ζ H s 2 () + z 3 (x) dx. Consequently, z 1 + z 2 + z 3 H 3 /2 () and z = z 1 + z 2 + z 3 + constant. Inequality (1.4.4) follows immediately from the above estimates for z i, i = 1, 2, 3. Corollary Suppose that Assumption 1.1 holds, f L p () with p > 4/3, g L 2 () and the condition (1.4.2) is satisfied. If y H 1 () is a solution of (1.4.1) then y H 3 /2 () and there exists a constant C > 0, independent of f and g, such that (1.4.10) y H 3/2 () ( f C ) L p () + g L 2 () + y(x) dx. Proof. It suffices to prove that L p () H s 2 () for some 3/2 < s < s. Then we can apply the previous theorem to obtain the desired regularity of y and (1.4.10). Thanks to the inclusion H 1 /2 () L 4 (), we can take ε > 0 small enough and s close to 3/2 such that p > (4 ε)/(3 ε) and H 2 s () L 4 ε (). Then it follows L p () L (4 ε) (3 ε) () = L 4 ε () H 2 s () = H s 2 (). The next proposition deals with the regularity of the trace and normal derivative of a function belonging to H 3 /2 (). It is known that for Lipschitz domains the trace operator is linear and continuous from H s () to H s 1/2 () if 1/2 < s < 3/2, as well as from H s () to H 1 () if s > 3/2, cf. Ding [48]. However, Jerison and Kenig [71, 3] have constructed a function in H 3 /2 (), whose trace is not in H 1 (). Furthermore, if is polygonal and y H s () with 3/2 < s < 2 then ν y L 2 (). Indeed, i y H s 1 () for i = 1, 2, hence there exists the trace ( i y) H s 3/2 (). Now ν y = y ν = ( 1 y) ν 1 +( 2 y) ν 2. The difficulty comes from the discontinuity of the normal vector ν. In polygons, ν is constant on every edge e of, therefore ν y e H s 3/2 (e) and 0 < s 3/2 < 1/2. Finally, we can apply Theorem (a) in Grisvard s book [60] to deduce, without additional conditions, that ν y L 2 ().
23 1.4. REGULARITY OF SOLUTIONS OF ELLIPTIC PDES IN DOMAINS WITH CORNERS 15 Proposition Under the Assumption 1.1, if y H 3 /2 () and y L p () with p > 4/3 then y H 1 (), ν y L 2 () and the following estimate holds (1.4.11) y H 1 () + ν y L 2 () C ( y H 3/2 () + y L p ()). Proof. Let y 1 H 1 () satisfy y 1 = y in and y 1 = 0 on. As in the proof of Theorem 1.12, y 1 H s () with some s > 3/2, hence ν y 1 L 2 () and (1.4.12) ν y 1 L 2 () C y 1 H s () C y L p (). Setting y 2 = y y 1, it follows that y 2 is harmonic and y 2 H 3 /2 (). Therefore, according to Jerison and Kenig [71, Theorem 5.6 and Corollary 5.7], y 2 H 1 (), ν y 2 L 2 () and (1.4.13) y 2 H 1 () + ν y 2 L 2 () C y 2 H 3/2 (). Furthermore, y 1 = 0 and y = y 2 H 1 (). Since ν y = ν y 1 + ν y 2 L 2 (), (1.4.11) follows from (1.4.12)(1.4.13) and the continuity of the trace operator: y H 1 () + ν y L 2 () y 2 H 1 () + ν y 1 L 2 () + ν y 2 L 2 () C ( y 2 H 3/2 () + y ) L p () C ( y H 3/2 () + y L p ()). Corollary Let Assumption 1.1 be fulfilled, p > 4/3 and c L p () satisfy c( ) 0 a.e. in and c( ) α > 0 at least on a subset of with positive measure. Then the problem { y + c(x)y = f in, (1.4.14) ν y = g on, with f L p () and g L 2 (), has a unique solution y H 3 /2 () and (1.4.15) y H 3/2 () C c ( f L p () + g L 2 ()), where C c > 0 depends on c L p () but it is independent of f and g. Proof. Under the assumptions of the corollary, the existence and uniqueness of a solution y H 1 () C( ) of (1.4.14) follows, for instance, from Alibert and Raymond [3, Theorem 2]. Moreover, ( ) y H 1 () + y C( ) C 1 f L p () + g L 2 () with C 1 > 0 independent of f, g and c. The H 3 /2 () regularity of y follows from Corollary 1.13 when replacing f by f cy L p (). To show (1.4.15) we apply
24 16 1. ANALYSIS OF QUASILINEAR ELLIPTIC PDES (1.4.10) and get y H 3/2 () C ( f cy L p () + g L 2 () C ( f L p () + c L p () y C( ) + g L 2 () ) C 2 ( f L p () + C 1 c L p () ( f L p () + g L 2 ()) + g L 2 () = C 2 ( 1 + C1 c L p ()) ( f L p () + g L 2 ()). Thus, (1.4.15) is obtained by setting C c = C 2 ( 1 + C1 c L p ()). Remark The H 3 /2 () regularity of the solution y of (1.4.1) if f L 2 () is studied by Casas et al. [29, Lemma 2.2]. Similarly to our technique, the Neumann problem is decomposed in two different problems and a combination of the results by Jerison and Kenig [70, 71] is used. For convex domains in R n, n 1, an analogous result to Proposition 1.14 is proved in Casas et al. [28, Lemma A.1]. ) ) 1.5. Higher regularity of solutions of the quasilinear equation In this section, we will see that stronger assumptions on the coefficient a of the differential operator yield higher regularity of the solution of (1.1.1). This higher regularity is crucial to confirm a high rate of convergence for the finite element approximation of (1.1.1); see Chapter 2. Since we will need in the sequel the differentiability of a a.e. in, we impose the following assumption. Assumption For every M > 0 there exists a constant C M > 0 such that, for all x i, y i M, i = 1, 2, the following local Lipschitz property is satisfied: a(x 2, y 2 ) a(x 1, y 1 ) C M ( x 2 x 1 + y 2 y 1 ). Since a is Lipschitz in x and y and C 0,1 ( ) = W 1, (), a is differentiable a.e. in with uniformly bounded weak partial derivatives and, given y H 1 () C( ), the chain rule is valid: j [a(x, y(x))] = [ x a] j (x, y(x)) + a y (x, y(x)) jy(x) for j = 1, 2, and a.a. x. Theorem Suppose that the Assumptions and 1.17 hold. Then for any u L 2 () the solution y u of (1.1.1) belongs to H 3 /2 (). Moreover, for any bounded set U L 2 () there exists a constant C U > 0 such that (1.5.1) y u H 3/2 () C U u U. Proof. From Theorem 1.9 we get that y u W 1,r () for some r > 3 (notice 2p that > 3 if 4/3 < p < 2). Let us show that y 2 p u H 3 /2 (). Thanks to the assumptions on a and the continuity of y u, it follows that a(, y u ( )) C( ) and
25 1.5. HIGHER REGULARITY OF SOLUTIONS 17 ( a/ y)(, y u ( )) L (). By expanding the divergence term of the equation (1.1.1) and dividing by a = a(, y u ( )) α a > 0, we verify in Remark 1.20 below that (1.5.2) (1.5.3) y u = 1 2 }{{} a f(, y u) + j a(, y u ) j y u + a } {{ } } {{ } } {{ } L L () p j=1 y (, y u) y u 2 } {{ } in, () L () L r () } {{ } L r /2 () L () ν y u = u a on. The righthand sides of (1.5.2) and (1.5.3) are in L min{p, r/2} () and L 2 (), respectively. Hence, we can apply Corollary 1.13 to obtain the H 3 /2 () regularity of y u. Inequality (1.5.1) simply follows from (1.4.10) along with the Assumptions , Remark 1.8 and (1.3.14). Conclusion In dimension two, there holds H 3 /2 () W 1,4 (). Therefore, if Assumption 1.17 is valid and u L 2 () then we can improve the regularity of y u to W 1,4 (), no matter if r < 4, where r is given in Theorem 1.9. Remark In the proof of the previous theorem we made use of the fact that the solution y u W 1,r () (r > 3) of (1.1.1) is also a solution of (1.5.2)(1.5.3). Let us verify this. For s = min{p, r/2} we define the functional F : L s () R by F (z) = 1 a 2 f(x, y u ) + j a(x, y u ) j y u + a j=1 y (x, y u) y u 2 z dx. We prove that y u is a solution of the equation y = F in, (1.5.4) ν y = u a on. For that purpose we pass to the weak formulation of (1.5.4) and consider the problem of finding y H 1 () such that, for every z H 1 () L s (), u (1.5.5) y z dx = z dσ(x) + F (z) a u = a z dσ(x) f(x, y u ) + j a(x, y u ) j y u + a a y (x, y u) y u 2 z dx. j=1 For arbitrary z H 1 () we have z j y u L 2 (), j = 1, 2, since H 1 () L 6 () and j y u L 3 (). Then owing to the boundedness of a(, y u ( )), [ x a] (, y u ( )) and ( a/ y)(, y u ( )), we deduce j [a(, y u )z] = [ x a] j (, y u )z + a y (, y u)z j y u + a(, y u ) j z L 2 () for j = 1, 2.
26 18 1. ANALYSIS OF QUASILINEAR ELLIPTIC PDES Hence, a(, y u )z H 1 () and substituting this for the test function z in (1.5.5), we see that y u satisfies (1.5.5). This proves that y u is a solution of (1.5.4) and any other solution y of (1.5.4) is of the form y = y u + constant. The last regularity result of this section is only valid for convex polygonal domains. To our best knowledge, for nonconvex corner domains, such a result does not exist in general. Related results for domains with smooth boundaries are addressed in Remark 1.22 below. Theorem Let the Assumptions and 1.17 be satisfied. We also assume that is convex and p 2. Then there exists r 0 > 2 depending on the measure of the angles in such that, for any 2 r min{ r, r 0, p} ( r ; see Theorem 1.9) and any u W 1 1/r,r (), the solution y u of (1.1.1) belongs to W 2,r (). Moreover, for any bounded set U W 1 1/r,r () there exists a constant C U > 0 such that y u W 2,r () C U u U. Proof. First, we prove the result for 2 r min{ r/2, p}. Since r 2, we have u W 1 1/r,r () H 1 /2 () L q () for any 1 q <. Taking q = r/2, we can apply Theorem 1.9 to deduce that y u W 1, r (). We have to show the W 2,r () regularity of y u. Repeating the steps of the proof of Theorem 1.18, we get that the righthand side of (1.5.2) is in L r (). Hence, it is enough to prove that u/a = (u/a(, y u ( ))) W 1 1/r,r (). Then a wellknown result by Grisvard [60, Corollary ] on maximal regularity yields the existence of r 0 > 2 such that y u W 2,r () if 2 r min{r 0, r/2, p}. This r 0 depends on the measure of the angles in. To verify the W 1 1/r,r () regularity of u/a we will use the following two facts: (1.5.6) If b C 0,µ (), u W 1 1/r,r () with µ > 1 1/r, then bu W 1 1/r,r () ; cf. Grisvard [60, Theorem ], and (1.5.7) W k,q () C 0,µ ( ) with µ = k 2/q if k 2/q < 1, < 1 if k 2/q = 1, = 1 if k 2/q > 1, provided that kq > 2, cf. Necǎs [82, 2, Theorem 3.8]. Thanks to Assumption 1.17 and (1.2.1), 1/a is Lipschitz w.r.t. both variables. Furthermore, from (1.5.7) we get y u W 1, r () C 0,1 2 r ( ), therefore (1/a) C 0,1 2 r (). Now there are two possibilities: (1) If r < r/2 or equivalently if 1 1/r < 1 2/ r then it follows from (1.5.6) that (u/a) W 1 1/r,r (). In particular, y u H 2 () if u H 1 /2 ().
27 1.6. DIFFERENTIABILITY OF THE SOLUTION OPERATOR 19 (2) If r = r/2 we cannot apply (1.5.6) directly. Nevertheless, because of the inclusion u W 1 1/r,r () H 1 /2 () we can use the above argumentation to obtain, in an intermediate step, that y u H 2 (). This fact, together with H 2 () C 0,µ ( ) for any µ (1 2 r, 1), yields (1/a) C0,µ (). By applying (1.5.6), we deduce (u/a) W 1 1/r,r () and y u W 2,r (). Finally, we assume that r/2 < r min{r 0, r, p} which implies p > 2; notice that r 6 3 > 4, hence r/2 > 2. From the first part of the proof we know that 5 y u W 2,min{r0, r/2,p} () C 0,1 ( ). Then (1/a) C 0,1 () and from (1.5.6) we get (u/a) W 1 1/r,r (), once again. Moreover, the righthand side of (1.5.2) is in L r (). By applying again Grisvard s result, we conclude y u W 2,r (). Remark The proof of Theorem 1.5 on the existence and uniqueness of a solution of (1.1.1) is valid for arbitrary Lipschitz domains in R n, n N, provided that s > n 1 and p > n/2. On the other hand, the statement of Theorem 1.5 holds true, assuming that is of class C 1 and that the leading coefficients of the elliptic differential operator are continuous, cf. Agmon et al. [2, Theorems 15.3 and 15.1 ]. If the boundary is of class C 1,1 then the W 2,r () regularity result of Theorem 1.21 is a consequence of Theorem in Grisvard [60] Differentiability of the solution operator This section is devoted to the analysis of a class of linear elliptic differential equations of nonmonotone type. This class includes, in particular, the linearization of the equation (1.1.1) around a solution y u. The necessity of studying the linearization of the quasilinear equation arises when differentiability properties of the solution operator u y u have to be investigated. Let us first consider the following linear elliptic equation in divergence form { div [ã(x) z(x) + b(x)z(x)] + c(x)z(x) = ζ(x) in, (1.6.1) [ã(x) z(x) + b(x)z(x)] ν = v(x) on, where ζ H 1 (), v H 1/2 () are given and ã, c : R, b : R 2 are specified in the next theorem. We say that z H 1 () is a solution of (1.6.1) if w H 1 () there holds {(ã(x) z + zb(x)) w + c(x)zw} dx = ζ, w H 1 (),H 1 () + v, w H 1/2 (),H 1 /2 (). In view of the occurrence of the nonmonotone part div[bz], the wellposedness of (1.6.1) is not obvious. The next theorem establishes the existence and uniqueness of a solution z H 1 () of the previous equation. Theorem Suppose that Assumption 1.1 holds. We also assume that β > 2 and γ > 1. Further, let ã L () with ã(x) α > 0 for a.a. x, b L β () 2 and c L γ (), satisfying c(x) 0 for a.a. x and c(x) α for all x E, where E is
28 20 1. ANALYSIS OF QUASILINEAR ELLIPTIC PDES a measurable subset of such that E > 0. Then the operator S : H 1 () H 1 () defined by (1.6.2) Sz, w H 1 (),H 1 () = {ã(x) z w + zb(x) w + c(x)zw} dx is an isomorphism. Proof. We follow here the ideas by Casas and Tröltzsch [36, Theorem 2.7], who study the linearized equation of (1.1.1) with homogeneous Dirichlet boundary conditions. Although (1.6.1) is a more general equation than the one considered by the previous authors and contains Neumann boundary data, the modification of the proof of [36, Theorem 2.7] is basically straightforward. Let us note the main differences. Since the operator S is linear, the continuity of S is equivalent to its boundedness. By Hölder s inequality and the embedding H 1 () L q () for every 1 q <, we have for every z H 1 () Sz H 1 () = sup Sz, w H 1 (),H 1 () w H 1 () 1 { sup ã L () z L 2 () w L 2 () w H 1 () 1 + b L β () z 2β L β 2 () w L 2 () + c L γ () z 2γ w L γ 1 () L C ( ã L () + b(x) L β () + c L γ ()) z H 1 (), thus S is bounded. It remains to prove that S is bijective. 2γ γ 1 () Injectivity of S. In order to prove that S is injective or equivalently that the equation Sz = 0 admits only the solution z = 0, we can follow the same steps of the proof of [36, Theorem 2.7] with obvious modifications. Surjectivity of S. To verify the surjectivity of S we modify conveniently the arguments of [36, Theorem 2.7]. For every t [0, 1] we define the linear operator S t : H 1 () H 1 () by S t z, w H 1 (),H 1 () = {ã(x) z w + tzb(x) w + c(x)zw} dx. Obviously, S 1 = S and the operator S 0 is monotone, hence S 0 is an isomorphism as follows from the LaxMilgram theorem. Let D be the set of points t [0, 1] for which S t defines an isomorphism. Then D, since 0 D. Let t max be the supremum of D. We are going to prove that S tmax is an isomorphism between H 1 () and H 1 (), i.e. t max D. We note that S tmax is continuous and injective; the fact of including t max in the equation does not matter for the proof of the continuity and injectivity. It remains to show that S tmax is surjective. Given ξ H 1 (), we have to find an }
29 1.6. DIFFERENTIABILITY OF THE SOLUTION OPERATOR 21 element z H 1 () such that S tmax z = ξ. Let {t k } k=1 D be a sequence such that t k t max when k and denote by z k the function in H 1 () such that S tk z k = ξ. Then, by the Poincaré inequality (1.3.3) and the assumptions on ã and c, we have α z k 2 H 1 () {ã(x) zk } 2 + c(x)zk 2 dx ( ) = ξ, z k H 1 (),H 1 () t k z k b(x) z k dx which implies C ( ξ H 1 () + t k z k b L ()) 2 zk H 1 (), ( ) C z k H 1 () C ξ H 1 () + t k z k ( 2β L ξ H 1 () + z k β 2 () b L β () 2β L β 2 () ). z k H 1 () Arguing as in [36, Theorem 2.7], we get that {z k } k=1 is bounded in H 1 () and the weak limit z of a subsequence of {z k } k=1 satisfies S tmax z = ξ. Therefore, we conclude that t max S. Finally, we prove that t max = 1. If it is false then we consider the operators S tmax+ε, S tmax L(H 1 (), H 1 () ), ε > 0 with t max + ε 1 and obtain S tmax+ε S tmax L(H 1 (),H 1 () ) ε sup sup z H 1 () 1 w H 1 () 1 sup sup z H 1 () 1 w H 1 () 1 εzb(x) w dx b L β () z 2β L β 2 () w L 2 () Cε. 1 Thus, taking 0 < ε < 1/( C S t max L(H 1 (),H 1 ())), we have by standard arguments of linear algebra that S tmax+ε is also an isomorphism, which contradicts the fact that t max is the supremum of D. Before establishing the analysis of the linearized equation of (1.1.1), we need some differentiability of the coefficients a and f. Assumption The functions a and f are of class C 2 with respect to the second variable and for any number M > 0 there exist constants D M, D M,a > 0, such that 2 j a (1) (x, y) j=1 yj + j f (1.6.3) (x, y) yj D M, k a (2) y (x 1, y k 1 ) k a y (x 2, y k 2 ) D M,a ( x 1 x 2 + y 1 y 2 ), for a.a. x, all x i and y, y i M, i = 1, 2, k = 1, 2.
30 22 1. ANALYSIS OF QUASILINEAR ELLIPTIC PDES Taking into account the differentiability of f w.r.t. the second variable and Assumption 1.3, we find that ( f/ y)(x, y) 0 for a.a. x and all y R. Moreover, (1.6.4) where E is given in Assumption 1.3. f y (x, y) α f > 0 (x, y) E R, Let us now come back to our quasilinear equation (1.1.1). The linearized equation of (1.1.1) around a solution y has the following form [ div a(x, y) z + a ] (x, y)z y + f (x, y)z = ζ in, y y (1.6.5) [ a(x, y) z + a ] (x, y)z y ν = v on. y By setting (1.6.6) ã(x) = a(x, y(x)), b(x) = a f (x, y(x)) y(x) and c(x) = (x, y(x)), y y we see that (1.6.5) is contained in the class of equations of the type (1.6.1). In analogy to Theorem 1.23, we may state the following existence and uniqueness theorem for a solution of (1.6.5). Theorem Suppose that the Assumptions and 1.24(1) hold. Given y W 1,r () with r > 2, for any v H 1/2 () and ζ H 1 (), the linearized equation (1.6.5) has a unique solution z H 1 () in the sense that { (1.6.7) a(x, y) z w + a } f (x, y)z y w + (x, y)zw dx y y Moreover, there holds the inequality = ζ, w H 1 (),H 1 () + v, w H 1/2 (),H 1 /2 () w H 1 (). z H 1 () C ( ζ H 1 () + v H 1/2 ()), with some positive constant C which may depend on a, f and y but not on ζ and v. Proof. In order to deduce the existence and uniqueness of a solution z H 1 () of (1.6.5), we will apply Theorem Therefore, it is enough to check if the assumptions of Theorem 1.23 are satisfied. For a fixed y W 1,r (), let us set the coefficients ã, b and c, as in (1.6.6). Since W 1,r () C( ) is valid for r > 2, y is uniformly bounded and we have, together with Assumption 1.2 and (1.6.3), that a(, y( )), ( a/ y)(, y( )) L () and ( f/ y)(, y( )) L (), hence ã L (), b L r () 2 and c L (). Moreover, by defining α = min{α a, α f }, it follows from (1.2.1) and (1.6.4) that ã( ) α and c( ) 0 a.e. in as well as c( ) α in E. Now Theorem 1.23 implies that the lefthand side of (1.6.7) defines an isomorphism
31 1.6. DIFFERENTIABILITY OF THE SOLUTION OPERATOR 23 between H 1 () and H 1 (). Given ξ H 1 (), this is equivalent to the existence and uniqueness of a solution z H 1 () to the equation { a(x, y) z w + a } f (x, y)z y w + (x, y)zw dx = ξ, w y y H 1 (),H 1 () w H 1 (). The proof of the wellposedness of (1.6.7) in H 1 () is then complete when taking ξ = ζ + Bv, where the operator B : H 1/2 () H 1 () is defined by (1.6.8) Bv, w H 1 (),H 1 () = v, w H 1/2 (),H 1 /2 (). The inequality in the assertion of the theorem is an immediate consequence of the linearity and continuity of the solution operator (ζ, v) z associated with (1.6.5). Remark The proof of Theorem 1.25 can be modified in an obvious way to verify that, for any given function y W 1,r (), with r > 2, and y i L (), i = 1, 2, 3, the equation [ div a(x, y 1 ) z(x) + a ] y (x, y 2)z y + f y (x, y 3)z = ζ in, has a unique solution z H 1 (). [ a(x, y 1 ) z(x) + a y (x, y 2)z y ] ν = v on, In the next step, we will prove that the solution z of (1.6.5) is continuous if v and ζ are regular functions. For this purpose, we state in the following lemma a useful result based on the wellknown Stampacchia truncation method, cf. Stampacchia [88] or Murthy and Stampacchia [81]. Further, this continuity will allow us to deduce higher regularity of z. This will be shown in Theorem 1.29 below. Lemma ([3, Theorem 2], see also [81, Theorem 2.9]) Suppose that Assumption 1.1 holds and that ã L () with ã(x) α > 0 for a.a. x, c L γ () with γ > 1, satisfying c(x) 0 for a.a. x and c(x) α for all x E, where E is a measurable subset of with E > 0. We also assume that ζ L γ (), v L θ () and f i L β (), i = 1, 2, with θ > 1 and β > 2. Then the unique solution z H 1 () of the equation { } 2 {ã(x) z w + c(x)zw} dx = ζw + f i i w dx + vw dσ w H 1 () is continuous in. Proposition Under the Assumptions and 1.24(1) and supposing that ζ L γ (), v L θ () and y W 1,r () with γ, θ > 1, and r > 2, the unique solution z of the equation (1.6.5) is continuous in. i=1
32 24 1. ANALYSIS OF QUASILINEAR ELLIPTIC PDES Proof. Obviously, the solution z H 1 () of (1.6.5) satisfies the equation { a(x, y) z w + f } (x, y)zw dx y { } 2 a = ζw y (x, y)z iy i w dx + vw dσ(x) w H 1 (). i=1 Since H 1 () L q () for all 1 q <, the functions ( a/ y)(, y( ))z i y, i = 1, 2, belong to L β () with some β > 2. Indeed, ( a/ y)(, y( )) L (), i y L r () and there exists q < such that 1 > Hence, setting 1 := 1 + 1, we have 2 r q β r q β > 2 and z i y L β (). The continuity of z follows then immediately from Lemma The next theorem states the W 1,r () regularity of the solution to the linearized equation (1.6.5). As we have seen in Theorem 1.9, in presence of nonzero Neumann boundary data, the proof of the W 1,r () regularity result requires a quite different approach compared to the case of homogeneous Dirichlet boundary conditions; see Casas and Tröltzsch [36, Theorem 2.4]. Theorem Suppose that the Assumptions and 1.24(1) hold. Assume further that a : R R is continuous. For any ζ L 2r r+2 (), v L r/2 () and y W 1,r () with r > 2 satisfying the condition (1.3.13), the solution z of (1.6.5) belongs to W 1,r (). Proof. Since v L r /2 () H 1/2 () and ζ L 2r r+2 () H 1 (), the existence and uniqueness of z H 1 () follow from Theorem Moreover, by applying Proposition 1.28, we deduce that z C( ). To obtain the W 1,r () regularity of z we follow the same steps as in Theorem 1.9 with obvious modifications. We only remark that equation (1.3.16) on page 10 has to be replaced by {ã(x) z j w + z j w} dx = F (w), for arbitrary w H 1 (), z j := zψ j, ã( ) := a(, y( )) and { F (w) := ã(x) (z w w z) ψ j a y (x, y)z y (ψ jw) ( + ζψ j w + 1 f ) } (x, y) zψ j w dx + y vψ j w dσ(x). By the same arguments as in the proof of Theorem 1.9 and thanks to the continuity of z and the embedding W 1,r () C( ), it can be easily checked that F W 1,r ().
33 1.6. DIFFERENTIABILITY OF THE SOLUTION OPERATOR 25 The next theorem states the differentiability of the relation between the boundary datum u and the associated solution y u of (1.1.1). Theorem Suppose that the Assumptions and 1.24(1) are fulfilled and a : R R is continuous. The mapping G : L r /2 () W 1,r (), G(u) = y u, is twice continuously Fréchetdifferentiable, i.e. of class C 2, for all r satisfying the condition (1.3.13). Moreover, for any v, v 1, v 2 L r /2 (), the functions z v = G (u)v and z v1,v 2 = G (u)[v 1, v 2 ] are the unique solutions in W 1,r () of the equations [ div a(x, y u ) z + a ] y (x, y u)z y u + f y (x, y u)z = 0 in, (1.6.9) and (1.6.10) [ div a(x, y u ) z + a ] y (x, y u)z y u + div [ a(x, y u ) z + a y (x, y u)z y u [ a y (x, y u) (z 1 z 2 + z 2 z 1 ) + 2 a [ a(x, y u ) z + a ] y (x, y u)z y u ν = = ] + f y (x, y u)z = 2 f ν = v on, y 2 (x, y u)z 1 z 2 y u [ a y (x, y u) (z 1 z 2 + z 2 z 1 ) + 2 a y 2 (x, y u)z 1 z 2 y u respectively, where z i = G (u)v i, i = 1, 2. y (x, y u)z 2 1 z 2 ] ] in, ν on, Proof. Let us introduce the mapping F : W 1,r () L r /2 () W 1,r (), F (y, u), w = {a(x, y) y w + f(x, y)w} dx uw dσ(x). As a consequence of the Assumptions 1.2, 1.3 and 1.24(1), and the embedding W 1 1/r,r () L r r 2 () = L (r/2) (), F is well defined, of class C 2 and F (y u, u) = 0 for every u L r /2 (). Our goal is to prove that y F (y u, u) : W 1,r () W 1,r () defined by { y F (y u, u)z, w = a(x, y u ) z w + a y (x, y u)z y u w + f } y (x, y u)zw dx is an isomorphism. Then we can apply the implicit function theorem (see Cartan [12]) to deduce the differentiability properties of G stated in the theorem. The representations (1.6.9) and (1.6.10) for G and G are then obtained by simple computations.
34 26 1. ANALYSIS OF QUASILINEAR ELLIPTIC PDES According to Theorem 1.25, for any v H 1/2 () there exists a unique element z v H 1 () such that y F (y u, u)z v = Bv, where the operator B : H 1/2 () H 1 () is defined in (1.6.8). It suffices to show that z v W 1,r () if v L r /2 (). More precisely, this means that the unique solution z v H 1 () of (1.6.5), with y replaced by y u, associated with v L r /2 () and ζ = 0, belongs to W 1,r (). But this regularity follows directly from Theorem The existence, uniqueness and W 1,r () regularity of the solution of (1.6.10) can be deduced analogously. If we assume that a is locally Lipschitz continuous w.r.t. both variables then we know from Theorem 1.18 that the solution y u of (1.1.1), associated with the boundary datum u L 2 (), belongs to H 3 /2 (). The next theorem shows that this regularity also holds for the solution of (1.6.5) provided that v L 2 () and ζ L p (), p > 4/3. Theorem Suppose that the Assumptions , 1.17 and 1.24 hold. We also assume that v, u L 2 (), ζ L p () (p > 4/3) and y = y u is the solution of (1.1.1). Then the solution z of (1.6.5) belongs to H 3 /2 () and there holds the estimate (1.6.11) z H 3/2 () C( v L 2 () + ζ L p ()), with some C > 0 which depends on y u H 3/2 () but not on v or ζ. Moreover, the mapping G : L 2 () H 3 /2 (), G(u) = y u, is of class C 2. Proof. According to Theorem 1.18, y u H 3 /2 () W 1,4 (), hence from Theorem 1.29 it follows z W 1,min{4, r} (); notice that 2r = p > 4 implies that r > 4. r+2 3 Let us show that z H 3 /2 (). Since the function a is locally Lipschitz w.r.t. both variables and strictly positive, we may expand the divergence in the first equality of (1.6.5) and divide by a to deduce (1.6.12) z + 1 f a y z = 1 a 2 ζ + ( j a j z) + a j=1 y z y u + div = 1 ( ) 2 ζ + j a j z + 2 a a j=1 j y z jy u + 2 a y z y u 2 + a ) 2 y z y u. [ ] a y z y u + 2 a y z y u Analogously, from (1.1.1) we have y u = 1 2 f + j a j y u + a a y y u 2 L min{p,2} (). j=1
35 1.6. DIFFERENTIABILITY OF THE SOLUTION OPERATOR 27 By combining this with the W 1,min{4, r} () regularity of z, we find that the righthand side of (1.6.12) is in L q () with some q > 4/3. On the other hand, taking into account that ν y u = (u/a) L 2 () and z L (), we get from (1.6.5) that a(, y u ) ν z = v ( a/ y)(, y u )z ν y u L 2 (). Finally, the H 3 /2 () regularity of z is a consequence of Corollary Now let us prove (1.6.11). By virtue of Corollary 1.15 and (1.6.12), we easily obtain z H 3/2 () C ( v L 2 () + ζ L p () + z W 1,r ()), where r := min{4, r}. It remains to estimate z W 1,r (). For this purpose, we make use of the linear operator y F (y u, u) : W 1,r () W 1,r () defined in the proof of Theorem 1.30 and B : L 2 () W 1,r () given by Bv, w = vw dσ(x). Setting ξ = ζ + Bv, we have the identity y F (y u, u)z = ξ. Since y F (y u, u) is an isomorphism, it follows that its inverse is an isomorphism, too. This fact yields the estimate z W 1,r () C ξ W 1,r () C ( ζ L p () + v L 2 ()). The differentiability of the mapping G as stated in the Theorem is then a consequence of Theorem 1.30 and the preceding regularity result. From the mean value theorem and the differentiability of the mapping G from L r /2 () to W 1,r () (Theorem 1.30) or from L 2 () to H 3 /2 () W 1,4 () (Theorem 1.31) we obtain Corollary For any bounded set U L r /2 () with r satisfying the condition (1.3.13) or r = 4 there exists a constant C U > 0 such that y u2 y u1 W 1,r () C U u 2 u 1 L r/2 () for any u i U, i = 1, 2. If is convex then from Theorem 1.21 we know that the solution y u corresponding to u W 1 1/r,r (), with 2 r min{ r, r 0, p} for some r 0 > 2 and r 6 3, 5 belongs to W 2,r (). In this context, a natural question arises: Can we prove a result analogous to Theorem 1.30 with G : W 1 1/r,r () W 2,r ()? The next theorem gives a positive answer to it. Theorem Let the Assumptions , 1.17 and 1.24 be satisfied and assume that is convex and p 2. There exists r 0 > 2 depending on the measure of the angles in such that the mapping G : W 1 1/r,r () W 2,r (), G(u) = y u, is of class C 2 for 2 r min{r 0, r, p} ( r ; see Theorem 1.9). Moreover, for any v, v 1, v 2 W 1 1/r,r (), the functions z v := G (u)v and z v1,v 2 := G (u)[v 1, v 2 ] are the unique solutions in W 2,r () of the equations (1.6.9) and (1.6.10), respectively.
36 28 1. ANALYSIS OF QUASILINEAR ELLIPTIC PDES Proof. Consider the Banach space V () = {y W 2,r () ν y W 1 1/r,r ()} endowed with the graph norm y V () = y W 2,r () + ν y W 1 1/r,r (). Then all the elements y u = G(u) belong to this space provided that u W 1 1/r,r (); see Theorem Let us introduce the mapping F : V () W 1 1/r,r () L r () W 1 1/r,r (), F (y, u) = ( div[a(, y) y] + f(, y), a(, y) ν y u). Next we verify that F is well defined. By expanding the divergence term, we find div [a(, y) y] = [ x a] (, y) y + a y (, y) y 2 + a(, y) y. Since y W 2,r () with r 2, the righthand side of the previous equality is in L r (), therefore it remains to show that a(, y) ν y W 1 1/r,r () to deduce that F is well defined. To this end, we use the Lipschitz property of a with respect to x and y as well as the embedding H 2 () C 0,µ ( ) for every µ (0, 1); see Eq. (1.5.7). In particular, a(, y) C 0,µ () for all µ (1 1, 1), which along with (1.5.6), yields r a(, y) ν y W 1 1/r,r (). On the other hand, it is obvious that F is a C 2 mapping. Next we are going to apply the implicit function theorem. Thus, we need to prove that the linear operator y F (y, u) : V () L r () W 1 1/r,r (), y F (y, u)z = ( [ = div a(, y) z + a ] (x, y)z y y + f [ (, y)z, y a(, y) z + a (, y)z y y is an isomorphism. To this aim, we have to show for any ζ L r () and any v W 1 1/r,r () the existence of a unique solution z V () of (1.6.5). The existence and uniqueness of a solution z in H 1 () C( ) follows from Theorem 1.25 and Proposition Further, due to the inclusion W 1 1/r,r () L q () for every q <, the W 1, r () regularity of z is deduced from Theorem Finally, we prove that z W 2,r () and ν z W 1 1/r,r () by transforming (1.6.5) to z = 1 ( [ ] a ζ + div (, y)z y +[ x a](, y) z + a ) (, y) z y f (, y)z a y y y = 1 ( [ ] a ζ + z x (, y) y + 2 a a y y (, y)z 2 y a (, y) z y y + a y (, y)z y + [ xa] (, y) z f ) (1.6.13) (, y)z in, y ν z = 1 ( v a ) (1.6.14) a y (, y)z νy on. ] ν ),
37 1.7. THE ADJOINT PROBLEM 29 In the sequel, we will follow the steps of the proof of Theorem 1.21 and consider first the case when r r/2. The righthand side of (1.6.13) belongs to L min{p,r} (). Let us show that the righthand side of (1.6.14) is in W 1 1/r,r (). Once this is proved, the wellknown regularity result by Grisvard [60, Corollary ] yields the existence of r 0 > 2 introduced in Theorem 1.21 such that z W 2,r () for 2 r min{r 0, r/2, p}. In particular, the proof would be complete if p = 2. Thanks to the inclusion (1.5.7), we have z W 1, r () C 0,1 2/ r ( ) as well as y H 2 () C 0,µ ( ) for every µ [1 2/ r, 1). Together with Assumption 1.24(2) 1 and (1.2.1) we find that the functions and 1 a (, y( ))z are in a(,y( )) a(,y( )) y C0,1 2/ r ( ). Now if r < r/2 we can apply (1.5.6) to deduce ( 1 (1.6.15) v a ) a y (, y)z νy W 1 1/r,r (). In particular, z H 2 () if u, v H 1 /2 (). If r = r/2 we will improve the regularity of z to W 2,r (). Since z H 2 () and taking into account (1.5.7), we obtain 1 a(,y( )), 1 a (, y( ))z a(,y( )) y C0,µ ( ) with some µ > 1 2/ r, hence once again (1.5.6) yields (1.6.15). Finally, let us discuss the case when r/2 < r min{r 0, r, p} which implies that p > 2. From the above arguments we know that y u W 2,min{r0, r/2,p} () C 0,1 ( ). Therefore, the righthand sides of (1.6.13) and (1.6.14) are in L r () and W 1 1/r,r (), respectively. Consequently, as above, we conclude that z W 2,r (). Remark In bounded polygonal domains, the space V () defined in the proof of the previous theorem is in general a proper subspace of W 2,r (). In particular, the normal derivative of a function in H 2 () is not necessarily an element of H 1 /2 () but it belongs only to H 1 /2 ε () for all ε > 0, cf. Grisvard [60, 1.5.3]. A simple counterexample is y(x) = x 2 on the unit square = (0, 1) 2, cf. Casas et al. [28, p. 800]. This is the reason why we have introduced the Banach space V (). If is of class C 1,1 we do not have this difficulty, cf. [60, Theorem ], hence the functional F given on page 28 can be defined in W 2,r () W 1 1/r,r () The adjoint problem In the previous section, we analyzed the differentiability of the mapping u y u. As pointed out at the beginning of this chapter, there are several reasons which motivate the study of the adjoint operator associated to the one defined in (1.6.2). We will proceed as in the previous section, by stating first a general result on the wellposedness of the adjoint problem. Further, we will turn to our specific problem (1.1.1) and consider the corresponding adjoint equation.
38 30 1. ANALYSIS OF QUASILINEAR ELLIPTIC PDES Theorem Supposing that the assumptions of Theorem 1.23 hold true, the operator T : H 1 () H 1 () defined by (1.7.1) T ϕ, z H 1 (),H 1 () = {ã(x) ϕ z + zb(x) ϕ + c(x)ϕz} dx is an isomorphism. Moreover, the following regularity properties hold: (1) Assume that ã C( ). For any 2 < r r with r > 3 given in Theorem 1.9, b L r () 2 and c, ζ L 2r r+2 (), there exists a unique element ϕ W 1,r () satisfying (1.7.2) T ϕ, z H 1 (),H 1 () = ζz dx z H 1 (). (2) If ã W 1,r (), b L r () 2 and c, ζ L r /2 () with some r > 8/3, then the solution ϕ of (1.7.2) belongs to H 3 /2 () and there exists a constant C > 0, dependent on ã, b and c, but not on ζ, such that (1.7.3) ϕ H 3/2 () C ζ L r/2 (). (3) If is convex, ã W 1,4 (), b L 4 () 2 and c, ζ L 2 (), then the solution ϕ of (1.7.2) is in H 2 () and there exists C > 0 independent of ζ such that (1.7.4) ϕ H 2 () C ζ L 2 (). Proof. Since T is the adjoint of the operator S defined in (1.6.2) and S is an isomorphism, this property holds also true for T. Proof of (1). The existence and uniqueness of a solution ϕ H 1 () of (1.7.2) is an immediate consequence of the fact that T is an isomorphism. Moreover, we know that the inverse of an isomorphism is also an isomorphism, hence (1.7.5) ϕ H 1 () C 1 ζ L r/2 () with some C 1 > 0 independent of ζ. Rewriting (1.7.2) in the form {ã(x) ϕ z + c(x)ϕz} dx = (ζ b(x) ϕ) z dx z H 1 (), we see that Lemma 1.27 is applicable here, yielding ϕ C( ). In order to prove the W 1,r () regularity of ϕ, we follow the steps of the proof of Theorem 1.9. Let us point out the main differences. This time, we consider the equation {ã(x) ϕ j z + ϕ j z} dx = F (z) for arbitrary z H 1 (), ϕ j := ϕψ j and (1.7.6) F (z) := {ã(x) (ϕ z z ϕ) ψ j + (ζ + ϕ b(x) ϕ c(x)ϕ) ψ j z} dx.
39 1.7. THE ADJOINT PROBLEM 31 By the same arguments as in the proof of Theorem 1.9, along with the continuity of ϕ, one can check that F W 1,r (). To verify this, we consider exemplarily only the term (b ϕ)z: (b ϕ)z L 1 () C b ϕ L 2r r+2 () z L 2r r 2 () C b L r () ϕ L 2 () z W 1,r (). Proof of (2). Repeating the arguments of the first part of the theorem, there holds F W 1,s () with s = min{r, r} > 8/3, consequently ϕ W 1,s (). Now using the assumptions on ã and s > 1 8 = 4 as well as ã b Lr () 2 L s () 2 and ϕ L s () 2, we can pass to the Neumann formulation ϕ + 1 (1.7.7) ã cϕ = 1 ã (ζ + ( ã b) ϕ) Ls /2 () in, ν ϕ = 0 on, and obtain the desired H 3 /2 () regularity of ϕ from Corollary Let us now prove (1.7.3). Exploiting Corollary 1.15, we get (1.7.8) ϕ H 3/2 () C ( 2 ζ L r/2 () + ( ) ã W 1,r () + b L ()) r ϕ W 1,s () with C 2 > 0 dependent on c. Let us now derive an estimate for ϕ W 1,s (). Passing to the weak formulation of (1.7.7) and rearranging the terms, there holds [ ] 1 { ϕ z + ϕz} dx = ã (ζ + ã ϕ b ϕ cϕ) + ϕ z dx =: F (z) z H 1 (). As pointed out on page 11, the lefthand side of the previous identity defines an isomorphism from W 1.s () to W 1.s (). Consequently, its inverse is also an isomorphism and ϕ W 1,s () C F W 1.s () C 3 ( ζ L r/2 () + ( ã W 1,r () + b L r () + c L r/2 () + 1 ) ϕ H 1 ()). Combining the last inequality with (1.7.5) and (1.7.8), we conclude (1.7.3). Proof of (3). Thanks to the second part of the theorem, ϕ H 3 /2 () W 1,4 (). The H 2 () regularity of ϕ is deduced by applying the wellknown result by Grisvard for elliptic problems in convex domains; cf. [60, Corollary ]. It is enough to remark that ϕ L 2 () as follows after passing the term cϕ in (1.7.7) to the righthand side. Prior to demonstrating (1.7.4), let us recall a classical result for convex domains which can be found e.g. in [60]. Let w H 2 () be the unique solution of the equation { w + w = f in, ν w = 0 on, where f L 2 (). Then there exists C > 0, depending only on, such that w H 2 () C f L 2 ().
40 32 1. ANALYSIS OF QUASILINEAR ELLIPTIC PDES Adding (1 1 c)ϕ to both sides of the first equation in (1.7.7), we can apply the ã last result to (1.7.7) and get, along with the positivity of ã, Hölder s inequality, the embedding H 3 /2 () W 1,4 () and (1.7.3), ( ϕ H 2 () C ϕ L 2 () + 1 ) α ζ + ( ã b) ϕ cϕ L 2 () C ( ζ L 2 () + ( ) 1 + c L 2 () + ã W 1,4 () + b L ()) 4 ϕ H 3/2 () C ζ L 2 (). Let us formulate now the adjoint problem associated with the quasilinear equation (1.1.1). We have seen in the previous section that the linearization of (1.1.1) around a solution leads to the equation (1.6.5). The following equation is the adjoint equation of (1.6.5): (1.7.9) div[a(x, y) ϕ] + a f (x, y) y ϕ + (x, y)ϕ = ζ y y in, a(x, y) ν ϕ = v on. In the next theorem, we establish the existence, uniqueness and regularity of the solution of (1.7.9). Theorem Suppose that the Assumptions and 1.24(1) hold. Given y W 1,r () with r > 2, for any v H 1/2 () and ζ H 1 (), the equation (1.7.9) has a unique solution ϕ H 1 (). (1) Assume that a : R R is continuous. If r satisfies the condition (1.3.13), ζ L 2r r+2 () and v L r/2 (), then there holds ϕ W 1,r (). (2) Supposing that Assumption 1.17 holds, r > 8/3, ζ L p () (p > 4/3) and v L 2 (), then ϕ H 3 /2 (). (3) Suppose that Assumption 1.17 is satisfied and is convex. Further, let 2 r r 0, with r 0 given as in Theorem 1.21, ζ L r () and either v = 0 and y W 1,2r () or v W 1 1/r,r () and y W 1,q () with q > 2r. Then the function ϕ belongs to W 2,r (). Proof. From Theorem 1.35 we know that T L(H 1 (), H 1 () ) is an isomorphism. Setting the coefficients ã, b and c as in (1.6.6), this is equivalent to the wellposedness of the adjoint equation (1.7.9) in H 1 (), since ϕ satisfies the equation T ϕ = ζ + Bv H 1 (), where B is defined in (1.6.8). Proof of (1). Thanks to the inclusion W 1,r () C( ), the function y is bounded, a = a(, y( )) C( ) and ( a/ y)(, y( )) y L r () 2. The result follows then
41 1.7. THE ADJOINT PROBLEM 33 by similar arguments as in Theorem 1.35(1). Let us only remark that, taking into account (1.6.6), the functional F defined in (1.7.6) has now the form { ( F (z) = a(x, y) (ϕ z z ϕ) ψ j + ζ a ) (x, y) y ϕ ψ j z y ( + 1 f ) } (x, y) ϕψ j z dx + vψ j z dσ(x) y and that the functional z vψ jz dσ(x) is an element of W 1,r () ; compare also page 10. Proof of (2). The H 3 /2 () regularity of ϕ is an immediate consequence of Theorem 1.35(2); the presence of the boundary datum v L 2 () does not matter for the proof. Proof of (3). To prove that ϕ W 2,r () we reformulate (1.7.9) as follows ϕ = 1 ( ζ f (1.7.10) a y (, y)ϕ + [ xa](, y) ϕ 2 a ) (, y) y ϕ y ν ϕ = v a in, on. If we show that ϕ L r () and ν ϕ W 1 1/r,r () then the result follows from Grisvard [60, Corollary ]. Let us consider first the case when r = 2. From the second part of the theorem we know that ϕ H 3 /2 () W 1,4 () which, together with y W 1,4 (), leads to ϕ L 2 (). This finishes the proof if v = 0. Consider now the case when v H 1 /2 (), v 0 and y W 1,q () with q > 4. From (1.5.7) we have W 1,q () C 0,1 2/q ( ), hence 1/a(, y( )) C 0,1 2/q ( ). Then due to the fact that 1 2 q > 1 1 r = 1 2, we can apply (1.5.6) to conclude (v/a) H 1 /2 (). If r > 2 then ϕ H 2 () W 1,2r (), as proved above. Together with y W 1,2r (), this yields ϕ L r () which is sufficient to deduce the W 2,r () regularity of ϕ when v = 0. Further, let us assume v 0, v W 1 1/r,r (), and y W 1,q () with q > 2r. Once again, 1/a(, y( )) C 0,1 2/q ( ) and (1.5.6) leads to (v/a) W 1 1/r,r () which completes the proof. The following result for the adjoint problem is the analog to Corollary Corollary Suppose that the Assumptions , 1.17 and 1.24(1) hold. Given a bounded set Y W 1,r () with r > 8/3 and v i L 2 (), ζ i L p () (p > 4/3), i = 1, 2, there exists C Y > 0 such that, for any y i Y, there holds the estimate (1.7.11) ϕ 2 ϕ 1 H 3/2 () C Y ( ) y2 y 1 W 1,r () + ζ 2 ζ 1 L p () + v 2 v 1 L 2 (),
42 34 1. ANALYSIS OF QUASILINEAR ELLIPTIC PDES where ϕ i H 3 /2 () is the solution to div[a(x, y i ) ϕ i (x)] + a y (x, y i) y i ϕ i + f y (x, y i)ϕ i = ζ i in, a(x, y i ) ν ϕ i = v i on. Proof. Since W 1,r () C( ), the set Y is bounded in C( ). Subtracting the equations satisfied by ϕ i, we get div [a(x, y 2 ) (ϕ 2 ϕ 1 )] + a y (x, y 2) y 2 (ϕ 2 ϕ 1 ) + f y (x, y 2)(ϕ 2 ϕ 1 ) = g in, a(x, y 2 ) (ϕ 2 ϕ 1 ) ν = g on, where g L s (), with some 4r r+4 s > 4 3, and g L 2 () are given by g = ζ 2 ζ 1 div [(a(, y 1 ) a(, y 2 )) ϕ 1 ] ( a + y (, y 1) y 1 a ) y (, y 2) y 2 ϕ 1 + ( f y (, y 1) f ) y (, y 2) ϕ 1 and g = v 2 v 1 + (a(, y 1 ) a(, y 2 )) ν ϕ 1, respectively. Notice that y i ϕ j L s (), i, j = 1, 2, because y i L r () 2, ϕ j L 4 () 2 and < 3. Now it is easy to modify the proof of (1.7.3) such 4 r s 4 that (1.7.11) holds true.
43 CHAPTER 2 Finite element approximation 2.1. Introduction This chapter is devoted to the finite element based approximation of the quasilinear equation (1.1.1) and the adjoint equation (1.7.9). We impose minimal regularity assumptions on the data, see Assumptions , and use finite elements of degree one to approximate the equations under consideration. One major difficulty in the analysis is that the uniqueness of solutions of the discrete quasilinear equation is an open problem, even though the continuous equation has a unique solution. This is due to the nonmonotone character of the equation (1.1.1). In contrast to the continuous case, the comparison principle that we exploited in the proof of Theorem 1.5 cannot be applied to the discrete equation. Moreover, the nonlinear term a introduces a strong difficulty in the analysis of error estimates for the discretization of the equations (1.1.1) and (1.7.9). Our main aims of this chapter are threefold. First, we derive estimates in different norms for the error between the solution of the continuous problem (1.1.1) and solutions of the corresponding approximate problems. We assume again that is a polygonal set of R 2 ; see Barrett and Elliot [6] for a finite element approximation of a Neumann type problem in a curved domain. Some results for polyhedral domains of dimension three are presented in Chapter 5. We will distinguish two cases: whether is convex or not. These two different situations yield different orders of convergence in the L 2 () norm of solutions of the discrete problem to the solution of (1.1.1); see Section 2.3. Second, we focus on the uniqueness of the solution of the discrete version of (1.1.1). Applying Brouwer s fixed point theorem, we can prove the existence of a discrete solution, but, as far as we know, the uniqueness is an open question until now. However, for a wide class of equations, we are able to prove uniqueness of the discrete solutions, provided that the discretization is fine enough. For equations which do not fall into this class, we give a useful classification of the behavior of solutions of the discrete version of (1.1.1); see Section 2.4. The reader is also referred to Hlaváček [66] and Hlaváček et al. [67] for some uniqueness results if h is large enough or y and f(, 0) are sufficiently small. Third, we study the differentiability of the mapping which associates with each boundary datum a solution of the discrete quasilinear equation which is, in some 35
44 36 2. FINITE ELEMENT APPROXIMATION sense, locally unique. An important consequence of this result is that the discrete adjoint equation has a unique solution in spite of its nonmonotone character; see Theorem Furthermore, we prove some error estimates for the numerical approximation of the adjoint equation. All these results play a crucial role in the proof of error estimates for optimal control problems associated with (1.1.1); see Chapter 4. Let us relate our results to the previous ones in the literature. The corresponding Dirichlet problem associated with equation (1.1.1) was first studied by Douglas and Dupont [50] in dimension n 3, with f independent on y. They assumed a to be a C 2 function in R such that ( j a/ y j ) was bounded in R for j = 0, 1, 2. The proof of the error estimates is based on the AubinNitsche trick. To this end, they consider the equation (2.1.1) L ϕ = ζ in, ϕ = 0 on, where ζ L 2 () and L is the adjoint operator of [ (Lw)(x) = div a(x, y(x)) w(x) + a ] (x, y(x))w(x) y(x). y Then they use the regularity ϕ H 2 () which holds true because of their assumptions about the C 2+α ( ) regularity of y and the smooth property of. Later, Liu et al. [78] considered an extension of equation (1.1.1) to functions a(x, y) = (a ij (x, y)) n i,j=1. They still assumed a to be C 2 and bounded along with their derivatives, as mentioned above. They also extended the estimates to solutions y H0() H 1 k+1 (), k 1 (an homogeneous Dirichlet problem was studied), hence they did not require classical solutions as in [50]. Moreover, they assumed to be a polygonal or polyhedral domain. In the proof of the error estimates, they followed the AubinNitsche approach, too. The difficulty was that under their assumptions they could not deduce the H 2 () regularity of the solution ϕ of (2.1.1), even the existence and uniqueness was not proved. They just assumed the existence, uniqueness and regularity of ϕ. Brenner and Scott [11, pp ] studied the Dirichlet problem for small data, proving existence and uniqueness of a solution of the discrete quasilinear equation and deriving error estimates. Unfortunately, their method relies deeply on a fixpoint method that cannot be extended to general data. Casas and Tröltzsch [37] focused on the piecewise linear finite element approximation of the equation (1.1.1) with homogeneous Dirichlet boundary condition in the context of optimal control problems. They obtained L r () and W 1,r () (r 2) error estimates without the boundedness assumption on a and its derivatives. They also proved existence, uniqueness and H 2 () regularity for the solution of (2.1.1). In Section 2.3, we will see that even the C 2 regularity of a w.r.t. y is not necessary to derive error estimates; we only require a local Lipschitz property of a.
45 2.2. ASSUMPTIONS AND PRELIMINARY RESULTS 37 The outline of this chapter is as follows: In the next section, we recall standard results concerning the finite element method. In Section 2.3, the discrete version of (1.1.1) is introduced and error estimates in different norms are proved in the case of a nonconvex and convex domain, respectively. The uniqueness of discrete solutions is the topic of Section 2.4. In Section 2.5, some numerical experiments are presented which confirm the theoretical results of Section 2.3. Section 2.6 is devoted to the study of some useful properties of the discrete solution operator. Finally, this chapter is ended with the error analysis of the finite element approximation of the adjoint equation (1.7.9) Assumptions and preliminary results In this section, we introduce some useful notations and recall some preliminary results concerning the finite element method. We start with the introduction of a family of triangulations {T h } h>0 of, where is supposed to satisfy Assumption 1.1. The mesh T h consists of open and pairwise disjoint triangles T T h such that = T T h T. With each element T Th we associate two parameters ρ(t ) and δ(t ), where ρ(t ) denotes the diameter of the triangle T and δ(t ) is the diameter of the largest ball contained in T. Define the size of the mesh by h := max T Th ρ(t ). This triangulation is supposed to be regular in the following sense (Brenner and Scott [11], see also Casas et al. [29]): There exist two positive constants ρ and δ such that ρ(t ) δ(t ) δ and h ρ(t ) ρ hold for any T T h and any h > 0. Associated with this triangulation we set Y h = { y h C( ) y h T P 1 (T ) for all T T h }, where P 1 (T ) stands for the space of polynomials of degree at most one defined in T. Clearly, the finite element space Y h consists of continuous functions which are linear in every triangle T T h. Throughout this chapter, we denote by Π h : H 1 () Y h the interpolation operator introduced by Scott and Zhang in [87] having the following two important properties: (2.2.1) z Π h z H k () Ch m k z H m () for z H m () and 2 m k 0 and (2.2.2) Π h z W 1,q () C q z W 1,q () for z W 1,q () and q (1, ), where C > 0 and C q > 0 are independent of h and z. Moreover, for 2 > m k 0 there holds the stronger result 1 (2.2.3) lim h 0 h z Π hz k m H k () = 0,
46 38 2. FINITE ELEMENT APPROXIMATION cf. Brenner and Scott [11, 4.8 and Theorem ] or Bramble and Scott [10]. For the particular case when m = 3/2 and k = 1 we are going to deduce a formula which is easier to apply than the previous one. Lemma 2.1. Let K H 3 /2 () be a compact set. Then there exists a sequence {ε h } h>0 of positive real numbers, dependent on K, with ε h 0 as h 0 such that (2.2.4) z Π h z H 1 () ε h h 1 /2 z H 3/2 () z K. Proof. Let us consider the linear mapping A h : H 3 /2 () H 1 () defined by A h (z) = 1 h (z Π 1 /2 hz). In view of (2.2.1), the mapping z A h z H 1 () is continuous in H 3 /2 () and, together with the compactness of K, we deduce the existence of an element z h in K such that ε h := sup A h z H 1 () = A h z h H 1 (). z K Moreover, (2.2.1) yields that 0 ε h C for any h > 0. Let us denote by ε 0 the upper limit of {ε h } h>0. Again, due to the compactness of K, there exists a subsequence of { z h } h>0, denoted in the same way, such that { z h } h>0 converges strongly in H 3 /2 () to some z H 3 /2 (). Now from (2.2.3) we obtain 0 = lim h 0 A h z H 1 () lim sup h 0 { Ah z h H 1 () A h ( z h z) H 1 ()} { } {ε h } C lim sup zh z H 3/2 () = ε0, h 0 lim sup h 0 hence ε 0 = 0 and consequently the whole sequence {ε h } h>0 converges to zero. We will rename Cε h by ε h, therefore all the constants are included in ε h. Obviously, this convention does not change the fact that ε h 0 when h 0. Let us also consider the standard nodal interpolation operator I h : C( ) Y h defined by (I h z)(x i ) = z(x i ) at every node x i of the triangulation T h. Lemma 2.2. ([42, Theorem 3.1.6]) Let k 0, m 0 and t, q [1, ], be given such that the embeddings W m,q (T ) C( T ) and W m,q (T ) W k,t (T ) hold for every T T h. Then there exists a constant C > 0 independent of h such that (2.2.5) z I h z W k,t (T ) Ch 2( 1/t 1/q)+m k z W m,q (T ) z W k,t (T ). Finally, we recall an inverse estimate for functions belonging to Y h : C (2.2.6) z h W m,q () h 2 max{0, 1/t 1/q}+m k z h W k,t () z h Y h if k m and t, q [1, ], where C > 0 is independent of h, cf. Ciarlet and Lions [43, Theorem 17.2] or Bramble and Scott [10, Theorem ].
47 2.3. NUMERICAL ANALYSIS OF THE QUASILINEAR EQUATION 39 In the subsequent sections, we will suppose that the Assumptions and 1.17 hold and, unless otherwise said, u L 2 () is arbitrary but fixed. In view of Theorem 1.18 on page 16, the solution y u of (1.1.1) belongs to H 3 /2 () W 1,4 (). Since u is fixed, we will only write y instead of y u. We close this preparatory section by recalling an inequality that will be often used in the sequel. Lemma 2.3. ([19, Lemma A.1]) There exists a constant C > 0 such that z L 4 () C z 1 /2 L 2 () z 1 /2 H 1 () z H 1 () Numerical analysis of the quasilinear equation In this section, we study the approximation of the quasilinear equation (1.1.1) by finite elements of degree one. The discrete version of (1.1.1) is given by Find y h Y h such that, for all φ h Y h, (2.3.1) {a(x, y h (x)) y h (x) φ h (x)+f(x, y h (x))φ h (x)} dx= u(x)φ h (x) dσ(x). The next theorem shows that this problem has at least one solution and that all solutions of (2.3.1) are bounded in H 1 (). Theorem 2.4. The equation (2.3.1) has at least one solution. Moreover, there exists a positive constant C > 0 such that, for any solution y h Y h of (2.3.1), the following inequality holds (2.3.2) y h H 1 () C ( ) f(, 0) L p () + u L 2 () (p > 4/3). Proof. To show the existence of a solution y h of (2.3.1) we are going to apply Brouwer s fixed point theorem. Let us take h > 0 arbitrary but fixed, M > 0, ε > 0 and consider the mapping L ε : Y h Y h, L ε w h =: z h,ε, where z h,ε satisfies the equation {a M (x, w h ) z h,ε φ h + εχ E z h,ε φ h } dx = uφ h dσ(x) f M (x, w h )φ h dx φ h Y h. Here, a M and f M are the truncations of a and f, respectively, defined on page 4 and E is introduced in Assumption 1.3. Taking into account that Y h H 1 () C( ), the mapping L ε is uniquely determined as follows from the LaxMilgram theorem. Next we show that the mapping L ε has a fixed point. Arguing as on page 4, we have z h,ε H 1 () C a,ε ( u L 2 () + f M (, w h ) L p ()),
48 40 2. FINITE ELEMENT APPROXIMATION where C a,ε depends only on, α a and ε, but neither on a M nor on f M. Let us now prove that L ε is continuous. Once this is verified, we can apply Brouwer s fixed point theorem to obtain a function y h,ε Y h such that y h,ε = L ε y h,ε, i.e. satisfying (2.3.3) {a M (x, y h,ε ) y h,ε φ h + εχ E y h,ε φ h } dx = uφ h dσ(x) f M (x, y h,ε )φ h dx φ h Y h. Let us take w h Y h and a sequence {w h,k } k=1 Y h converging to w h. It is enough to prove that z k h,ε := L ε w h,k L ε w h =: z h,ε as k. Since {z k h,ε} k=1 is bounded in Y h and we are in finite dimensions, there exists a subsequence, denoted in the same way, converging strongly in any norm to some z h,ε. Now we can pass to the limit in the equation satisfied by z k h,ε and observe that z h,ε = L ε w h = z h,ε. Since any converging subsequence of {z k h,ε} k=1 converges to z h,ε, it is a standard consequence that the whole sequence converges to z h,ε, too; see e.g. Gajewski et al. [54, Lemma 5.4 on page 10]. The convergence L ε w h,k L ε w h implies the continuity of L ε and this concludes the existence of y h,ε Y h satisfying (2.3.3). Furthermore, the boundedness of {y h,ε } ε>0 in H 1 () independent of ε can be obtained as in the proof of Theorem 1.5; see pages 56. Again, since all norms are equivalent in Y h, the sequence { y h,ε C( ) } ε>0 is bounded by some constant C > 0. Hence, we can choose M C to deduce that y h,ε satisfies (2.3.3) with a and f instead of a M and f M, respectively. On the other hand, there exists a subsequence {y h,εk } k=1 (ε k 0 as k ) of {y h,ε } ε>0 which converges strongly in Y h to some y h when k. By passing to the limit in (2.3.3), with ε replaced by ε k, it follows that y h is a solution of (2.3.1). It remains to prove inequality (2.3.2). To this end, we consider the following identity { } a(x, yh ) y h 2 +(f(x, y h ) f(x, 0)) y h dx = uy h dσ(x) f(x, 0)y h dx. From this equality, along with the positivity of a and the monotonicity of f, we infer C 0 y h 2 H 1 () α a y h 2 L 2 () + α f y h 2 L 2 (E) which implies (2.3.2) with C = C 1 /C 0. C 1 ( f(, 0) L p () + u L 2 ()) yh H 1 () Remark 2.5. The existence of a solution of the corresponding problem with mixed Dirichlet and Neumann boundary conditions was proved by Hlaváček et al. [67] under stronger assumptions on a and f, namely supposing that a and f are uniformly bounded. The issue of the uniqueness of solutions of (2.3.1) is much more difficult and it is the scope of Section 2.4.
49 2.3. NUMERICAL ANALYSIS OF THE QUASILINEAR EQUATION Error estimates in nonconvex domains. In this subsection, we consider the case when is not convex and derive estimates of the error y y h in the L 2 () and H 1 () norms. Theorem 2.6. There exists h 0 > 0 such that, for any h < h 0, the equation (2.3.1) has at least one solution y h that obeys (2.3.4) y y h L 2 () + h 1 /2 y y h H 1 () ε h h, where ε h 0 when h 0. If {y h } h>0 is a family of solutions of (2.3.1) that is bounded in L (), then (2.3.4) holds as well. Proof. Let us take M = 1 + y L (), c 0 (x) = α f 2 χ α f E(x) and f(x, t) = f(x, t) 2 χ E(x)t for x and t R. The monotonicity of f w.r.t. the second component can be seen in the following way. If x E then for any real t 1 t 2 we have from inequality (1.2.2) on page 3 that f(x, t 2 ) f(x, t 1 ) = f(x, t 2) f(x, t 1 ) α f t 2 t 1 α f t 2 t 1 t 2 t 1 2 t 2 t 1 2. If x \ E the monotonicity of f follows immediately from the monotonicity of f. Let y h Y h be any function satisfying φ h Y h the equation { (2.3.5) am (x, y h ) y h φ h + ( c 0 (x)y h + f M (x, y h ) ) } φ h dx = uφ h dσ. Here, a M and f M are the truncations of a and f, respectively, described on page 4. As in the proof of Theorem 2.4, the existence of y h follows from Brouwer s fixed point theorem. Observing that f M (, y h ) L p () ( φ M L p () + α f /2 ) M + f(, 0) L p (), with p > 4/3 and φ M introduced in Assumption 1.3, the boundedness of { y h H 1 ()} h>0 follows by the same arguments as (2.3.2). In the sequel, we prove that y h satisfies (2.3.4), provided that h is small enough. Certainly, if y h L () M then y h is a solution of (2.3.1). We will split the proof of (2.3.4) in several steps. Step 1: Preparations. Consider the function B M : L 2 () H 1 () H 1 () R defined by B M (w, z, φ) = {a M (x, w(x)) z φ + (c 0 (x) + c M (x)) zφ} dx, where f M (x, y(x)) f M (x, y h (x)) if y(x) y (2.3.6) c M (x) := h (x), y(x) y h (x) α f otherwise.
50 42 2. FINITE ELEMENT APPROXIMATION From the definition of M and the monotonicity of f w.r.t. the second variable it holds c M 0 a.e. in and (2.3.7) c 0 (x) + c M (x) α f 2 x E. To see (2.3.7) we consider the most difficult case when y h (x) > M and x E. Then (1.2.2) implies c M (x) = f M (x, y(x)) f M (x, y h (x)) y(x) y h (x) f(x, y(x)) f(x, M) = α f y(x) M y(x) y h (x) 2 y(x) y h (x) = y(x) M ( f(x, y(x)) f(x, M) α f y(x) y h (x) y(x) M 2 ) y(x) M α f y(x) y h (x) 2 0. If y h (x) < M we can proceed analogously. If x E and y h (x) M then c M (x) = f M (x, y(x)) f M (x, y h (x)) y(x) y h (x) = f(x, y(x)) f(x, y h(x)) y(x) y h (x) α f 2 χ E(x) α f 2. On the other hand, analogous arguments and inequality (1.2.3) on page 3 lead to (2.3.8) c M (x) φ M (x) + α f 2 =: φ M (x) and φ M L p (). Let us now provide some useful properties of the function B M. From the positivity of a, (2.3.7) and the Poincaré inequality (1.3.3) on page 4, we get B M (w, z, z) α a z 2 dx + α f z 2 dx C 0 z 2 H 2 1 () and, together with (2.3.8), B M (w, z, φ) sup a(x, t) z L 2 () φ L 2 () + α f x, t M 2 z L 2 () φ L 2 () + φ M L p () z 2p φ L p 1 () L C M z H 1 () φ H 1 () E 2p p 1 () with C M > 0 only dependent on M but not on h. Moreover, invoking the definition of M, (1.1.1) and (2.3.5), one finds for all φ H 1 () and φ h Y h ( (2.3.9) B M (y, y, φ) = uφ dσ(x) + cm (x)y f M (x, y) ) φ dx, ( (2.3.10) B M (y h, y h, φ h ) = uφ h dσ(x) + cm (x)y h f M (x, y h ) ) φ h dx.
51 2.3. NUMERICAL ANALYSIS OF THE QUASILINEAR EQUATION 43 Furthermore, with the aid of (2.3.6), (2.3.9) and (2.3.10), we obtain for every φ h Y h (2.3.11) B M (y, y, φ h ) B M (y h, y h, φ h ) { = cm (x)(y y h ) ( fm (x, y) f M (x, y h ) )} φ h dx = 0. Step 2: Proof of (2.3.4) with y h satisfying (2.3.5). Step 2.1: H 1 () error estimate. Using (2.3.11), the Lipschitz property of a, (2.2.4) (we take here K = {y}) and (2.2.2) we have the estimate (2.3.12) Thus, C 0 y h Π h y 2 H 1 () B M (y h, y h Π h y, y h Π h y) = B M (y h, y h, y h Π h y) B M (y h, Π h y, y h Π h y) = B M (y, y, y h Π h y) B M (y h, Π h y, y h Π h y) = B M (y, y Π h y, y h Π h y) + (B M (y, Π h y, y h Π h y) B M (y h, Π h y, y h Π h y)) = B M (y, y Π h y, y h Π h y) + (a M (x, y) a M (x, y h )) Π h y (y h Π h y) dx C M y Π h y H 1 () y h Π h y H 1 () + C a,m y y h Π h y (y h Π h y) dx ( ε h h 1 /2 y H 3/2 () + C y y h L 4 () y W 1,4 ()) yh Π h y H 1 (). y h Π h y H 1 () ε h h 1 /2 + C y y h L 4 (). From this inequality and the estimate (2.2.4) we infer (2.3.13) y y h H 1 () y Π h y H 1 () + y h Π h y H 1 () Further, according to Lemma 2.3, it follows ε h h 1 /2 + C y y h L 4 (). y y h L 4 () C y y h 1 /2 L 2 () y y h 1 /2 H 1 () and, for q = q = 2 and any ε > 0, Young s inequality leads to y y h 1 /2 L 2 () y y h 1 /2 H 1 () 1 2ε 2 y y h L 2 () + ε2 2 y y h H 1 (). Taking ε small enough in the previous inequality, we conclude from (2.3.13) that (2.3.14) y y h H 1 () ε h h 1 /2 + C y y h L 2 ().
52 44 2. FINITE ELEMENT APPROXIMATION Step 2.2: L 2 () error estimate. In order to estimate y y h L 2 (), we use a duality argument based on the AubinNitsche trick. To this aim, we introduce the function a M (x, y(x)) a M (x, y h (x)) if y(x) y h (x), b M (x) = y(x) y h (x) α a otherwise. Exploiting Assumption 1.2, we easily see that b M (x) C M x, with C M > 0 being independent of h. Next we apply Theorem 1.35(2) on page 30, by setting ã(x) = a M (x, y(x)), b(x) = b M (x) y(x) and c(x) = c 0 (x) + c M (x), to obtain a function ϕ H 3 /2 () such that, for every w H 1 () (2.3.15) {a M (x, y) ϕ w + b M w y ϕ + (c 0 + c M ) ϕw} dx = (y y h )w dx. By taking w = y y h in (2.3.15) and using the definitions of c M and b M as well as the positivity of a and (2.3.5), we derive y y h 2 L 2 () = {a M (x, y) ϕ ( y y h ) + b M (x) (y y h ) y ϕ + (c 0 (x) + c M (x)) ϕ(y y h )} dx { = am (x, y) ϕ y + ( c 0 (x)y + f M (x, y) ) ϕ } dx { am (x, y) ϕ y h + ( c 0 (x)y h + f M (x, y h ) ) ϕ } dx + (a M (x, y) a M (x, y h )) y ϕ dx { = am (x, y)( ϕ Π h ϕ) y + ( c 0 (x)y + f M (x, y) ) (ϕ Π h ϕ) } dx { + am (x, y h ) Π h ϕ y h + ( c 0 (x)y h + f M (x, y h ) ) Π h ϕ } dx { am (x, y) ϕ y h + ( c 0 (x)y h + f M (x, y h ) ) ϕ } dx + (a M (x, y) a M (x, y h )) y ϕ dx = a M (x, y) ( ϕ Π h ϕ) ( y y h ) dx ( + c0 (x) (y y h ) + f M (x, y) f M (x, y h ) ) (ϕ Π h ϕ) dx + (a M (x, y) a M (x, y h )) ( y ϕ y h Π h ϕ) dx
53 (2.3.16) 2.3. NUMERICAL ANALYSIS OF THE QUASILINEAR EQUATION 45 = a M (x, y) ( ϕ Π h ϕ) ( y y h ) dx ( + c0 (x) (y y h ) + f M (x, y) f M (x, y h ) ) (ϕ Π h ϕ) dx + (a M (x, y) a M (x, y h )) ( y y h ) Π h ϕ dx + (a M (x, y) a M (x, y h )) ( ϕ Π h ϕ) y dx. With the aid of the assumptions on a and f and the estimates (2.2.1) and (2.2.2), it follows y y h 2 L 2 () C ( h 1 /2 + y y h H 1 ()) y yh H 1 () ϕ H 3/2 (). In view of inequality (1.7.3) on page 30, we have ϕ H 3/2 () C y y h L 2 (), hence (2.3.17) y y h L 2 () C ( h 1 /2 + y y h H 1 ()) y yh H 1 (). Utilizing (2.3.14), we get from the previous inequality (2.3.18) y y h L 2 () ε h h + C y y h 2 L 2 (). In the next step, we prove the strong convergence y h y in L 2 () as h 0, implying C y y h 2 L 2 () 1 2 y y h L 2 (), for sufficiently small h. Thus, we deduce from (2.3.14) and (2.3.18) that y h satisfies (2.3.4) for every h > 0 small enough. Step 3: Convergence y h y in L 2 (). The boundedness of {y h } h>0 in H 1 () implies the existence of an element w H 1 () and a subsequence of {y h } h>0, denoted again by {y h } h>0, such that y h w weakly in H 1 (). Now we show that w = y. To this end, let z H 2 () be arbitrarily chosen. Then Π h z Y h and there holds z Π h z H 1 () 0 as h 0. Knowing that y h satisfies { am (x, y h ) y h Π h z + ( c 0 (x)y h + f M (x, y h ) ) Π h z } dx = uπ h z dσ(x), and due to the boundedness of a M (, y h ) and the domination of f M (, y h ) by a function in L p (), the strong convergence Π h z z in H 1 (), (2.2.2) and the weak convergence y h w in H 1 (), we can pass to the limit in the previous equation and find that w satisfies { (2.3.19) am (x, w) w z + ( c 0 (x)w + f M (x, w) ) z } dx = uz dσ(x). From this equality and the density of H 2 () in H 1 () we deduce that (2.3.19) holds for any z H 1 (). It is immediate to check that (2.3.19) also holds with y instead of w. Hence, from the uniqueness of the solution of (2.3.19), see Lemma 2.9 below,
54 46 2. FINITE ELEMENT APPROXIMATION we conclude that w = y. Now, as argued in the proof of Theorem 2.4, the whole original sequence {y h } h>0 converges weakly to y in H 1 (). Since H 1 () is compactly embedded in L 2 (), the convergence y h y is strong in L 2 (). Step 4: Convergence y h y in L (). In the last step of the proof, we demonstrate the convergence y h y in L () which implies that y h is a solution of (2.3.1) for h sufficiently small. To this aim, we use the estimates (2.2.5) and (2.2.6) to obtain (2.3.20) z I h z L 2 () + h z I h z L () Ch 3 /2 z H 3/2 () z H 3 /2 () and (2.3.21) z h L () C h z h L 2 () z h Y h. From (2.3.4), (2.3.20) and (2.3.21), we deduce y y h L () y I h y L () + I h y y h L () Ch 1 /2 y H 3/2 () + C h I hy y h L 2 () Ch 1 /2 y H 3/2 () + C ( ) Ih y y L h 2 () + y y h L 2 () 0. The above uniform convergence ensures the existence of a value h 0 > 0 such that y h L () y L () + 1 = M for all h < h 0. This implies a M (x, y h ) = a(x, y h ) and f M (x, y h ) = f(x, y h ), thus y h is a solution of (2.3.1) for every h < h 0 and (2.3.4) holds. Finally, if {y h } h>0 is a family of solutions of (2.3.1) bounded in L (), then we can argue as above taking (2.3.22) M = sup y h L () + y L () + 1. h>0 Remark 2.7. In step 2.2 of the previous proof, we can also use (2.2.4) to estimate the error ϕ Π h ϕ H 1 (), although ϕ is dependent on h because of its dependence on y h. This is due to the fact that {y y h } h>0 is bounded in H 1 (), hence relatively compact in L 2 (). This implies that the set containing the solutions ϕ = ϕ(h) of (2.3.15) corresponding to y y h, h > 0, is relatively compact in H 3 /2 (). According to Lemma 2.1, ε h can be chosen independently of y y h in the interpolation estimate of ϕ. Remark 2.8. The introduction of the equation (2.3.5) is not standard. The reason for this is that the truncation f M of the function f does not satisfy in general inequality (1.2.2) on page 3. To preserve this property we have replaced f M (x, t) by c 0 (x)t + f M (x, t), (x, t) R. This difficulty does not occur in the case of homogeneous Dirichlet boundary conditions, since (1.2.2) is not a necessary assumption, cf. Casas and Tröltzsch [37].
55 2.3. NUMERICAL ANALYSIS OF THE QUASILINEAR EQUATION 47 Next we prove a result that we have already used in the proof of the previous theorem. Lemma 2.9. The solution of the equation (2.3.19) is unique. Proof. First, the solution y = y u of (1.1.1) is also a solution of (2.3.19), with M given on page 41. To deduce its uniqueness we follow the lines of Theorem 1.5 on page 3. Let w H 1 () be another solution of (2.3.19). The continuity of w in can be shown as in the proof of Theorem 1.5. Let ε > 0. By setting y 1 = y and y 2 = w, we define the sets 0, ε, and the function z ε H 1 () as on pages 67. Thanks to the monotonicity of f, we find ( fm (x, w) f M (x, y) ) z ε 0. Indeed, if z ε (x) 0 it follows by computations similar to the ones on page 42 that ( fm (x, w) f M (x, y) ) z ε = f M (x, w) f M (x, y) zε 2 f M (x, w) f M (x, y) zε 2 0. z ε w y Now a straightforward modification of the estimates (1.3.10) and (1.3.11) on page 7 yields z ε 2 L 2 () C ε 2 y 2 L 2 ( 0 \ ε). The rest of the proof is identical with that of Theorem 1.5. As a consequence of the estimate (2.3.4), we obtain the following result. Corollary For every sequence {y h } h<h0 (2.3.4) it holds (2.3.23) y y h W 1,q () Ch 2 /q 1/2 of solutions to (2.3.1) satisfying q (2, 4]. Proof. Invoking Theorem 2.6 and the inequalities (2.2.5) and (2.2.6), it follows that y y h W 1,q () y I h y W 1,q () + I h y y h W 1,q () C 1 h 2 /q 1/2 y H 3/2 () + C 2 I h q 2 h y y h H 1 () q C 1 h 2 /q 1/2 y H 3/2 () + C 2 h q 2 q C 1 h 2 /q 1/2 y H 3/2 () + C 2h 1 2 q 2 q ( Ih y y H 1 () + y y h H 1 ()) ( C1 y H 3/2 () + ε h ) Ch 2/q 1/2 and (2.3.23) is concluded Error estimates in convex domains. Our goal in this subsection is to derive error estimates in the L 2 () and H 1 () norms in the case when is convex. These estimates will improve the ones obtained in the nonconvex case, due to the higher regularity of the solution ϕ of the adjoint problem (2.3.15) in convex domains.
56 48 2. FINITE ELEMENT APPROXIMATION Theorem Assuming that is convex, the conclusions of Theorem 2.6 remain valid with (2.3.24) y y h L 2 () + h y y h H 1 () ε h h 3 /2 instead of (2.3.4). Proof. By Theorem 2.6, there exists h 0 > 0 such that, for any h < h 0, the equation (2.3.1) has at least one solution y h that obeys (2.3.4). The proof will be complete if we show the error estimate in the L 2 () norm as stated in the theorem. According to Corollary 2.10 and to the embedding W 1,q () C( ) for any q > 2, y h converges uniformly to y as h tends to zero. Choosing M as in (2.3.22) (the supremum is now taken over {y h } h<h0 ), we can skip the truncation of the coefficients a and f and consider equation (2.3.1) directly. In the convex case, Theorem 1.35(3) on page 30 (remember that y W 1,4 ()) ensures the H 2 () regularity of the solution ϕ of the equation (2.3.25) {a(x, y) ϕ w +(b y ϕ + cϕ) w} dx= (y y h )w dx w H 1 (), where b(x) := a(x, y(x)) a(x, y h(x)) y(x) y h (x) and c(x) := f(x, y(x)) f(x, y h(x)) y(x) y h (x) if y(x) y h (x) and b(x) := α a and c(x) := α f otherwise. Moreover, inequality (1.7.4) on page 30 implies the existence of C > 0 such that ϕ H 2 () C y y h L 2 (). Furthermore, from (2.2.1) it follows that ϕ Π h ϕ is of order h 2 and h in the L 2 () and H 1 () norms, respectively. In the sequel, we show that the above results lead to the desired error estimate in the L 2 () norm. From inequality (2.3.4) we get (2.3.26) y y h L 2 () ε h h and (2.3.27) y y h L 6 () ε h h 1 /2, where we have used that H 1 () L 6 (). Now (2.3.16) can be replaced by y y h 2 L 2 () = a(x, y) (ϕ Π h ϕ) (y y h ) dx + (f(x, y) f(x, y h )) (ϕ Π h ϕ) dx + (a(x, y) a(x, y h )) (y y h ) Π h ϕ dx (2.3.28) + (a(x, y) a(x, y h )) (ϕ Π h ϕ) y dx.
57 2.3. NUMERICAL ANALYSIS OF THE QUASILINEAR EQUATION 49 With the aid of (2.3.28), (2.3.4) and (2.2.1), we are going to show that (2.3.29) y y h L 2 () ε h h 3 /2 + ε h h 1 /2 y y h L 3 (). To prove this inequality let us estimate each integral on the righthand side of (2.3.28). The boundedness of a and the CauchySchwarz inequality lead to a(x, y) (ϕ Π h ϕ) (y y h ) dx C (ϕ Π h ϕ) L 2 () (y y h ) L 2 () C ϕ Π h ϕ H 1 () y y h H 1 () ε h h 3 /2 ϕ H 2 () ε h h 3 /2 y y h L 2 (). Using (1.2.3) on page 3 and Hölder s inequality, we can estimate the second term on the righthand side of (2.3.28) as follows (f(x, y) f(x, y h )) (ϕ Π h ϕ) dx φ M (x) y y h ϕ Π h ϕ dx φ M L 4/3 () y y h ϕ Π h ϕ L 4 () φ M L 4/3 () y y h L 8 () ϕ Π h ϕ L 8 (). The embedding H 1 () L 8 (), along with (2.3.4), yields (f(x, y) f(x, y h )) (ϕ Π h ϕ) dx ε h h 3 /2 ϕ H 2 () ε h h 3 /2 y y h L 2 (). Thanks to the stability of the interpolation operator Π h in W 1,6 () (Eq. (2.2.2)) and the embedding H 2 () W 1,6 (), we obtain for the third term on the righthand side of (2.3.28) (a(x, y) a(x, y h )) (y y h ) Π h ϕ dx C a,m y y h Π h ϕ L 2 () (y y h ) L 2 () ε h h 1 /2 y y h L 3 () Π h ϕ L 6 () ε h h 1 /2 y y h L 3 () y y h L 2 (). Concerning the last integral on the righthand side of (2.3.28), we have, along with (2.3.27), (a(x, y) a(x, y h )) (ϕ Π h ϕ) y dx C a,m y y h L 6 () (ϕ Π h ϕ) L 2 () y L 3 () ε h h 3 /2 y y h L 2 (). Summing up all inequalities above, we conclude (2.3.29).
58 50 2. FINITE ELEMENT APPROXIMATION Further, let us take h small enough such that h < 1 and ε h < 1. We prove now the following inequality (2.3.30) y y h L 2 () ε h 1 ε h h 3 /2 which yields the desired error estimate in the L 2 () norm. To verify (2.3.30) we deduce that k 1 (2.3.31) y y h L 2 () ε h h 3 /2 h 1 2 k for every k N. ε j h j=0 Taking the limit k in the last inequality, we arrive at (2.3.30). To obtain (2.3.31) we proceed by induction on k. For k = 1 (2.3.31) is the same as (2.3.26). Let us assume that (2.3.31) holds for some integer k and let us prove it for k + 1. First, employing Hölder s inequality, we find (2.3.32) y y h L 3 () y y h 1 /2 L 2 () y y h 1 /2 L 6 (). Next using (2.3.31) and (2.3.27), we get from (2.3.32) 1/2 k 1 y y h L 3 () ε 1 /2 h h3 /4 ε j h h 1 2 k+1 ε 1 /2 h h1 /4 = ε h h j=0 1/2 k 1 ε j h h 1 j=0 k 1 2 k+1 ε h h ε j h j=0 h 1 2 k+1. Inserting this estimate in (2.3.29), it follows k 1 k y y h L 2 () ε h h 3 /2 1 + ε h ε j h h 1 2 k+1 ε h h 3 /2 ε j h h 1 2 k+1 j=0 j=0 which yields (2.3.31) and concludes the proof Further results. In many applications, the regularity of the solution of the equation (1.1.1) is better than H 3 /2 (). This is the case when, e.g., is convex and u is sufficiently regular; see Theorem 1.21 on page 18. In such a situation, we obtain better error estimates as the following theorem shows. Theorem Let y H 2 () be the solution of (1.1.1). Then the conclusions of Theorem 2.6 remain valid with (2.3.33) y y h L 2 () + h 1 /2 y y h H 1 () Ch 3 /2 instead of (2.3.4), where C > 0 is independent of h. Moreover, if is convex then (2.3.34) y y h L 2 () + h y y h H 1 () Ch 2.
59 2.3. NUMERICAL ANALYSIS OF THE QUASILINEAR EQUATION 51 Proof. According to Theorem 2.6, for any h < h 0 there exists at least one solution y h of (2.3.1) satisfying (2.3.4). Since {y h } h<h0 is bounded in W 1,4 () C( ) (Corollary 2.10), we can choose M as in (2.3.22) (the supremum is now taken over {y h } h<h0 ) to skip the truncation of the coefficients a and f. Therefore, we can consider equation (2.3.1) directly. Thanks to the higher regularity of y, we have by (2.2.1) the following estimate for the interpolation error (2.3.35) y Π h y L 2 () + h y Π h y H 1 () Ch 2 y H 2 (). Let us first prove (2.3.33) if is nonconvex. By the same arguments as in the proof of Theorem 2.6, the previous estimate allows us to deduce (2.3.36) y y h H 1 () C ( ) h + y y h L 2 () h < h 0. We apply (2.3.36) to improve the order of convergence up to that given in (2.3.33). Combining (2.3.36) with (2.3.17), we find for sufficiently small h that y y h L 2 () + h 1 /2 y y h H 1 () C ( h 3 /2 + y y h 2 L 2 ()). Finally, the convergence y y h L 2 () 0 as h 0 leads to (2.3.33). If is convex then (2.3.24) holds true. Moreover, the fact that y H 2 () W 1,4 () and Theorem 1.35(3) on page 30 imply that the solution ϕ of (2.3.25) is in H 2 () and ϕ H 2 () C y y h L 2 () is valid with some constant C > 0. With the aid of the assumptions on a and f, it follows from (2.3.28) that y y h L 2 () C ( h y y h H 1 () + y y h 2 H 1 ()). This inequality, combined with (2.3.36), gives y y h L 2 () C ( h 2 + y y h 2 L 2 ()). The last estimate, along with the convergence y y h L 2 () 0 as h 0, completes the proof of (2.3.34). Remark For an extension of the statement of Theorem 2.12 to higher order finite elements the reader is referred to [18, Theorem 3.7]. The next result is a simple consequence of the previous theorem in the case when u H 1 /2 (), p 2 and convex. Then from Theorem 1.21 it follows that y H 2 (). Corollary Let u H 1 /2 () and assume that is convex and p 2. Then the conclusions of Theorem 2.6 remain valid with instead of (2.3.4). y y h L 2 () + h y y h H 1 () Ch 2
60 52 2. FINITE ELEMENT APPROXIMATION Up to now we have established error estimates only for a fixed boundary datum u. Notice that the sequence {ε h } h>0 given in the statement of Theorem 2.6 is dependent on u. With the aid of this sequence, we were able to deduce the convergence in L () of solutions of (2.3.5) to y. Therefore, h 0 given in Theorem 2.6 depends also on u. In the context of optimal control problems, we frequently are faced with the situation in which we have to apply error estimates for the solution of (1.1.1) with u being an arbitrary element of a given bounded set K of L 2 (). In such a situation, an essential assumption in Lemma 2.1 does not hold in general, namely the compactness of the set G(K), where G : u y u is the solution operator introduced in Theorem 1.31 on page 26. However, we should remark that Lemma 2.1 was only important to obtain the convergence in L () of solutions of (2.3.5) to y u for every u K. Having this converge in mind, we can proceed as follows: we truncate a and f with M = 1 + sup y u C( ) < u K (Eq. (1.3.1) on page 3) and we get error estimates in the H 1 () and W 1,q () norms, q (2, 4), using (2.2.1) instead of (2.2.4), hence replacing ε h in both estimates by a constant C dependent on K. Then a posteriori the convergence in W 1,q () implies the uniform convergence of discrete solutions to the solution of the continuous equation. Consequently, the truncation does not play any role for h < h 0 which depends now only on K. The following result extends the estimates obtained so far to the above situation. Corollary Let K L 2 () be bounded. Then there exist h 0 > 0 and C Ki > 0, i = 0, 1, such that, for any u K and h < h 0, equation (2.3.1) has at least one solution y h (u) that obeys (2.3.37) (2.3.38) y u y h (u) L 2 () + h 1 /2 y u y h (u) H 1 () C K0 h, y u y h (u) W 1,q () C K1 h 2 /q 1/2 q (2, 4], where y u is the solution of (1.1.1). If is convex then there holds (2.3.39) y u y h (u) L 2 () + h y u y h (u) H 1 () C K2 h 3 /2 instead of (2.3.37). Moreover, if is convex, p 2 and u H 1 /2 (), then (2.3.40) y u y h (u) L 2 () + h y u y h (u) H 1 () C K3 h Uniqueness of solutions of the discrete equation In this section, we are going to study the uniqueness of solutions of (2.3.1). As far as we know, the issue of uniqueness is an open problem until now. In the sequel, we establish uniqueness results in two different situations. If a is bounded in R and f is dominated by a L p () function then we are able to prove uniqueness of (2.3.1) provided that h is small enough; see Theorem If the above assumptions
61 2.4. UNIQUENESS OF SOLUTIONS OF THE DISCRETE EQUATION 53 on a and f are not fulfilled, then we have uniqueness in a more restrictive class of functions; see Theorem 2.18 for a precise formulation of the latter result. As mentioned at the end of Section 2.2, u L 2 () is fixed and y := y u. Theorem Suppose that there exist a constant C > 0 and a function φ in L p () (p > 4/3) such that a(x, t) C and f(x, t) φ (x) for a.a. x, t R. Then there exists h 0 > 0 such that (2.3.1) has a unique solution for every h < h 0. Proof. According to Corollary 2.10, there exists h 0 > 0 such that, for any h < h 0, (2.3.1) has at least one solution ŷ h with ŷ h y in L () as h tends to zero. To show the uniqueness of ŷ h we will argue by contradiction. To this aim, we assume the existence of a sequence {h k } k=1, with 0 < h k < h 0 and h k 0 as k, such that (2.3.1) has another solution y hk with y hk ŷ hk. The function y hk satisfies for every φ hk Y hk the equation (2.4.1) {a(x, y hk ) y hk φ hk + f(x, y hk )φ hk } dx = uφ hk dσ(x). By virtue of Theorem 2.4, {y hk } k=1 is bounded in H 1 (), hence we can extract a subsequence, denoted in the same way, such that y hk ỹ weakly in H 1 () as k. Let us show that y = ỹ. Because of the compactness of the embedding H 1 () L q () q [1, ), the convergence y h to ỹ is strong in L q (). Moreover, y hk (x) ỹ(x) for a.a. x and, by the continuity of a and f w.r.t. the second component, f(x, y hk (x)) and a(x, y hk (x)) converge to f(x, ỹ(x)) and a(x, ỹ(x)), respectively. Furthermore, thanks to the assumptions of the theorem on a and f, we have f(, y hk ) f(, ỹ) in L p (), a(, y hk ) φ hk a(, ỹ) φ in L 2 () φ, φ hk H 1 () with φ hk φ in H 1 (). By taking φ H 2 () arbitrarily and putting φ hk = Π hk φ in (2.4.1), using the convergence Π hk φ φ in H 1 () as h k 0 and passing to the limit in (2.4.1), we end up with the equation {a(x, ỹ) ỹ φ + f(x, ỹ)φ} dx = uφ dσ(x). Finally, as a result of the density of H 2 () in H 1 () and the uniqueness of the solution y of (1.1.1), we conclude that y = ỹ. Let us now set z hk = y hk ŷ hk and w hk = z hk / z hk L 6 (). Our next goal is to show that {w hk } k=1 is bounded in H 1 (). Subtracting both equations satisfied by y hk and ŷ hk and dividing by z hk L 6 (), we arrive at (2.4.2) {a(x, y hk ) w hk φ hk + b hk w hk ŷ hk φ hk + c hk w hk φ hk } dx = 0
62 54 2. FINITE ELEMENT APPROXIMATION for all φ hk Y hk, where b hk (x) := a(x, y h k (x)) a(x, ŷ hk (x)) y hk (x) ŷ hk (x) and c hk (x) := f(x, y h k (x)) f(x, ŷ hk (x)) y hk (x) ŷ hk (x) if y hk (x) ŷ hk (x) and b hk (x) := α a and c hk (x) := α f otherwise. The boundedness of {b hk } k=1 in L () and {c hk } k=1 in L p () can be seen in the following way. Define M = sup ŷ hk L () and hk = {x y hk (x) M + 1}. h k Then by the assumptions of the theorem on a and f, we find b hk (x) C a,m+1, c hk (x) φ M+1 (x) if x hk, and taking into account that y hk (x) ŷ hk (x) 1 x \ hk, b hk (x) a(x, y hk (x)) a(x, ŷ hk (x)) 2C, c hk (x) f(x, y hk (x)) f(x, ŷ hk (x)) 2φ (x), if x \ hk. By taking subsequences if necessary, b hk b weakly in L () and c hk c weakly in L p (). Moreover, c is nonnegative and c(x) α f x E. From (2.4.2), along with the monotonicity of f and Hölder s inequality, we get { } C 0 w hk 2 H 1 () a(x, yhk (x)) w hk 2 + c hk (x)wh 2 k dx = b hk (x)w hk ŷ hk w hk dx C w hk L 6 () ŷ hk W 1,3 () w hk H 1 (). Thus, w hk H 1 () C ŷ hk W 1,3 () which shows the boundedness of w hk H 1 (), because {ŷ hk } k=1 is bounded in W 1,3 (); see Corollary Hence, taking a subsequence, denoted again by {w hk } k=1, there exists w H 1 () such that w hk w weakly in H 1 () as k. Once again, due to the compactness of the embedding H 1 () L 6 (), the convergence w hk w is strong in L 6 (). Since w hk L 6 () = 1, it follows that w L 6 () = 1, too. Taking in (2.4.2) φ hk = Π hk φ, with φ H 2 () arbitrarily chosen, and passing to the limit, we obtain, along with the strong convergences ŷ h y in W 1,3 () (Corollary 2.10), Π hk φ φ in H 1 () and the convergence y hk y a.e. in, that (2.4.3) {a(x, y(x)) w φ + b(x)w y φ + c(x)wφ} dx = 0. Finally, since H 2 () is dense in H 1 (), equation (2.4.3) holds for every φ H 1 (). The lefthand side of (2.4.3) equals Sw, φ H 1 (),H 1 (), where S is the isomorphism defined on page 20 with ã = a(, y( )) and b = b y. Consequently, w = 0 which contradicts the fact that w L 6 () = 1. Proposition For any M > y L (), there exists h M > 0 such that (2.3.5) has a unique solution for every h < h M.
63 2.5. NUMERICAL EXPERIMENTS 55 Proof. We know already from the last step of the proof of Theorem 2.6 that every solution of (2.3.5) converges to y in L (). In the sequel, we will follow the lines of the proof of Theorem 2.16 and assume again the existence of a sequence {h k } k=1 of positive real numbers, with h k 0 as k, such that (2.3.5) has two solutions y hk and ŷ hk with y hk ŷ hk. Because of the uniform convergence mentioned above, there exists k N such that y hk L () M and ŷ hk L () M k k. Thus, for any k k the functions y hk and ŷ hk satisfy (2.3.1). Setting z hk = y hk ŷ hk and w hk = z hk / z hk L 6 (), we consider again equation (2.4.2). The boundedness of {b hk } k k in L () and {c hk } k k in L p () are easy to verify. By taking subsequences, it follows that b hk b weakly in L () and c hk c weakly in L p (). Finally, the boundedness of {w hk } k k in H 1 () and the contradiction is obtained in the same way as in the proof of Theorem Theorem For every M > y L () there exists h M > 0 such that the equation (2.3.1) has at most one solution y h with the property y h L () M h < h M. Moreover, if for any h < h M there is another solution ỹ h of (2.3.1) such that ỹ h L () > M then lim h 0 ỹ h L () = +. Proof. Let us assume that y h and ỹ h are solutions of (2.3.1) with y h L () M and ỹ h L () M. Then y h and ỹ h are also solutions of the equation (2.3.5). According to the preceding proposition, there exists h M > 0 such that y h = ỹ h h < h M. In order to show that lim h 0 ỹ h L () = + if ỹ h is a solution of (2.3.1) such that ỹ h L () > M, we assume the contrary. Then there exists a subsequence {ỹ hk } k=1 bounded in L () with h k < h M and h k 0 as k +. For M := sup ỹ hk L () k the function ỹ hk satisfies (2.3.5) with M and ỹ hk instead of M and y hk, respectively. By virtue of the last step of the proof of Theorem 2.6, ỹ hk converges to y strongly in L (). Thus, ỹ hk L () < M for h k sufficiently small which contradicts the assumption that ỹ hk L () > M for all h k < h M Numerical experiments This section is devoted to the verification of the error estimates obtained in Section 2.3 by three numerical test examples. In the first two examples, we consider the Neumann problem { y + y = 0 in, (2.5.1) ν y = u on, where is convex or not. In both examples, the boundary data u belong to L 2 () but u L s () for any s > 2. In the construction of such a function u, we have
64 56 2. FINITE ELEMENT APPROXIMATION incorporated the function g : (0, 1/2) R, g(t) = 1/(t 1 /2 log(t)). Making use of the substitution z = log(t), we have 1/2 1 s log(2) e s 2 2 z < if s = 2, dt = dz t 1 /2 log(t) z s = if s > 2. 0 Hence, the function g, and consequently u, too, has exactly the regularity mentioned above. Therefore, the solution y of (2.5.1) is in H 3 /2 (); see Corollary 1.15 on page 15. Since we do not know y exactly, we take as reference solution the numerical solution y fine of (2.5.1) computed on a very fine mesh whose mesh size is denoted in the sequel by h fine. Hence, instead of studying the behavior of the error y y h X as h becomes small, we investigate the error y fine y h X for X = L 2 () and X = H 1 (). The experimental order of convergence is computed by EOC X (y fine ) := log( y fine y h1 X ) log( y fine y h2 X ) log(h 1 ) log(h 2 ) for two consecutive mesh sizes h 1 and h 2. To motivate the above formula let us assume that y fine y h X is of order O(h σ ) which is denoted by y fine y h X Ch σ. Then y fine y hi X Ch σ i, i = 1, 2, hence log( y fine y hi X ) log(c) + σ log(h i ) and finally log( y fine y h1 X ) log( y fine y h2 X )/(log(h 1 ) log(h 2 )) σ. The third example is concerned with a particular quasilinear equation of the form (1.1.1). We follow the same procedure as above and report on both situations, convex and nonconvex domains, where we consider the same boundary datum u as in the first and second example, respectively. Let us briefly describe how we have performed the computations. To solve problem (2.5.1) numerically we make use of the finite element solver of the MATLAB PDE Toolbox. For the evaluation of the singular integrals involving the function g we have used the adaptive GaussKronrod quadrature quadgk of MATLAB. Concerning the quasilinear equation (1.1.1), a Newton method is implemented. To this aim, we transform the problem of finding y satisfying (1.1.1) into the problem of finding a solution of the equation F (y) = 0, with F appropriately chosen. To solve this problem we set up the following iterative method: Given an initial iterate y 0, determine the iterate y n+1 by solving the equation F (y n ) + F (y n )(y n+1 y n ) = 0, n = 0, 1, 2... More precisely, the algorithm is given by the following steps. Algorithm (Algorithm for (1.1.1)) (I) Initialization: Choose an initial datum y 0 H 3 /2 () and set n = 0. (II) Compute the solution y of the linearized equation { (2.5.2) a(x, y n ) y φ + a y (x, y n)y y n φ + f } y (x, y n)yφ dx = F n (φ)
65 2.5. NUMERICAL EXPERIMENTS 57 φ H 1 (), where {( ) f F n (φ) := y (x, y n)y n f(x, y n ) φ + a } y (x, y n)y n y n φ dx + uφ dσ(x). (III) Stop if y y n L () < T OL or set y n+1 = y, n = n + 1 and go to (II). For our purposes, we have chosen T OL = 10 8 and y 0 1. Again, we have taken the finite element solver of the MATLAB PDE Toolbox to deal with the linear equation (2.5.2). This solver had to be extended, since nonstandard expressions are present such as the term a (x, y y n)y y n φ dx. A study of the previous algorithm as well as convergence issues are not discussed here, since these would go beyond the scope of this thesis. Example We fix = (0, 1) 2 and denote by 1 to 4 the four sides of the square, starting at the bottom side and moving counterclockwise. We construct the boundary datum as follows. Let c i : (0, 1) i be the parametrization of i, i = 1,..., 4, with c 1 (t) = {t} {0} and the remaining c i are defined analogously, taking into account the aforementioned running direction on. Then we define u(c i (t)) = 1 t 1 /2 log(t), t (0, 0.5], and u(c i(1 t)) = 1, t (0, 0.5), t 1 /2 log(t) i.e. on every boundary side i, u has a peak shape concentrated at both endpoints of i. Figure 2.1 shows u and y h for h = 2 7. Table 2.1 below illustrates the distance Figure 2.1. Data from Example 2.20; u (left frame) and y h with h = 2 7 (right frame). between y h and the reference solution y fine corresponding to the mesh size h ref = 2 10 as well as the convergence speed represented by EOC X (y fine ) for X = L 2 () and X = H 1 (). We point out that h ref is in the range of mesh sizes associated with the finest meshes we have been able to manage with a PC with MATLAB. Apparently,
66 58 2. FINITE ELEMENT APPROXIMATION h y fine y h L 2 () EOC L 2 ()(y fine ) y fine y h H 1 () EOC H 1 ()(y fine ) e e e e e e e e e e e e e e Table 2.1. Convergence behavior of y fine y h X and experimental order of convergence EOC X (y fine ) for X = L 2 () and X = H 1 (). the order of convergence in the L 2 () and H 1 () norms are better than O(h 3 /2 ) and O(h 1 /2 ), respectively, as expected from (2.3.24). However, this is not surprising, since our theoretical results exhibit a worstcase scenario which is hardly to realize numerically. Nevertheless, it is obvious that the convergence rates in the L 2 () and H 1 () norms are worse than O(h 2 ) and O(h) as predicted in Corollary 2.14 in the case of regular boundary data. This indicates that y H 2 (). Example Let be the Lshaped domain = (0, 1) (0, 0.5) (0.5, 1) (0, 1). Analogous to Example 2.20, we denote by 1 to 6 the six boundary sides of, starting at the bottom side and moving counterclockwise. Now define the mappings c i : (0, 1) i, i = 1, 2, and c j : (0, 0.5) j, j = 3,..., 6, by c 1 (t) = {t} {0}, c 2 (t) = {1} {t}, c 3 (t) = {1 t} {1}, c 4 (t) = {0.5} {1 t}, c 5 (t) = {0.5 t} {0.5} and c 6 (t) = {0} {0.5 t}. We set and 1 u(c 1 (t)) = t 1 /2 log(t), t (0, 0.5], and u(c 1(1 t)) = u(c 2 (t)) = 1, t (0, 0.5], t 1 /2 log(t) and 1, t (0, 0.5), t 1 /2 log(t) u(c 1 2(1 t)) =, t (0, 0.5), t 1 /2 log(t) 1 u(c j (t)) = (0.5 t) 1 /2 log(0.5 t) 1, t (0, 0.5), j = 3,..., 6. t 1 /2 log(t) Certainly, on every boundary side i, u has a peak shape concentrated at both endpoints of i. The function u and the numerical solution y h of (2.5.1) for h = 2 7 are shown in Figure 2.2. In Table 2.2 we report on the convergence history for this test example. Again, it can be seen that the numerical computations provide a better convergence
67 2.5. NUMERICAL EXPERIMENTS 59 Figure 2.2. Data from Example 2.21; u (left frame) and y h with h = 2 7 (right frame). h y fine y h L 2 () EOC L 2 ()(y fine ) y fine y h H 1 () EOC H 1 ()(y fine ) e e e e e e e e e e e e e e Table 2.2. Convergence behavior of y fine y h X and experimental order of convergence EOC X (y fine ) for X = L 2 () and X = H 1 (). behavior than our prediction for nonconvex domains; theoretically, we expect to see the orders O(h) and O(h 1 /2 ) in the L 2 () and H 1 () norms, respectively (Eq. (2.3.4)). However, compared to the first example, it is obvious that the nonconvexity of Lshaped domain has indeed an effect on the order of convergence of the solutions in the L 2 () norm, thus confirming our theoretical investigations in Section 2.3. Example We consider the quasilinear equation { div [(1 + (x1 + x 2 ) 2 + y 2 (x)) y(x)] + y + y 3 (x) = 0 in, (2.5.3) (1 + (x 1 + x 2 ) 2 + y 2 (x)) ν y(x) = u(x) on, where and u are first given as in Example 2.20 and next as in Example Figure 2.3 shows the numerical solution y h for h = 2 7 in both cases. Tables 2.3 and 2.4 give a detailed insight into the behavior of y fine y h L 2 () and y fine y h H 1 ()
68 60 2. FINITE ELEMENT APPROXIMATION Figure 2.3. Data from Example 2.22; y h with h = 2 7 when is convex (left frame) and nonconvex (right frame). for h = 2 i, i = 2,..., 8. Now y fine is the solution of the discrete version of (2.5.3) h y fine y h L 2 () EOC L 2 ()(y fine ) y fine y h H 1 () EOC H 1 ()(y fine ) e e e e e e e e e e e e e e Table 2.3. Convergence behavior of y fine y h X and experimental order of convergence EOC X (y fine ) for convex, X = L 2 () and X = H 1 (). corresponding to the mesh with mesh size h ref = 2 9. In order to accelerate the convergence of the Newton method for computing y hi with h i = 2 i, i = 3,..., 8, we have taken as initial iterate y 0 the solution y hi 1, since the latter is close to the solution of the continuous equation (2.5.3). A close look at the Tables 2.3 and 2.4 shows that the convergence behavior in both cases, when is convex or not, is the same as that observed in Example 2.20 and Example 2.21, respectively The discrete solution operator In this section, we establish some important properties of the discrete solution operator u y h (u). Since this mapping is possibly multivalued, our analysis has only local character. However, we are able to prove that for a fixed datum ū L 2 ()
69 2.6. THE DISCRETE SOLUTION OPERATOR 61 h y fine y h L 2 () EOC L 2 ()(y fine ) y fine y h H 1 () EOC H 1 ()(y fine ) e e e e e e e e e e e e e e Table 2.4. Convergence behavior of y fine y h X and experimental order of convergence EOC X (y fine ) for nonconvex, X = L 2 () and X = H 1 (). there exists a L 2 () ball centered at ū such that, for any element u of it, there exists a unique discrete solution y h (u) in a certain W 1,q () ball, q (2, 4), centered at yū. Throughout this section, we suppose that the Assumptions , 1.17 and 1.24(1) hold. Further, we fix ū L 2 () and ȳ := yū H 3 /2 (); see Theorem 1.18 on page 16. Theorem Let q (2, 4) be fixed. There exist h 0 > 0 and constants ρ > 0 and κ ρ > 0, dependent on h 0 and q, such that, for any h < h 0 and any u B L 2 ()(ū, ρ), the discrete quasilinear equation (2.3.1) has a unique solution y h (u) Y h in the closed ball B W 1, q ()(ȳ, κ ρ ). Proof. Let us first assume ρ = 1. A smaller radius will be introduced later. Applying Corollary 2.15, we deduce the existence of h 0 > 0 such that, for any ū u L 2 () ρ and h < h 0, there exists a solution y h (u) Y h of (2.3.1) verifying the estimate y u y h (u) W 1, q () C 1 h 2 / q 1/2 C 1 h 2 / q 1/2 0. Then, due to the Lipschitz continuity of G (Corollary 1.32 on page 27), there holds ȳ y h (u) W 1, q () ȳ y u W 1, q () + y u y h (u) W 1, q () C 2 ū u L 2 () + C 1 h 2 / q 1/2 0. Thus, taking Ĉ = max{c 1, C 2 } and κ ρ = Ĉ ( ) ρ + h 2 / q 1/2 0, there exists at least one element y h (u) B W 1, q ()(ȳ, κ ρ ) for any u B L 2 ()(ū, ρ) and h < h 0. In the rest of the proof, we show the existence of a number ρ such that y h (u) is the unique solution of (2.3.1) in the ball B W 1, q ()(ȳ, κ ρ ) for any u B L 2 ()(ū, ρ). To this end, we will argue by contradiction. Let us assume that there exists a sequence {h k } k=1 of positive real numbers with h k 0 as k, a sequence {u hk } k=1 in
70 62 2. FINITE ELEMENT APPROXIMATION L 2 () with u hk ū L 2 () < 1 and functions k y1 h k (u hk ), yh 2 k (u hk ) B W 1, q ()(ȳ, η k ) with η k = Ĉ ( 1 + ) / q 1/2 k h2 k and y 1 hk (u hk ) yh 2 k (u hk ), such that for any φ hk Y hk, i = 1, 2, { } a(x, y i hk (u hk )) yh i k (u hk ) φ hk + f(x, yh i k (u hk ))φ hk dx = u hk φ hk dσ(x). To simplify the notation we set y i k = y i h k (u hk ). Notice that the existence of y i k with the property y i k ȳ in W 1, q () is a consequence of Corollary Further, define y k = y 2 k y 1 k y 2 k y1 k L 2 q q 2 () We show first that {y k } k=1 is bounded in H 1 (). To this end, we subtract both equations satisfied by yk 2 and yk, 1 divide by yk 2 yk 1 2 q and apply the mean L q 2 () value theorem, to get the existence of measurable functions θ hk and η hk with values in [0, 1] such that { a(x, yk) y 1 k φ hk + a y (x, v h k )y k yk φ 2 hk + f } (2.6.1) y (x, w h k )y k φ hk dx = 0, where v hk := y 1 k + θ hk (y 2 k y 1 k) and w hk := y 1 k + η hk (y 2 k y 1 k). Notice that the measurability of θ hk and η hk can be shown by applying [53, Theorem 1.2 on page 236 and Proposition 1.1 on page 234] to the positive functions and g 1 : [0, 1] R, g 1 (x, t) = a(x, y2 k(x)) a(x, yk(x)) 1 a y (x, y1 k(x) + t(yk(x) 2 yk(x))) 1, g 2 : [0, 1] R, g 2 (x, t) = f(x, y2 k(x)) f(x, yk(x)) 1 f y (x, y1 k(x) + t(yk(x) 2 yk(x))) 1, respectively. Exploiting the uniform boundedness of yk, i v hk and w hk, and the assumptions on a and f, we get from (2.6.1) C 0 y k 2 H 1 () α a y k 2 L 2 () + α f y k 2 L 2 () C y k yk y 2 k dx C y k L 2 q q 2 () y2 k(u hk ) L q () y k H 1 (). Hence, the boundedness of y k H 1 () is an immediate consequence of the boundedness of yk 2 in W 1, q () and the identity y k 2 q = 1. Taking a subsequence, denoted in L q 2 () the same way, there exists ŷ H 1 () such that y k ŷ weakly in H 1 () as k. Due to the compactness of the embedding H 1 () L 2 q q 2 (), the convergence yk.
71 2.6. THE DISCRETE SOLUTION OPERATOR 63 to ŷ is strong in L 2 q q 2 () and, because of yk 2 q = 1, we have ŷ 2 q = 1. L q 2 () L q 2 () Using similar arguments as on page 54, we pass to the limit in (2.6.1) and arrive at (2.6.2) { a(x, ȳ) ŷ φ + a } f (x, ȳ)ŷ ȳ φ + (x, ȳ)ŷφ dx = 0 y y φ H 1 (). According to Theorem 1.25 on page 22, (2.6.2) implies that ŷ = 0 which contradicts the fact that ŷ 2 q = 1. L q 2 () Remark Exploiting the arguments on page 61, it is easy to see that, given h 0 > 0 and q (2, 4), there exist ρ > 0 and κ ρ > 0 such that, for any h < h 0 and any u B L 2 ()(ū, ρ), the equation (2.3.1) has a unique solution y h (u) Y h in the closed ball B W 1,q ()(ȳ, κ ρ ) q [ q, 4). In particular ρ > 0 and κ ρ > 0 are the same for all q [ q, 4). As a result of Theorem 2.23 we may define the discrete counterpart of the solution operator G, namely G h : B L 2 ()(ū, ρ) B W 1, q ()(ȳ, κ ρ ) Y h, G h (u) = y h (u), where q (2, 4) and y h (u) is the unique solution of (2.3.1) in B W 1, q ()(ȳ, κ ρ ). Our next goal is to study the differentiability of G h. Theorem There exists h 1 h 0 such that, for any h < h 1, the mapping G h : B L 2 ()(ū, ρ) B W 1, q ()(ȳ, κ ρ ) Y h, u y h (u), is of class C 2. If we denote z h (v) = G h(u)v, with v L 2 (), then z h (v) is the unique solution of the problem (2.6.3) Find z h (v) Y h such that, for all φ h Y h, { a(x, y h (u)) z h (v) φ h + a y (x, y h(u))z h (v) y h (u) φ h + f } y (x, y h(u))z h (v)φ h dx = vφ h dσ(x). Proof. To achieve the regularity of G h stated in the theorem we will use the implicit function theorem by considering F h : L 2 () Y h Yh, F h (u, y h ), φ h = {a(x, y h ) y h φ h + f(x, y h )φ h } dx uφ h dσ(x). It is obvious that F h is of class C 2 in B L 2 ()(ū, ρ) Y h and F h (u, y h (u)) = 0 for every u B L 2 ()(ū, ρ) and h < h 0. Therefore, it remains to prove that the mapping
72 64 2. FINITE ELEMENT APPROXIMATION ( F h / y h )(u, y h (u)) : Y h Yh Fh (u, y h (u))z h, φ h = y h = { defined by a(x, y h (u)) z h φ h + a y (x, y h(u))z h y h (u) φ h + f y (x, y h(u))z h φ h is an isomorphism. The representation (2.6.3) of G h is obtained by a simple computation. The mapping ( F h / y h )(u, y h (u)) is linear and Y h is finite dimensional, thus it suffices to prove its injectivity or equivalently that the equation ( F h / y h )(u, y h (u))z h = 0 admits only the solution z h = 0. For this purpose, we will follow the approach proposed by Schatz [86]. In order to shorten the notation, we denote y h = y h (u). Introduce the function B h : H 1 () H 1 () R defined by (2.6.4) B h (w, φ) = { a(x, y h ) w φ + a y (x, y h)w y h φ + f } y (x, y h)wφ dx. Since {y h } h<h0 is bounded in W 1,4 () C( ) (Eq. (2.3.38)), we get from the Assumptions 1.2 and 1.24(1) that (2.6.5) B h (w, φ) C w H 1 () φ H 1 () w, φ H 1 (). We divide the remaining part of the proof of z h = 0 in three steps. Step 1: Proof of a Gårding s inequality. We prove that there exist constants C 1 and C 2 with C 1 > 0 such that (2.6.6) B h (w, w) C 1 w 2 H 1 () C 2 w 2 L 2 () w H 1 (). If C 2 0, thanks to (2.6.5)(2.6.6), an application of the LaxMilgram theorem yields that the unique solution z h of ( F h / y h )(u, y h (u))z h = 0 is zero. Let us now study the more general case when C 2 > 0. Using the assumptions on a and f, we have { B h (w, w) = a(x, y h ) w 2 + a y (x, y h)w w y h + f } y (x, y h)w 2 dx min{α a, α f } w 2 H 1 () C w w y h dx (2.6.7) min{α a, α f } w 2 H 1 () C w L 4 () w H 1 () y h W 1,4 (). Further, Lemma 2.3 and Young s inequality with q = q = 2 imply w L 4 () C 3 w 1 /2 L 2 () w 1 /2 H 1 () C 3 ( 1 2ε 2 w L 2 () + ε2 2 w H 1 () ) } dx
73 2.6. THE DISCRETE SOLUTION OPERATOR 65 for any ε > 0. Combining the last inequality with (2.6.7), it follows (2.6.8) B h (w, w) min{α a, α f } w 2 H 1 () ( ) 1 C 4 2ε w 2 L 2 () + ε2 2 w H 1 () w H 1 (). Again, by Young s inequality, there holds w L 2 () w H 1 () 1 4σ w 2 L 2 () + σ w 2 H 1 () σ > 0 which, along with (2.6.8), leads to (2.6.6) by setting C 1 = min{α a, α f } C 4 ( ε2 + σ 2 C 2 = C 4, taking σ = ε 3 and ε small enough such that C 8σε 2 1 > 0. Step 2: z h = 0. By the definition of B h and inequality (2.6.6), we have C 1 z h 2 H 1 () C 2 z h 2 Fh L 2 () B h (z h, z h ) = (u, y h (u))z h, z h = 0, y h hence (2.6.9) z h H 1 () ( C2 C 1 )1/2 z h L 2 (). The last step of the proof consists in verifying the following result (2.6.10) ĥ1 > 0 such that z h L 2 () C 5 h 1 /2 z h H 1 () h < ĥ1, where C 5 > 0 is independent of z h. Once (2.6.10) is shown, taking { } h 1 = min C 1 ĥ 1, h 0, C 2 C5 2 we deduce from (2.6.9) and (2.6.10) that z h = 0 for all h < h 1. Step 3: Proof of (2.6.10). Let us denote y = y u and by ϕ the unique solution in H 3 /2 () of the equation div [a(x, y) ϕ] + a f (x, y) y ϕ + (2.6.11) y y (x, y)ϕ = z h in, a(x, y) ν ϕ = 0 on. Let us also define the function B : H 1 () H 1 () R by { (2.6.12) B(w, φ) = a(x, y) w φ + a } f (x, y)w y φ + (x, y)φw dx. y y, 2ε 2 ),
74 66 2. FINITE ELEMENT APPROXIMATION Passing to the weak formulation of equation (2.6.11), along with (2.3.37), (2.2.1), (2.2.2) and the fact that B h (z h, Π h ϕ) = 0, we find (2.6.13) z h 2 L 2 () = B(z h, ϕ) = B(z h, ϕ Π h ϕ) + B(z h, Π h ϕ) = B(z h, ϕ Π h ϕ) + (B(z h, Π h ϕ) B h (z h, Π h ϕ)) C ( ϕ Π h ϕ H 1 () + y y h H 1 () Π h ϕ W 1,4 ()) zh H 1 () Ch 1 /2 ϕ H 3/2 () z h H 1 (). Then (2.6.10) follows from (2.6.13) and inequality ϕ H 3/2 () C z h L 2 () (Eq. (1.7.3) on page 30). Proposition Let U B L 2 ()(ū, ρ) be bounded. Then there exist C U > 0 and h 2 h 1 such that for any h < h 2, u U and v L 2 (), there holds (2.6.14) G h(u)v W 1,4 () C U v L 2 (). Proof. Let us introduce the following auxiliary problem of finding z H 1 () such that { a(x, y h ) z φ + f } y (x, y h)zφ dx a = vφ dσ(x) y (x, y h)z h (v) y h φ dx =: F (φ) φ H 1 (). From Theorem 2.25 we know that {z h (v)} h<h1 is bounded in W 1, q () C( ) for q (2, 4). This fact, along with the boundedness of {y h } h<h0 in W 1,4 (), implies that F W 1, 4/3 (). Hence, z W 1,4 (), as follows from Theorem 1.9 on page 9. Now we apply a wellknown estimate by Brenner and Scott [11, p. 171] which is also valid in the case of Neumann boundary conditions and leads to the estimate (2.6.14): there exist a constant C > 0 and 0 < h 2 h 1 such that for any h < h 2 z h (v) W 1,4 () C z W 1,4 (). We should remark that the linear elliptic operators in the equations satisfied by z and z h (v) depend on y h and consequently on h. However, we may use the result by Brenner and Scott [11, p. 171] due to the following reason. The previous inequality holds true for y h instead of y h, with h < h 1 arbitrary but fixed and C may depend on the norm of y h in W 1,4 (). Thanks to the boundedness of {y h } h<h1 in W 1,4 (), this inequality remains valid, in particular, when taking h = h Numerical analysis of the adjoint equation Throughout this section, we suppose that the Assumptions , 1.17 and 1.24 hold.
75 2.7. NUMERICAL ANALYSIS OF THE ADJOINT EQUATION 67 Let us denote by ū L 2 () a fixed boundary datum and by ȳ H 3 /2 () the associated solution of (1.1.1). According to Theorem 2.23, there exist h 0 > 0 and, for q (2, 4) fixed, ρ > 0 and κ ρ > 0, such that, for any h < h 0 and any u B L 2 ()(ū, ρ), the equation (2.3.1) has a unique solution y h = y h (u) Y h B W 1, q ()(ȳ, κ ρ ). Notice that {y h } h<h0 is bounded in W 1,4 () because of the boundedness of B L 2 ()(ū, ρ) in L 2 (). Let us also denote by y the function y u = G(u) associated with a fixed element u B L 2 ()(ū, ρ). From Theorem 1.18 on page 16 we know that y H 3 /2 (). Our next step is to carry out the numerical analysis of the adjoint equation div [a(x, y) ϕ] + a f (x, y) y ϕ + (x, y)ϕ = ζ in, (2.7.1) y y a(x, y) ν ϕ = v on, for v L 2 () and ζ L p () with p > 4/3. Using the triangulation T h introduced in Section 2.3, we approximate ϕ by solutions of the discrete version of (2.7.1): Find ϕ h Y h such that for all φ h Y h { (2.7.2) a(x, y h ) ϕ h φ h + a y (x, y h)φ h y h ϕ h + f } y (x, y h)ϕ h φ h dx = ζφ h dx + vφ h dσ(x). Theorem For every h < h 1, with h 1 given in Theorem 2.25, v L 2 () and ζ L p () (p > 4/3), the variational problem (2.7.2) has a unique solution ϕ h Y h. Proof. Since the mapping ( F h / y h )(u, y h ) defined in the proof of Theorem 2.25 is an isomorphism, the same holds true for its adjoint. This leads immediately to the existence and uniqueness of a solution ϕ h Y h of (2.7.2) for any h < h 1. Before establishing error estimates for the approximation of (2.7.1), we prove an auxiliary result concerning a partial discretization of equation (2.7.1). In a first step, we consider a continuous problem of the form (2.7.1) with y h substituted for y. In a second step, we pass into the fully discretized problem (2.7.2). In the sequel, we make use of the following inequality (2.7.3) z L 2 () C ( ε 1/2 z 2 L 2 () + ε 1/2 z 2 L 2 ()) 1/2 z H 1 (), ε (0, 1), cf. Grisvard [60, Theorem ]). Lemma For any h < h 0, v L q () and ζ L p (), with q > 2 and p > 4/3, the equation div [a(x, y h ) ϕ] + a (2.7.4) y (x, y h) y h ϕ + f y (x, y h) ϕ = ζ in, a(x, y h ) ν ϕ = v on,
76 68 2. FINITE ELEMENT APPROXIMATION has a unique solution ϕ H 3 /2 () and it obeys the estimate (2.7.5) ϕ ϕ H 1 () Ch 3 /4 ( v L q () + ζ L p ()), where C > 0 is independent of h. Moreover, if is convex then (2.7.6) ϕ ϕ H 1 () Ch ( v L q () + ζ L p ()). Finally, if is convex, p 2 and u H 1 /2 (), then (2.7.5) is replaced by (2.7.7) ϕ ϕ H 1 () Ch 3 /2 ( v L q () + ζ L p ()). We point out that q > 2 in the statement of the lemma is only necessary for deriving the error estimates; for the regularity result it is enough to take q = 2. Proof. Taking into account the boundedness of {y h } h<h0 in W 1,4 (), the existence, uniqueness and regularity of ϕ is a consequence of Theorem 1.36 on page 32. Let us derive two useful estimates concerning ϕ. In view of ϕ H 3 /2 () W 1,4 () and replacing in (1.7.10) on page 33 y and ϕ by y h and ϕ, respectively, we obtain ϕ L min{p,2} () C ( ζ L p () + ϕ H 3/2 ()). Moreover, we get by a straightforward modification of the proof of (1.7.3) on page 30 that (2.7.8) ϕ H 3/2 () C ( ζ L p () + v L q ()). Let us now prove (2.7.5). Subtracting (2.7.1) and (2.7.4), we have div [a(x, y) (ϕ ϕ)] + a (x, y) y (ϕ ϕ) y (2.7.9) + f y (x, y)(ϕ ϕ) = g in, a(x, y) ν (ϕ ϕ) = g on, where the functions g and g are given by ( a (2.7.10) g (x) = div [(a(x, y) a(x, y h )) ϕ] + y (x, y h) y h a ) (x, y) y ϕ y ( f + y (x, y h) f ) (x, y) ϕ y and g (x) = [a(x, y h ) a(x, y)] ν ϕ, respectively. It is easy to check that g L min{p,2} () and g L q () as a consequence of the W 1,4 () regularity of ϕ, y and y h, and the facts ϕ L min{p,2} () and
77 2.7. NUMERICAL ANALYSIS OF THE ADJOINT EQUATION 69 ν ϕ L q (); see the second equation of (2.7.4). From Theorem 1.36 we know that ϕ ϕ is the unique solution of (2.7.9) and it satisfies (2.7.11) ϕ ϕ H 1 () C ( g H 1 () + g ) H 1/2 () with C > 0 being independent of ϕ or ϕ. Next we derive an estimate for g H 1 (). An integration by parts yields for arbitrary φ H 1 () (2.7.12) div [(a(x, y) a(x, y h )) ϕ] φ dx = (a(x, y) a(x, y h )) ν ϕφ dσ(x) (a(x, y) a(x, y h )) ϕ φ dx. Let us estimate both terms. From the assumptions on a, using (2.3.37), (2.7.3) with ε = h, as well as the embedding H 1 /2 () L s () for all s [1, ), we obtain (a(x, y) a(x, y h )) ν ϕφ dσ(x) C y y h ν ϕφ dσ(x) (2.7.13) Invoking Lemma 2.3 and (2.3.37) again, we have C y y h L 2 () vφ L 2 () ε h h 3 /4 v L q () φ H 1 (). (2.7.14) y y h L 4 () C y y h 1 /2 L 2 () y y h 1 /2 H 1 () Ch3 /4, hence (2.7.15) (a(x, y) a(x, y h )) ϕ φ dx C y y h ϕ φ dx C y y h L 4 () ϕ W 1,4 () φ H 1 () Ch 3 /4 ϕ H 3/2 () φ H 1 (). Thus, from (2.7.12), (2.7.13) and (2.7.15), we deduce the following estimate for the first term of g (2.7.16) div [(a(x, y) a(x, y h )) ϕ] φ dx Ch 3 /4 ( ϕ H 3/2 () + v L q ()) φ H 1 (). Concerning the second term in the definition of g, we write ( a (2.7.17) y (x, y h) y h a ) (x, y) y ϕφ dx = y ( a = y (x, y h) a ) a (x, y) y h ϕφ dx + y y (x, y) (y h y) ϕφ dx.
78 70 2. FINITE ELEMENT APPROXIMATION Now we get, along with (2.7.14), ( a y (x, y h) a ) (x, y) y h ϕφ dx y C (y h y)φ L 2 () y h ϕ L 2 () (2.7.18) C y h y L 4 () ϕ H 3/2 () φ H 1 () Ch 3 /4 ϕ H 3/2 () φ H 1 (). To estimate the second term of (2.7.17) we take into account that [ ] a a y (x, y) ϕ L2 () 2 and div (x, y) ϕ L min{p,2} (). y Hence, we can perform again an integration by parts a (2.7.19) y (x, y) (y a h y) ϕφ dx = y (x, y) ν ϕ(y h y)φ dσ(x) [ ] a a y (x, y)(y h y) ϕ φ dx div (x, y) ϕ (y h y)φ dx. y For the first two terms we proceed as in (2.7.13) and (2.7.15). For the third one we find [ ] a div (x, y) ϕ (y h y)φ dx y [ ] a C div (x, y) ϕ (y y h y)φ L max{2,p } () L min{2,p} () C ( ϕ L min{2,p} () + ϕ H 3/2 ()) yh y L 4 () φ H 1 () (2.7.20) Ch 3 /4 ( ζ L p () + ϕ H 3/2 ()) φ H 1 (), where we used (2.7.14), the facts that p < 4 (because p > 4/3) and H 1 () L s () for any s [1, ). Therefore, the inequalities (2.7.19) and (2.7.20) lead to (2.7.21) a y (x, y) (y h y) ϕφ dx Finally, (2.7.17), (2.7.18) and (2.7.21), imply (2.7.22) Ch 3 /4 ( ζ L p () + v L q () + ϕ H 3/2 ()) φ H 1 (). ( a y (x, y h) y h a ) (x, y) y ϕφ dx y Ch 3 /4 ( ζ L p () + v L q () + ϕ H 3/2 ()) φ H 1 ().
79 2.7. NUMERICAL ANALYSIS OF THE ADJOINT EQUATION 71 The last term on the righthand side of (2.7.10) is easy to estimate: ( f y (x, y h) f ) (x, y) ϕφ dx y C y h y L 2 () ϕ L 4 () φ L 4 () Ch ϕ H 3/2 () φ H 1 (). Let us now estimate g H 1/2 (). Taking ψ H 1 /2 (), we get analogous to (2.7.13) (a(x, y h ) a(x, y)) ν ϕψ dσ(x) y h y L 2 () ν ϕ L q () ψ 2q L (2.7.23) Ch 3 /4 v L q () ψ H 1/2 (). q 2 () From (2.7.11), along with the inequalities (2.7.16) and (2.7.22)(2.7.23), we arrive at ( ) ϕ ϕ H 1 () Ch 3 /4 ζ L p () + v L q () + ϕ H 3/2 () ( ) Ch 3 /4 ζ L p () + v L q () and conclude (2.7.5). It remains to prove (2.7.6) and (2.7.7). To deduce (2.7.6) we repeat the above steps. Some estimates can be improved because we can use (2.3.39) for the error y h y. As a consequence of (2.3.39) and Lemma 2.3, we have y y h L 4 () C y y h 1 /2 L 2 () y y h 1 /2 H 1 () Ch. Further, an application of (2.7.3) with ε = h 2 yields y y h L 2 () C ( h y yh 2 H 1 () + h 1 y y h 2 L 2 ()) 1/2 Ch. Taking into account these inequalities, we get the order O(h) for the expressions in (2.7.16) and (2.7.22)(2.7.23), and finally for ϕ ϕ H 1 (). Let us finish the proof by considering the case of a convex domain, p 2 and a boundary datum u H 1 /2 (). With the aid of (2.3.40), we observe that (2.7.3) applied with ε = h 2 leads this time to the inequality y y h L 2 () C ( h y yh 2 H 1 () + h 1 y y h 2 L 2 ()) 1/2 Ch 3 /2. Further, we have y y h L 4 () C y y h 1 /2 L 2 () y y h 1 /2 H 1 () Ch3 /2. With these inequalities we can improve the previous estimates to conclude (2.7.7). Theorem For any h < h 1, v L q () and ζ L p (), with h 1 given in Theorem 2.25, q > 2 and p > 4/3, we have ( ) (2.7.24) ϕ ϕ h L 2 () + h 1 /4 ϕ ϕ h H 1 () Ch 3 /4 v L q () + ζ L p ()
80 72 2. FINITE ELEMENT APPROXIMATION with C > 0 independent of h. Moreover, if is convex then (2.7.25) ϕ ϕ h L 2 () + h 1 /2 ϕ ϕ h H 1 () Ch ( v L q () + ζ L p ()). Finally, if is convex, p 2 and v, u H 1 /2 (), then (2.7.24) is replaced by (2.7.26) ϕ ϕ h L 2 () + h 1 /2 ϕ ϕ h H 1 () Ch 3 /2 ( v H 1/2 () + ζ L p ()). Proof. First, we show that (2.7.27) ϕ ϕ h L 2 () Ch 1 /2 ϕ ϕ h H 1 () and (2.7.28) ϕ ϕ h H 1 () Ch 1 /2 ϕ H 3/2 (). The first estimate can be deduced in an analogous way as we proved (2.6.10). Let us only comment the differences. Let z H 3 /2 () be the unique solution of the equation [ div a(x, y) z + a ] (x, y)z y + f y y (x, y)z = ϕ ϕ h in, (2.7.29) [ a(x, y) z + a ] (x, y)z y ν = 0 on. y Notice that the H 3 /2 () regularity of z follows directly from Theorem 1.31 on page 26. Making use of the mappings B and B h defined in (2.6.12) and (2.6.4), respectively, (2.3.37) and (2.2.1), and taking into account that B h (Π h z, ϕ ϕ h ) = 0, we obtain ϕ ϕ h 2 L 2 () = B(z, ϕ ϕ h ) (2.7.30) = B(z Π h z, ϕ ϕ h ) + B(Π h z, ϕ ϕ h ) = B(z Π h z, ϕ ϕ h ) + (B(Π h z, ϕ ϕ h ) B h (Π h z, ϕ ϕ h )) C ( z Π h z H 1 () + y y h H 1 () Π h z W 1,4 ()) ϕ ϕh H 1 () Ch 1 /2 z H 3/2 () ϕ ϕ h H 1 (). Hence, (2.7.27) follows from (2.7.30) and the inequality z H 3/2 () C ϕ ϕ h L 2 (); compare inequality (1.6.11) on page 26. To show (2.7.28) let us introduce the function { S h : H 1 () H 1 () R, S h (w, φ) = a(x, y h ) w φ + f } y (x, y h)wφ dx. Utilizing the assumptions on a and f, it is immediate that S h (w, w) C 0 w 2 H 1 () and S h (w, φ) C 1 w H 1 () φ H 1 () w, φ H 1 (). Consequently, C 0 ϕ ϕ h 2 H 1 () S h ( ϕ ϕ h, ϕ ϕ h ) = S h ( ϕ ϕ h, ϕ Π h ϕ) + S h ( ϕ ϕ h, Π h ϕ ϕ h ) =: I 1 + I 2.
81 2.7. NUMERICAL ANALYSIS OF THE ADJOINT EQUATION 73 Exploiting (2.2.1), the estimate for I 1 is obvious: I 1 = S h ( ϕ ϕ h, ϕ Π h ϕ) Ch 1 /2 ϕ H 3/2 () ϕ ϕ h H 1 (). Invoking the equations satisfied by ϕ and ϕ h, we infer I 2 = S h ( ϕ ϕ h, Π h ϕ ϕ h ) a = y (x, y h) (Π h ϕ ϕ h ) y h ( ϕ ϕ h ) dx C (Π h ϕ ϕ h ) y h L 2 () ϕ ϕ h H 1 () C ( Π h ϕ ϕ L 4 () + ϕ ϕ h L 4 ()) ϕ ϕh H 1 (). From the last three inequalities it follows ϕ ϕ h 2 H 1 () C ( h 1 /2 ϕ H 3/2 () + ϕ ϕ h L 4 ()) ϕ ϕh H 1 () which, as already shown on page 43, leads to (2.7.31) ϕ ϕ h H 1 () C ( h 1 /2 ϕ H 3/2 () + ϕ ϕ h L 2 ()). Combining (2.7.31) and (2.7.27), we finally conclude (2.7.28). The estimate for ϕ ϕ h H 1 () as given in (2.7.24) and (2.7.25) follows from (2.7.28), (2.7.8) and Lemma 2.28: ϕ ϕ h H 1 () ϕ ϕ H 1 () + ϕ ϕ h H 1 () ϕ ϕ H 1 () + Ch 1 /2 ϕ H 3/2 () Ch 1 /2 ( v L q () + ζ L p ()). To complete the proof of (2.7.24) and (2.7.25) we have to estimate ϕ ϕ h L 2 (). To this end, we use again the previous lemma, the embedding H 1 () L 2 (), (2.7.27) and the estimate in the H 1 () norm already proved, and get ϕ ϕ h L 2 () ϕ ϕ L 2 () + ϕ ϕ h L 2 () C ( ϕ ϕ H 1 () + h 1 /2 ϕ ϕ h H 1 () C ( ) ϕ ϕ H 1 () + h ϕ H 3/2 () Ch ( ) σ v L q () + ζ L p () with σ = 3/4 if is nonconvex and σ = 1 if is convex. It remains to show (2.7.26). Under the assumptions of the theorem, we know that y = y u H 2 (). Moreover, ϕ H 2 () which can be deduced from Theorem 1.36(3) on page 32, because {y h } h<h1 is bounded in W 1,6 (). The latter boundedness result )
82 74 2. FINITE ELEMENT APPROXIMATION follows from (2.2.2), the inverse inequality (2.2.6), (2.3.40) and (2.3.35), y h Π h y W 1,6 () C h y 2 /3 h Π h y H 1 () C ( yh y h 2 /3 H 1 () + y Π h y H ()) 1 C h h ( ) 1 + y 2 /3 H 2 () <. Furthermore, the solution z of (2.7.29) is in H 2 (), too. This can be shown along the lines of the proof of Theorem 1.33; see pages The higher regularity of y and z allows us to use (2.3.40) and (2.3.35). Therefore, we can improve (2.7.27) to ϕ ϕ h L 2 () Ch ϕ ϕ h H 1 (). On the other hand, thanks to the H 2 () regularity of ϕ, we can replace (2.7.28) by ϕ ϕ h H 1 () Ch ϕ H 2 (). This is easily obtained, by repeating the proof of inequality (2.7.28) and taking into account (2.3.35). Combining the last two inequalities and we get ϕ H 2 () C ( v H 1/2 () + ζ L p ()), ϕ ϕ h L 2 () + h ϕ ϕ h H 1 () Ch 2 ( v H 1/2 () + ζ L p ()). By following the steps above and using (2.7.7), the proof of (2.7.26) is complete. Remark Lemma 2.28 contains estimates for the error ϕ ϕ in the H 1 () norm only. Unfortunately, we are not able to prove a higher order of convergence in the L 2 () norm. For this reason we have used on page 73 error estimates in the H 1 () norm to obtain estimates for ϕ ϕ h L 2 (). Indeed, this procedure delivers very rough estimates in general. The following result will be used in the context of error estimates for control problems associated with the quasilinear equation (1.1.1); see Chapter 4. Corollary For any h < h 1, v L q () and ζ L p (), with h 1 given in Theorem 2.25, q > 2 and p > 4/3, we have (2.7.32) ϕ h W 1,4 () C ( v L q () + ζ L p ()), where C is only dependent on u but not on h, v or ζ. Proof. By a simple modification of the proof of estimate (1.7.3) on page 30, we see that the solution ϕ of (2.7.1) satisfies ϕ H 3/2 () C ( v L q () + ζ L p ()). Combining this result with the inequality ϕ ϕ h W 1,4 () C ( v L q () + ζ L p ()), which can be deduced along the lines of the proof of (2.3.23), we conclude (2.7.32).
83 CHAPTER 3 The optimal control problem 3.1. Introduction In this chapter, we investigate a wide class of optimal boundary control problems governed by quasilinear elliptic equations of the type (1.1.1). Our main goal is to establish first and secondorder conditions for local optimality in the presence of pointwise constraints on the control. For convex problems firstorder necessary optimality conditions are even sufficient for global optimality. In contrast to this, for nonlinear problems higher order conditions such as secondorder sufficient conditions should be employed to verify local optimality. The latter ones are proved to be indispensable for several reasons. First, they play an important role in the stability and numerical analysis of the optimal control problems, in particular in the error analysis for local solutions of the finite element approximation of the control problems. Secondly, the convergence analysis of higher order numerical optimization algorithms such as SQPtype methods rely heavily on secondorder sufficient conditions, see Alt and Malanowski [4], Dontchev et al. [49] or Ito and Kunisch [68]. Likewise, secondorder necessary conditions should also be studied since they serve to measure the gap between them and the sufficient ones. In turn, this gap shows how restrictive the sufficient conditions under consideration are. This explains why we are concerned in formulating sufficient secondorder conditions which are the closest to the associated necessary ones. There are two common techniques to verify that certain secondorder conditions are sufficient for local optimality. The first way is to apply some abstract methods for optimization problems in function spaces, see Casas and Tröltzsch [35], while the other method uses Pontryagin s principle, cf. Casas and Mateos [25]. In [30], Casas, Mateos and Tröltzsch, have shown that both methods are equivalent. Meanwhile, there exists a very extensive literature devoted to secondorder optimality conditions for control problems governed by partial differential equations. We mention only the textbook by Tröltzsch [91] for an overview, Goldberg and Tröltzsch [56, 57, 58] for boundary control of parabolic equations, Casas, Tröltzsch and Unger [39, 40], as well as Casas and Tröltzsch [34], for elliptic boundary control problems with nonlinear boundary conditions. The list of contributions concerning the Pontryagin s principle is very large. For elliptic problems this principle was investigated by Bonnans and Casas [9] and Casas 75
84 76 3. THE OPTIMAL CONTROL PROBLEM [13], while the parabolic case was studied by Casas [15], Casas et al. [32] and Raymond and Zidani [83]. In the context of quasilinear equations with nonlinearity of gradient type, Pontryagin s principle was considered by Casas [14] and Casas and Yong [41]. There is some recent progress in the case of optimal control problems governed by quasilinear equations. The first step towards a corresponding analysis was recently made by Casas and Tröltzsch in [36], where first and secondorder optimality conditions as well as a Pontryagin s principle for the distributed optimal control of quasilinear elliptic equations are discussed. Other publications concerning quasilinear equations, in which the leading coefficient of the differential operator depends on the gradient of the solution, have already been commented in Section 1.1. The plan of this chapter is as follows. In the first section, the control problem is formulated and the existence of an optimal solution is shown. Next firstorder necessary optimality conditions are derived. These conditions lead to a useful characterization for optimal controls, which in turn allows us to deduce a corresponding regularity result; see Theorem For the derivation of the secondorder optimality conditions a Pontryagin s principle is proved in Section 3.4. Finally, in Section 3.5 we establish necessary and sufficient secondorder optimality conditions Problem formulation and main assumptions We associate with the quasilinear equation (1.1.1) the following optimal control problem min J(u) = L(x, y u (x)) dx + l(x, y u (x), u(x)) dσ(x), (P) subject to u U ad := {v L () u a (x) v(x) u b (x) a.e. x }, (y u, u) satisfying the equation (1.1.1), where L : R R and l : (R R) R are Carathéodory functions and u a, u b L (), with u a u b a.e. on. In the context of optimal control, the PDE (1.1.1) is denoted as state equation, u is called control function and y u is the associated state, J the cost function or objective function and U ad the set of admissible controls. Example 3.1. In control theory, a frequent example for the choice of the functional is that of tracking type (see, for instance, Ito and Kunisch [68]) J(u) = 1 (y u (x) y d (x)) 2 dx + λ (u(x) u d (x)) 2 dσ(x) 2 2 with given functions y d L 2 (), u d L 2 () and λ 0. The functions y d and u d denote the desired state and control of the problem, respectively. If λ > 0 then
85 3.2. PROBLEM FORMULATION AND MAIN ASSUMPTIONS 77 it is wellknown that the socalled Tikhonov term λ 2 regularizing effect on the optimal control. (u(x) u d(x)) 2 dσ(x) has a Here we do not consider the case when l does not depend on u. In such a situation optimal controls are often of bangbang type. Definition 3.2. A control ū U ad is said to be (globally) optimal with optimal state ȳ = yū if J(ū) J(u) u U ad. The control ū is called locally optimal in the sense of L () if there exists ε > 0 such that the previous inequality holds for all u U ad with ū u L () < ε. The next theorem concerns the existence of a global solution for problem (P). Although the proof of this theorem is standard (compare also Casas et al. [23] or Casas and Mateos [26]), we will sketch it here for convenience of the reader. Theorem 3.3. Suppose that the Assumptions hold true, a : R R is continuous and, for every (x, y) R, l(x, y, ) : R R is convex. Assume also that, for any M > 0, there exist functions ψ L,M L 1 () and ψ l,m L 1 () such that L(x, y) ψ L,M (x) and l(s, y, u) ψ l,m (s) for a.a. x, s and y, u M. Then (P) admits at least one optimal solution ū. Proof. Let {u k } k=1 U ad be a minimizing sequence for (P), i.e. J(u k ) inf(p) when k. Since {u k } k=1 is bounded in L (), it is also bounded in L r /2 () for any r > 2 satisfying the condition (1.3.13) on page 9. Therefore, we can take a subsequence, denoted in the same way, which converges weakly in L r /2 () to some ū. Moreover, ū is a admissible control for (P), because the set U ad is closed and convex in L r /2 (), hence it is weakly closed. Further, Theorem 1.9 on page 9 implies that {y uk } k=1 is bounded in W 1,r (). Again, we can extract a subsequence, denoted again in the same way, such that y uk ỹ weakly in W 1,r () and strongly in C( ), due to the compactness of the embedding W 1,r () C( ). It suffices to show that yū = ỹ and that ū is optimal for (P). The solution y uk of the state equation corresponding to u k satisfies (3.2.1) {a(x, y uk ) y uk φ + f(x, y uk )φ} dx = u k φ dσ(x) φ H 1 (). Thanks to the Assumptions and the previous convergences established, we can pass to the limit in (3.2.1) and obtain a(x, ỹ) ỹ φ dx + f(x, ỹ)φ dx = ūφ dσ(x) φ H 1 (). The uniqueness of the solution of the previous equation implies yū = ỹ. Finally, we can prove that J(ū) = inf(p) by applying Mazur s theorem (see, for instance,
86 78 3. THE OPTIMAL CONTROL PROBLEM Ekeland and Temam [53]). Since this proof is very similar to that of Casas and Mateos [26, Theorem 8], we omit it here. Remark 3.4. An important property of the admissible set U ad is that it is closed and convex. In the event that U ad is not bounded, for instance in the unconstrained case U ad = L 2 (), some coercivity assumption on J is necessary to ensure the existence of a solution for (P). Remark 3.5. The convexity of l with respect to the control variable plays an essential role in the proof of Theorem 3.3. In the absence of the convexity, some compactness arguments are necessary to deduce the existence of a minimum. Otherwise, this existence is not guaranteed in general. In the sequel, we will consider locally optimal solutions, since they are very interesting from the numerical point of view; optimization algorithms mostly provide local minima. Moreover, from the theoretical point of view there are no criteria to distinguish local and global minima except their definitions. Throughout this chapter, we suppose that a : R R is continuous and the Assumptions and 1.24(1) are satisfied Firstorder optimality conditions The goal of this section is to derive firstorder necessary optimality conditions. These optimality conditions satisfied by ū U ad can be obtained from the standard variational inequality (3.3.1) J (ū)(u ū) 0 u U ad. To proceed in this way the differentiability of J is needed. Since we also aim to discuss secondorder optimality conditions, we shall impose the following assumption. Assumption 3.6. Let r > 2 satisfy (1.3.13), β 2r r + 2 and γ r 2. The functions L : R R and l : (R R) R are of class C 2 with respect to the second variable and to the last two variables, respectively. For any M > 0 there exist constants C L,M, C l,m > 0, and functions ψ,m L β (), ψ 1,,M L γ ()
87 3.3. FIRSTORDER OPTIMALITY CONDITIONS 79 and ψ 2,,M L 2 (), such that L (x, y) y ψ 2 L,M(x), (x, y) y2 C L,M, 2 L y (x, y 2) 2 L 2 y (x, y 1) 2 C L,M y 2 y 1, l y (x, y, u) ψ 1,,M(x l ), u (x, y, u) ψ 2,,M(x ), D 2 (y,u) l(x, y, u) R C 2 2 l,m, D(y,u)l(x 2, y 2, u 2 ) D(y,u)l(x 2, y 1, u 1 ) R C 2 2 l,m ( y 2 y 1 + u 2 u 1 ), for a.a. x, x and y, y i, u, u i M, i = 1, 2, where D(y,u) 2 l denotes the second derivative of l w.r.t. (y, u), i.e. the associated Hessian matrix, and R 2 2 is any matricial norm. Theorem 3.7. The functional J : L () R is of class C 2 u, v, v 1, v 2 L (), we have ( ) l (3.3.2) J (u)v = u (x, y u, u) + ϕ u v dσ(x) and for every and { J 2 l (u)v 1 v 2 = y (x, y u, u)z 2 v1 z v2 + 2 l y u (x, y u, u)(z v1 v 2 + z v2 v 1 ) } ( + 2 l 2 u (x, y L u, u)v 2 1 v 2 dσ(x) + y (x, y 2 ) f u) ϕ 2 u y (x, y u) z 2 v1 z v2 dx ( 2 a ϕ u y (x, y u)z 2 v1 z v2 y u + a ) (3.3.3) y (x, y u) (z v1 z v2 + z v2 z v1 ) dx, where ϕ u W 1,r () is the unique solution of the adjoint equation div[a(x, y u ) ϕ u ]+ a y (x, y u) y u ϕ u + f y (x, y u)ϕ u = L y (x, y u) in, (3.3.4) a(x, y u ) ν ϕ u = l y (x, y u, u) on, and z vi = G (u)v i is the solution of (1.6.9) with v = v i, i = 1, 2. If, in addition, Assumption 1.17 holds, β > 4/3 and γ 2 (see Assumption 3.6), then ϕ u H 3 /2 (). Proof. The first and secondorder derivatives of J can be obtained from Theorem 1.30 on page 25 and the chain rule. By taking into account that y u W 1,r () (Theorem 1.9 on page 9), the existence, uniqueness and W 1,r () regularity of ϕ u follow from Theorem 1.36(1) on page 32. The last statement of the theorem is a consequence of Theorem 1.36(2) and the fact that y H 3 /2 (); see Theorem 1.18 on page 16.
88 80 3. THE OPTIMAL CONTROL PROBLEM The function ϕ u is called the adjoint state associated with u. As one can see from the previous theorem, it allows us to get a simple expression of the derivatives of J. Remark 3.8. From the expressions (3.3.2)(3.3.3) for J (u) and J (u) it is easy to check that the functionals J (u) and J (u) can be extended from L () to L 2 (). Indeed, it is enough to remark that, thanks to the W 1,r () regularity of z vi, the integrals in (3.3.3) are well defined for every v i L 2 (), i = 1, 2. Moreover, these integrals are continuous w.r.t. the topology of L 2 () because of the continuous dependence of z vi on v i ; notice that the inverse operator of y F (y u, u) given on page 25 is an isomorphism. The firstorder necessary optimality conditions stated in the next theorem follow from the variational inequality (3.3.1), along with the expression of the derivative of J given in (3.3.2) and (3.3.4). Theorem 3.9. Assume that ū is a local minimum of (P) and ȳ the associated state. Then there exists ϕ W 1,r () such that div[a(x, ȳ) ϕ] + a f L (x, ȳ) ȳ ϕ + (x, ȳ) ϕ = (x, ȳ) in, y y y (3.3.5) a(x, ȳ) ν ϕ = l (x, ȳ, ū) on, y (3.3.6) ( ) l (x, ȳ(x), ū(x)) + ϕ(x) (u(x) ū(x)) dσ(x) 0 u U ad. u If we define the Riesz representation of J by l (3.3.7) d(x) = (x, ȳ(x), ū(x)) + ϕ(x) u for a.a. x then we get from the variational inequality (3.3.6) that = 0 if u a (x) < ū(x) < u b (x), (3.3.8) d(x) = 0 if ū(x) = u b (x), 0 if ū(x) = u a (x). We finish this section by proving a regularity result for optimal solutions of (P) which is deduced from the firstorder necessary conditions. For a slightly more general result we refer to [20]. Theorem Suppose that (3.3.9) Λ l > 0 such that 2 l u 2 (x, y, u) Λ l for a.a. x and (y, u) R 2.
89 3.3. FIRSTORDER OPTIMALITY CONDITIONS 81 Let us also assume that, for every M > 0, there exists C l,m > 0 such that l (3.3.10) u (x 2, y, u) l u (x 1, y, u) + l y (x 2, y, u) l y (x 1, y, u) C l,m x 2 x 1 x 1, x 2 and y, u M. Then for every x the equation (3.3.11) l (x, ȳ(x), t) + ϕ(x) = 0 u has a unique solution t =: s(x). The function s : R is related to ū by the formula (3.3.12) ū(x) = Proj [ua(x),u b (x)] { s(x)} = max {min {u b (x), s(x)}, u a (x)}. Moreover, under the Assumption 1.17, the following regularity results are valid: (1) If β > 4/3 and γ 2 (see Assumption 3.6) then s H 1 (). Moreover, if u a, u b C 0, 1/2 () (respectively H 1 ()), then ū C 0, 1/2 () (respectively H 1 ()). (2) If is convex, 2 < min{p, β, γ} (see Assumption 3.6) and u a, u b C 0,1 (), then s, ū C 0,1 (), and ȳ, ϕ W 2,ϱ (), with some ϱ > 2. Proof. Let us recall that ȳ, ϕ W 1,r () C( ). We fix x and consider the real function g : R R defined by g(t) = ϕ(x) + l (x, ȳ(x), t). u Then g is a C 1 function with g (t) Λ l > 0. Therefore, g is strictly increasing and lim g(t) = and lim g(t) = +. t t + Hence, there exists a unique element t R satisfying g( t) = 0, i.e. s is well defined. According to the definition of d in (3.3.7) and using (3.3.8) as well as the strict monotonicity of ( l/ u) with respect to the third variable, we obtain if d(x) = 0 then ū(x) = s(x), if d(x) < 0 then u b (x) = ū(x) < s(x), if d(x) > 0 then u a (x) = ū(x) > s(x), which implies (3.3.12). Proof of (1). First we observe that ȳ, ϕ H 3 /2 () C( ). The continuity of s at every point x follows from the continuity of the functions ȳ, ϕ and ( l/ u), by
90 82 3. THE OPTIMAL CONTROL PROBLEM using the estimate (3.3.13) s(x) s(x ) 1 l Λ l u (x, ȳ(x ), s(x)) l u (x, ȳ(x ), s(x )) 1 ( ϕ(x l ) ϕ(x) + Λ l u (x, ȳ(x ), s(x)) l ) (x, ȳ(x), s(x)) u for x. Next we show that s H 1 () C 0, 1/2 (). Remark that is a 1dimensional manifold, hence a function v is in H 1 () if and only if v is continuous at every vertex x j of, j = 0,..., N() and x 0 = x N(), and on every edge e of there holds v e H 1 (e). Therefore, it is enough to show that s e H 1 (e) on every edge e of. To this aim, we will prove that s is absolutely continuous on e. Then it is known that s is differentiable a.e. on e and that s coincides with the weak derivative of s. Let us take a, b e, arbitrarily. From (3.3.9)(3.3.11) we get s(b) s(a) 1 l l (a, ȳ(a), s(b)) (a, ȳ(a), s(a)) Λ l u u C Λ l ( b a + ϕ(b) ϕ(a) + ȳ(b) ȳ(a) ). Hence, the absolute continuity of s follows from the absolute continuity of the restriction of ȳ and ϕ to. Here we have used the fact that ȳ, ϕ H 3 /2 (), thus by Proposition 1.14 on page 15, ȳ, ϕ H 1 (), and every function belonging to H 1 (I) on a given interval I R is absolutely continuous; see, for instance, Rudin [85]. Therefore, s is differentiable a.e. on and by differentiating (3.3.11) with s(x) substituted for t, we find for a.a. x that s (x) 1 l 2 (x, ȳ(x), s(x)) + l2 Λ l x u y u (x, ȳ(x), s(x))ȳ (x) + ϕ (x). Since the tangential derivatives ȳ and ϕ belong to L 2 (), we deduce from the previous inequality that s L 2 (), consequently s H 1 (). The rest of the statement (1) follows immediately from the identity (3.3.12) and the regularity of s, provided that u a and u b belong to C 0, 1/2 () (respectively H 1 ()). Proof of (2). Let us take ϱ = min{r 0, r, p, β, γ, 4} > 2, where r 0 is introduced in Theorem 1.21 on page 18 and r is given in Assumption 3.6. Now let us prove that s, ū W 1 1/ϱ,ϱ (). The traces of ȳ and ϕ belong to W 1 1/ϱ,ϱ (). Taking the intrinsic norm in W 1 1/ϱ,ϱ (), the W 1 1/ϱ,ϱ () regularity of ū and s follows from (3.3.12) (3.3.13). Indeed, (3.3.13) implies that s(x) s(x ) ϱ ( C 1 + ϕ(x) ϕ(x ) x x ϱ x x + ȳ(x) ) ϱ ȳ(x ) =: z(x, x ) x x
91 and 3.3. FIRSTORDER OPTIMALITY CONDITIONS 83 z(x, x ) dσ(x)dσ(x ) <, hence s W 1 1/ϱ,ϱ (). On the other hand, the projection Proj [ua( ),u b ( )] is Lipschitz with Lipschitz constant 1, therefore (3.3.14) ū(x) ū(x ) = Proj [ua(x),u b (x)]{ s(x)} Proj [ua(x ),u b (x )]{ s(x )} Proj [ua(x),u b (x)]{ s(x)} Proj [ua(x),u b (x)]{ s(x )} + Proj [ua(x),u b (x)]{ s(x )} Proj [ua(x ),u b (x )]{ s(x )} s(x) s(x ) + u a (x) u a (x ) + u b (x) u b (x ). Consequently, ū(x) ū(x ) ϱ ( s(x) s(x ) + u a(x) u a (x ) x x ϱ x x x x ( s(x) s(x ) ϱ ) + L x x a + L b, + u b(x) u b (x ) ϱ ) x x where L a and L b are the Lipschitz constants of u a and u b, respectively. This leads to the W 1 1/ϱ,ϱ () regularity of ū. Since is assumed convex, Theorem 1.21 yields ȳ W 2,ϱ () C 0,1 () (Eq. (1.5.7) on page 18). The function ϕ has also the same regularity. Next we show that ( l/ y)(, ȳ, ū) W 1 1/ϱ,ϱ () which implies, along with ( L/ y)(, ȳ) L ϱ () and Theorem 1.36(3) on page 32, the W 2,ϱ () regularity of ϕ. From (3.3.10) and Assumption 3.6 we get ( l/ y)(, ȳ, ū) L ϱ () and l l (x, ȳ(x), ū(x)) y y (x, ȳ(x ), ū(x )) C ( x x + ȳ(x) ȳ(x ) + ū(x) ū(x ) ), for a.a. x, x. As above, it follows l l ϱ (x, ȳ(x), ū(x)) y y (x, ȳ(x ), ū(x )) x x ϱ ( C ϱ 1 + ȳ(x) ȳ(x ) ϱ + ū(x) ū(x ) ϱ ), x x ϱ x x ϱ therefore ( l/ y)(, ȳ, ū) W 1 1/ϱ,ϱ (). Since 2 < ϱ, we can use again the inclusion W 2,ϱ () C 0,1 ( ) and the inequalities (3.3.13)(3.3.14) to deduce the Lipschitz regularity of s and ū. Remark that inequality (3.3.9) implies the strict convexity of l with respect to the third variable. Remark The previous theorem is also valid for nonconvex and nonpolygonal domains, assuming the C 1,1 regularity of ; see Grisvard [60].
92 84 3. THE OPTIMAL CONTROL PROBLEM 3.4. Pontryagin s principle In this section, we derive the Pontryagin s principle satisfied by a local solution of (P). This principle is needed for the secondorder analysis which we will carry out in Section 3.5. In contrast to the firstorder necessary optimality conditions of integral form (Eq. (3.3.6)), Pontryagin s principle does not require neither the convexity nor the differentiability of the cost function J w.r.t. the controls. Since equation (1.1.1) is not monotone, we have to adapt the known results for monotone equations to our situation. To overcome the difficulties arising from the lack of monotonicity we shall rely on the following assumption. Assumption The functions L : R R and l : (R R) R are of class C 1 with respect to the second variable, 2 < r satisfies (1.3.13) and, for any M > 0, there exist functions ψ,m L 2r r+2 () and ψ,m L r /2 () such that L (x, y) y ψ l,m(x) and (s, y, u) y ψ,m(s) hold for a.a. x and all s, y, u M. Let us introduce the Hamiltonian H associated with the control problem (P): H : (R R R) R, H(x, y, u, ϕ) = l(x, y, u) + ϕu. Pontryagin s principle says that a local solution of (P) minimizes the Hamiltonian at almost every point x, hence it is a stronger condition than the variational inequality (3.3.6). Theorem Let ū be a local solution of (P) with associated state ȳ and suppose that Assumption 3.12 holds. Then there exists ϕ W 1,r () satisfying the adjoint equation (3.3.5) and the minimum condition (3.4.1) H(x, ȳ(x), ū(x), ϕ(x)) = min H(x, ȳ(x), s, ϕ(x)) for a.a. x, s [u a ε (x),u b ε (x)] where u a ε (x) := max{u a (x), ū(x) ε}, u b ε (x) := min{u b (x), ū(x) + ε}, and ε > 0 is the radius of the ball in L () where J achieves the (local) minimum value at ū among all admissible controls. Remark If l is convex with respect to the control variable then (3.4.1) follows immediately from the variational inequality (3.3.6). To prove the preceding theorem first the sensitivity of the state with respect to certain pointwise perturbations of the control is studied. The following auxiliary results are crucial to accomplish these perturbations.
93 3.4. PONTRYAGIN S PRINCIPLE 85 Lemma ([14, Proposition 2]) For ρ (0, 1) there exists a sequence of σ measurable sets {E k } k=1, with E k and σ(e k ) = ρσ(), such that (1/ρ)χ Ek 1 weakly in L () when k. Lemma Under the assumptions of Theorem 3.13, for any u L () there exists a number ρ (0, 1) and σmeasurable sets E ρ, with σ(e ρ ) = ρσ() for all ρ (0, ρ), having the following properties: If we define ū(x) if x \ E ρ, u ρ (x) = u(x) if x E ρ, then (3.4.2) (3.4.3) 1 y ρ = ȳ + ρz + r ρ with lim ρ 0 ρ r ρ W 1,r () = 0, J(u ρ ) = J(ū) + ρj 0 + rρ 0 1 with lim ρ 0 ρ r0 ρ = 0, hold true, where y ρ is the state associated with u ρ, z is the unique element of W 1,r () satisfying the linearized equation [ div a(x, ȳ) z + a ] (x, ȳ)z ȳ + f (x, ȳ)z = 0 in, y y (3.4.4) [ a(x, ȳ) z + a ] (x, ȳ)z ȳ ν = u ū on, y and J 0 := { } L l (x, ȳ)z dx + (x, ȳ, ū)z + l(x, ȳ, u) l(x, ȳ, ū) dσ(x). y y Proof. Since the proof is similar to that of Proposition 4.3 in Casas and Tröltzsch [36], we only comment upon the main differences. We define g L 1 () by g(x) = l(x, ȳ(x), u(x)) l(x, ȳ(x), ū(x)). Given ρ (0, 1), we take a sequence {E k } k=1 as in Lemma Since L () is compactly embedded in W 1/r,r (), we have (1/ρ)χ Ek 1 strongly in W 1/r,r () when k. Hence, there exists k ρ N such that ( (3.4.5) 1 1 ) ρ χ E k (x) g(x) dσ(x) + (1 1 ρ χ E k )(u ū) < ρ k k ρ. W 1/r,r () Inequality (3.4.5) is the analog of [36, Eq. (4.7)]. The same argumentation as in the proof of [36, Proposition 4.3] yields (3.4.2). In order to prove (3.4.3), we introduce z ρ := (y ρ ȳ)/ρ, L ρ (x) := 1 0 L y (x, ȳ(x) + θ(y ρ(x) ȳ(x))) dθ
94 86 3. THE OPTIMAL CONTROL PROBLEM and 1 l l ρ (x) := 0 y (x, ȳ(x) + θ(y ρ(x) ȳ(x)), u ρ (x)) dθ. Now recalling the definition of g and using (3.4.2) and (3.4.5), we have J(u ρ ) J(ū) = ρ L(x, y ρ (x)) L(x, ȳ(x)) l(x, y ρ (x), u ρ (x)) l(x, ȳ(x), ū(x)) = dx + dσ(x) ρ ρ L(x, y ρ (x)) L(x, ȳ(x)) l(x, y ρ (x), u ρ (x)) l(x, ȳ(x), u ρ (x)) = dx + dσ(x) ρ ρ l(x, ȳ(x), u ρ (x)) l(x, ȳ(x), ū(x)) + dσ(x) ρ { = L ρ (x)z ρ (x) dx + l ρ (x)z ρ (x) + 1 } ρ χ E ρ (x)g(x) dσ(x) L (x, ȳ(x))z(x) dx + y as ρ 0 which implies (3.4.3). { } l (x, ȳ(x), ū(x))z(x) + g(x) dσ(x) = J 0 y Proof of Theorem Since ū is a local solution of problem (P), there exists ε > 0 such that J achieves the minimum at ū among all admissible controls of B L ()(ū, ε). Let u be one of these controls, i.e. u B L ()(ū, ε) U ad. By virtue of Lemma 3.16, we consider sets E ρ, ρ > 0, such that (3.4.2) and (3.4.3) hold. Then u ρ B L ()(ū, ε) and (3.4.3) leads to 0 lim ρ 0 J(u ρ ) J(ū) ρ = J 0. By using the definition of J 0, the variational formulation of (3.4.4) and the adjoint state equation (3.3.5), we get from the previous inequality { } L l 0 (x, ȳ)z dx + (x, ȳ, ū)z + l(x, ȳ, u) l(x, ȳ, ū) dσ(x) y y { = a(x, ȳ) ϕ z + a } f (x, ȳ)z ϕ ȳ + (x, ȳ) ϕz dx y y + {l(x, ȳ, u) l(x, ȳ, ū)} dσ(x) = { ϕ(x) (u(x) ū(x)) + l(x, ȳ, u) l(x, ȳ, ū)} dσ(x) (3.4.6) = {H(x, ȳ(x), u(x), ϕ(x)) H(x, ȳ(x), ū(x), ϕ(x))} dσ(x).
95 3.4. PONTRYAGIN S PRINCIPLE 87 Since u B L ()(ū, ε) U ad is arbitrary and taking into account the definitions of u a ε and u b ε given in the statement of Theorem 3.13, we deduce from (3.4.6) (3.4.7) H(x, ȳ(x), ū(x), ϕ(x)) dσ(x) = min H(x, ȳ(x), u(x), ϕ(x)) dσ(x). u a ε u u b ε It remains to prove that (3.4.7) implies (3.4.1). To this end, we follow Casas and Tröltzsch [36, 4] and Casas [14]. Knowing that is a Lipschitz manifold, we consider a finite collection of σmeasurable sets { k } d k=1 and functions {ψ k } d k=1 with the following properties: (1) d k=1 k =, i j = if i j, and σ() = d k=1 σ( k ). (2) The functions ψ k : (0, 1) R are Lipschitz and for some coordinate system (x k, x k,2 ) = (x k,1, x k,2 ) in R 2 we have that k = {(x k, ψ k (x k)) x k (0, 1)} and, for every set E = {(x k, ψ k (x k)) x k F } with F (0, 1) Lebesgue measurable, there holds σ(e) = F 1 + ψ k (x k ) 2 dx k. Now let the sequence {q j } j=1 exhaust the rational numbers contained in [0, 1]. For every j we set u j (x) = q j u a ε (x) + (1 q j )u b ε (x), x. Then every function u j belongs to L () and u a ε (x) u j (x) u b ε (x) for every x. Next we consider the functions F 0, F j : R given by F 0 (x) = H(x, ȳ(x), ū(x), ϕ(x)) and F j (x) = H(x, ȳ(x), u j (x), ϕ(x)), j N. According to [14, Lemma 3], associated with these integrable functions, there exist σ measurable sets S 0 and {S j } j=1 of Lebesgue regular points which satisfy S i d k=1 k, σ(s i ) = σ() for i = 0, 1,..., and (3.4.8) lim ε 0 1 σ( ε (x 0 )) ε(x 0 ) F i (x) dσ(x) = F i (x 0 ) x 0 S i, where, given x 0 = (x 0k, ψ k (x 0k)) k, 1 k d, and ε > 0 small enough, we set ε (x 0 ) = {(x k, ψ k (x k)) x k (x 0k ε, x 0k + ε) (0, 1)}. Setting S = i=0 S i, we have σ(s) = σ() and (3.4.8) for every x 0 S. Taking x 0 S and ε > 0, we define ū(x) if x / ε (x 0 ), u j,ε (x) = u j (x) otherwise, j N. Then from (3.4.7) and the definition of u j,ε we deduce H(x, ȳ(x), ū(x), ϕ(x)) dσ(x) H(x, ȳ(x), u j,ε (x), ϕ(x)) dσ(x),
96 88 3. THE OPTIMAL CONTROL PROBLEM hence 1 σ( ε (x 0 )) ε(x 0 ) H(x, ȳ(x), ū(x), ϕ(x)) dσ(x) 1 H(x, ȳ(x), u j,ε (x), ϕ(x)) dσ(x). σ( ε (x 0 )) ε(x 0 ) Passing to the limit in the last inequality when ε 0, we get, along with (3.4.8), H(x 0, ȳ(x 0 ), ū(x 0 ), ϕ(x 0 )) H(x 0, ȳ(x 0 ), u j (x 0 ), ϕ(x 0 )). Since the function s H(x 0, ȳ(x 0 ), s, ϕ(x 0 )) is continuous (l is a Carathéodory function; see page 76) and {u j (x 0 )} j=1 is dense in [u a ε (x 0 ), u b ε (x 0 )], we infer H(x 0, ȳ(x 0 ), ū(x 0 ), ϕ(x 0 )) H(x 0, ȳ(x 0 ), s, ϕ(x 0 )) s [u a ε (x 0 ), u b ε (x 0 )]. Finally, (3.4.1) follows from the previous inequality and the fact that x 0 is an arbitrary point of S Secondorder optimality conditions In this section, we prove necessary and sufficient secondorder optimality conditions for the problem (P). As pointed out at the beginning of this chapter, secondorder optimality conditions are very important to analyze the convergence properties of numerical optimization algorithms which are used to solve the control problem. On the other hand, they are also a key tool to carry out error estimates for local solutions of the finite element approximated optimal control problem; see Chapter 4. Throughout this section, we suppose that Assumption 3.6 holds which implies Assumption 3.12, therefore we can apply Pontryagin s principle; see Theorem If ū is an admissible control for problem (P), with associated state ȳ and adjoint state ϕ satisfying (3.3.5) and (3.3.6), then the socalled cone of critical directions Cū is given by Cū = 0 if ū(x) = u a (x) v L () v(x) 0 if ū(x) = u b (x) = 0 if d(x) 0 for x, where d is defined in (3.3.7). In the previous section, we introduced the Hamiltonian H associated with our problem (P) which obviously satisfies H u (x, ȳ(x), ū(x), ϕ(x)) = d(x). In the sequel, we will use the shorter notations H u (x) = H u (x, ȳ(x), ū(x), ϕ(x)) and Huu (x) = 2 H (x, ȳ(x), ū(x), ϕ(x)). u2 The following theorem deals with the necessary secondorder optimality conditions.
97 3.5. SECONDORDER OPTIMALITY CONDITIONS 89 Theorem Let ū be a local optimal solution of (P). Then the following inequalities hold: J (ū)v 2 0 v Cū, (3.5.1) H uu (x) 0 for a.a. x such that H u (x) = 0. Proof. To prove the first inequality of (3.5.1) we will argue as in the proof of Theorem 5.1 in Casas and Tröltzsch [36]. Given v Cū arbitrary and ε (0, ε), with ε chosen as in Theorem 3.13, we define 0 if u a (x) < ū(x) < u a (x) + ε or u b (x) ε < ū(x) < u b (x), v ε (x) = max { 1 {+ ε, min 1 }} ε, v(x) otherwise. Obviously, v ε Cū and v ε v in L 2 () when ε 0. Moreover, u a (x) ū(x) + tv ε (x) u b (x) for a.a. x and 0 < t < ε 2. Hence, as a consequence of the local optimality of ū, we have for g ε : [0, ε 2 ] R, g ε (t) := J(ū + tv ε ), that g ε (0) = min t [0,ε 2 ] g ε(t). Thanks to our assumptions, it is clear that g ε is of class C 2. From the fact that v ε Cū we deduce that g ε(0) = J (ū)v ε = d(x)v ε (x) dσ(x) = 0. Since the first derivative of g ε is zero, the following secondorder necessary optimality condition must hold: (3.5.2) 0 g ε (0) = J (ū)vε 2 { 2 } l = y (x, ȳ, 2 ū)z2 v ε l y u (x, ȳ, ū)z v ε v ε + 2 l u (x, ȳ, 2 ū)v2 ε dσ(x) {( 2 ) L + y (x, ȳ) f 2 ϕ 2 (x, ȳ) z 2 y2 v ε ( 2 a ϕ y (x, 2 ȳ)z2 v ε ȳ + 2 a )} y (x, ȳ)z v ε z vε dx, where the last equality in (3.5.2) follows from (3.3.3) with z vε := G (ū)v ε. In order to prove the first inequality of (3.5.1), we will pass to the limit in (3.5.2). First, the convergence v ε v in L 2 () implies that z vε z v in H 1 () with z v := G (ū)v; notice that the inverse operator of y F (y u, u) introduced on page 25 is an isomorphism.
98 90 3. THE OPTIMAL CONTROL PROBLEM Next we estimate each term on the righthand side of (3.5.2). Taking into account the embedding H 1 () L q () q [1, ) and Assumption 1.24(1), we get 2 a y (x, 2 ȳ)z2 v ε ȳ ϕ dx D M z vε 2 ȳ L r 2 2r L () r () ϕ L r () C z vε 2 H 1 () ȳ W 1,r () ϕ W 1,r (), with r given in Assumption 3.6, and a y (x, ȳ)z v ε z vε ϕ dx D M z vε ϕ L 2 () z vε L 2 () D M z vε L 2r r 2 () ϕ L r () z vε L 2 () C z vε 2 H 1 () ϕ W 1,r (). The remaining terms in (3.5.2) can be estimated analogously, with the help of the Assumptions 1.24(1) and 3.6. We only remark that for the integral terms over we make additionally use of the embedding H 1 /2 () L q () for every q [1, ). Finally, we pass to the limit in (3.5.2) and deduce 0 lim ε 0 J (ū)v 2 ε = J (ū)v 2. This yields the first inequality of (3.5.1). The second inequality follows directly from (3.4.1). Indeed, it is an easy and well known conclusion of (3.4.1) that 0 if ū(x) = u a (x), (3.5.3) Hu (x) 0 if ū(x) = u b (x), for a.a. x = 0 if u a (x) < ū(x) < u b (x), and H uu (x) 0 for a.a. x such that H u (x) = 0. In optimization theory, secondorder optimality conditions are conveniently expressed in terms of a Lagrange function associated with (P) L : W 1,r () L () W 1,r () R defined by L(y, u, ϕ) = J (y, u) {a(x, y) y ϕ + ϕf(x, y)} dx + ϕu dσ(x) = H(x, y(x), u(x), ϕ(x)) dσ(x) + {L(x, y) a(x, y) y ϕ ϕf(x, y)} dx, where r is given in Assumption 3.6 and J (y, u) := L(x, y) dx + l(x, y, u) dσ(x).
99 3.5. SECONDORDER OPTIMALITY CONDITIONS 91 In the next step, we shall formulate the secondorder necessary optimality conditions involving the Lagrange function and Hamiltonian. Defining H y, Hyy and H yu, similarly to H u and H uu, we can write the first and second order derivatives of L w.r.t. (y, u) as follows { D (y,u) L(ȳ, ū, ϕ)(z, v) = Hy (x)z(x) + H u (x)v(x) } dσ(x) + { L y (x, ȳ)z ϕ f (x, ȳ)z ϕ y Invoking the adjoint equation (3.3.5), we obtain D (y,u) L(ȳ, ū, ϕ)(z, v) = H u (x)v(x) dσ(x). ( a(x, ȳ) z + a )} (x, ȳ)z ȳ dx. y Moreover, there holds { D(y,u)L(ȳ, 2 ū, ϕ)(z, v) 2 = Hyy (x)z 2 (x) + 2 H yu (x)zv + H uu (x)v 2 (x) } dσ(x) { 2 L + y (x, 2 ȳ)z2 ϕ f 2 ( 2 y (x, a 2 ȳ)z2 ϕ y (x, 2 ȳ)z2 ȳ + 2 a )} (x, ȳ)z z dx. y If we take z = G (ū)v we deduce from (3.3.3) (3.5.4) J (ū)v 2 = D 2 (y,u)l(ȳ, ū, ϕ)(z, v) 2, hence we can rewrite the condition (3.5.1) as follows D(y,u) 2 L(ȳ, ū, ϕ)(z, v)2 0 (z, v) H 1 () Cū satisfying z = G (ū)v H uu (x) 0 for a.a. x such that H u (x) = 0. The next theorem is the main result of this section and it provides the secondorder sufficient optimality conditions for (P). We will employ here the technique devised by Casas et al. [17] to obtain optimality conditions having a form similar to the ones in the theory of nonlinear optimization in finitedimensional spaces. For an analogous distributed control problem this technique is used by Casas and Tröltzsch [36]. Theorem Let ū be an admissible control for problem (P) and ϕ W 1,r () satisfying (3.3.5) and (3.3.6). We also assume that there exist µ > 0 and τ > 0 such that J (ū)v 2 > 0 v Cū \ {0}, (3.5.5) H uu (x) µ if H u (x) τ for a.a. x. Then there exist ε > 0 and δ > 0 such that J(ū) + δ 2 u ū 2 L 2 () J(u)
100 92 3. THE OPTIMAL CONTROL PROBLEM for every admissible control u for (P) with u ū L () ε. Proof. The proof follows basically the ideas by Casas and Tröltzsch [36, Theorem 5.2]. Step 1: Preparations. We will argue by contradiction. Let us assume that there exists a sequence {u k } k=1 L () of admissible controls for (P) such that (3.5.6) u k ū L () < 1 k and J(ū) + 1 k u k ū 2 L 2 () > J(u k ) k N. Let us define then y k = G(u k ) = y uk, ȳ = G(ū) = yū, ρ k = u k ū L 2 () and v k = 1 ρ k (u k ū), (3.5.7) lim k y k ȳ W 1,r () = 0, lim k ρ k = 0 and v k L 2 () = 1 k N, where r is given in Assumption 3.6. By taking a subsequence if necessary, we can assume that v k v weakly in L 2 (). In the second step of this proof, we are going to show that v Cū. To this aim, we prove first the following convergence result lim k 1 ρ k (y k ȳ) = z in H 1 (), where z = G (ū)v. By setting z k = (y k ȳ)/ρ k, subtracting the state equations satisfied by y k and ȳ, dividing by ρ k and applying the mean value theorem, we find [ div a(x, y k ) z k + a ] y (x, ȳ + θ k(y k ȳ))z k ȳ (3.5.8) + f y (x, ȳ + η k(y k ȳ))z k = 0 in, [ a(x, y k ) z k + a ] y (x, ȳ + θ k(y k ȳ))z k ȳ ν = v k on. Notice that θ k and η k are functions depending on the space variable and their measurability can be shown by following the argumentation on page 62. We multiply (3.5.8) by z k and make an integration by parts to get with the assumptions on a and f and the Poincaré inequality (1.3.3) on page 4 that { min{α a, α f } z k 2 H 1 () a(x, y k ) z k 2 + f } y (x, ȳ + η k(y k ȳ))zk 2 dx = v k z k dσ(x) a y (x, ȳ + θ k(y k ȳ))z k z k ȳ dx v k L 2 () z k L 2 () + C z k L 2r r 2 () ȳ L r () z k L 2 ().
101 3.5. SECONDORDER OPTIMALITY CONDITIONS 93 From the previous estimate and (3.5.7) we deduce ( ) z k H 1 () C 1 + z k 2r. L r 2 () As indicated on page 21, the previous inequality implies that {z k } k=1 is bounded in H 1 (). This leads to the existence of a subsequence, denoted in the same way, such that z k z weakly in H 1 (). Thanks to the compact embedding H 1 () L 2r r 2 (), the convergence z k z is strong in L 2r r 2 () L 2 (), a y (x, ȳ + θ k(y k ȳ))z k ȳ a y (x, ȳ)z ȳ in L2 () 2 and f y (x, ȳ + η k(y k ȳ))z k f y (x, ȳ)z in L2 (). Therefore, we can pass to the limit in (3.5.8) and deduce (3.5.9) [ div a(x, ȳ) z + a ] (x, ȳ)z ȳ y [ + f y a(x, ȳ) z + a (x, ȳ)z ȳ y (x, ȳ)z = 0 in, ] ν = v on, hence z = G (ū)v. It remains to show the strong convergence z k z in H 1 (). To this aim, it is enough to prove that a(x, ȳ) z k 2 dx a(x, ȳ) z 2 dx as k, which, along with the strong convergence z k z in L 2 (), implies the strong convergence z k z in H 1 (). Taking into account (3.5.8) and (3.5.9), we have a(x, ȳ) zk 2 z 2 dx a(x, ȳ) a(x, y k ) z k 2 dx f + v k z k vz dσ(x) + y (x, ȳ + η k(y k ȳ))zk 2 f (x, ȳ)z2 y dx a + ȳ y (x, ȳ + θ k(y k ȳ))z k z k a (x, ȳ)z z y dx 0, as an easy consequence of the strong convergences y k ȳ in W 1,r () C( ), z k z in L 2r r 2 () and the weak convergences zk z in H 1 () and v k v in L 2 (). Step 2: v Cū. From the fact that u a (x) u k (x) u b (x) for a.a. x, we have that v k 0 if ū(x) = u a (x) and v k (x) 0 if ū(x) = u b (x) a.e on. Since the set of functions satisfying these sign conditions is convex and closed in L 2 (), it is weakly closed. Therefore, the weak limit v of {v k } k=1 satisfies these sign conditions, too. It
102 94 3. THE OPTIMAL CONTROL PROBLEM remains to prove that v(x) = 0 for a.a. x such that d(x) 0. By using the mean value theorem, we obtain from (3.5.6) ρ k k = 1 u k ū 2 L kρ 2 () > J(u k) J(ū) L = k ρ k y (x, ȳ + ν k(y k ȳ))z k dx l + y (x, ȳ + θ k(y k ȳ), ū + θ k (u k ū))z k dσ(x) l + u (x, ȳ + θ k(y k ȳ), ū + θ k (u k ū))v k dσ(x), where ν k and θ k are measurable functions with values in [0, 1]. Taking the limits in both sides of the previous inequality, using the convergences z k z in H 1 (), y k ȳ in C( ), u k ū in L (), Assumption 3.6, the adjoint equation (3.3.5), (3.5.9), and integrating by parts, we infer 0 = L (x, ȳ)z dx + y ( ) l (x, ȳ, ū) + ϕ u { l l (x, ȳ, ū)z + y u v dσ(x) = (x, ȳ, ū)v } d(x)v(x) dσ(x) = dσ(x) d(x) v(x) dσ(x). The last equality is a consequence of the signs of v and d (Eq. (3.3.8)). The previous inequality implies that d(x)v(x) = 0 holds a.e. on, hence v(x) = 0 whenever d(x) 0, as we wanted to prove. Step 3: v = 0. In this step, our goal is to prove that v does not satisfy the first condition of (3.5.5). This leads immediately to the identity v = 0 and then to the final contradiction. By using the definition of the Lagrange function L, (3.5.6) and the fact that y k and ȳ are the states corresponding to u k and ū, respectively, we get L(y k, u k, ϕ) = J (y k, u k ) < J (ȳ, ū) + 1 k u k ū 2 L 2 () (3.5.10) = L(ȳ, ū, ϕ) + 1 k u k ū 2 L 2 (). Performing a Taylor expansion up to the second order, we obtain L(y k, u k, ϕ) = L(ȳ + ρ k z k, ū + ρ k v k, ϕ) = L(ȳ, ū, ϕ)+ρ k D (y,u) L(ȳ, ū, ϕ)(z k, v k )+ ρ2 k 2 D2 (y,u)l(ȳ+ν k ρ k z k, ū+θ k ρ k v k, ϕ)(z k, v k ) 2 with functions ν k and θ k having the same properties as in the second step of the proof. Employing (3.5.10) and (3.5.6), the last equality leads to ρ k D (y,u) L(ȳ, ū, ϕ)(z k, v k ) + ρ2 k 2 D2 (y,u)l(ξ k, w k, ϕ)(z k, v k ) 2 < 1 k u k ū 2 L 2 () = ρ2 k k,
103 3.5. SECONDORDER OPTIMALITY CONDITIONS 95 where ξ k := ȳ + ν k ρ k z k and w k := ū + θ k ρ k v k. Obviously, ξ k ȳ in W 1,r () and w k ū in L () as k. Taking into account the expressions obtained for the derivatives of L, it follows, after dividing the previous inequality by ρ 2 k, 1 H u (x)v k dσ(x) + 1 ( { } H k ρ k 2 yy (x)zk 2 + 2Hyu(x)(z k k, v k ) + Huu(x)v k k 2 dσ(x) { 2 L + y (x, ξ k)z 2 2 k ϕ f 2 y (x, ξ k)z 2 2 k ( 2 a ϕ y (x, ξ k)zk ξ 2 2 k + 2 a )} ) (3.5.11) y (x, ξ k)z k z k dx < 1 k, where H k yy(x) := H yy (x, ξ k (x), w k (x), ϕ(x)) with analogous definitions for H k yu(x) and H k uu(x). In view of the properties of D 2 (y,u) l given in Assumption 3.6, it is easy to check that (H k yy(x), H k yu(x), H k uu(x)) ( H yy (x), H yu (x), H uu (x)) as k and H k yy(x) + H k yu(x) + H k uu(x) C for a.a. x and some constant C > 0. The following convergence properties can also be verified easily j a y (, ξ k)z j k ϕ j a (, ȳ)z ϕ, j = 1, 2, yj z k z and z k ξ k z ȳ in L 2 () 2 and ϕ 2 f y (, ξ k)z 2 k ϕ 2 f y (, ȳ)z in 2 L2 (). Using the above properties, we can pass to the limit in (3.5.11) as follows { 1 lim sup H u (x)v k (x) dσ(x) + 1 } H k k ρ k 2 uu(x)vk(x) 2 dσ(x) + 1 ( { Hyy (x)z H yu (x)(z, v) } { 2 L dσ(x) + (x, ȳ)z2 y2 ϕ f 2 ( 2 y (x, a 2 ȳ)z2 ϕ y (x, 2 ȳ)z2 ȳ + 2 a )} ) (3.5.12) (x, ȳ)z z dx 0. y Now we prove that 1 H uu (x)v 2 (x) dσ(x) is a lower bound of the above upper limit. 2 Then from (3.5.12) and (3.5.4) it follows that J (ū)v 2 = D(y,u) 2 L(ȳ, ū, ϕ)(z, v)2 0. Finally, according to (3.5.5), this is possible only if v = 0. To show the lower estimate mentioned above we make use of a convexity argument for which the second condition of (3.5.5) plays an essential role. The difficulty in deducing this estimate is due to the fact that we only have a weak convergence v k v.
104 96 3. THE OPTIMAL CONTROL PROBLEM Using the convergence Huu k H uu in L (), the identity H u (x)v k (x) = H u (x) v k (x) (Eq. (3.5.3)) and v k L 2 () = 1, we get { 1 lim sup H u (x)v k (x) dσ(x) + 1 } H k k ρ k 2 uu(x)vk(x) 2 dσ(x) { 1 = lim sup k ρ H u (x) v k (x) dσ(x) + 1 } H uu (x)v 2 k 2 k(x) dσ(x) { { 1 lim sup k { H u(x) >τ} ρ H u (x) v k (x) + 1 } k 2 H uu (x)vk(x) 2 dσ(x) + 1 } (3.5.13) H uu (x)v 2 2 k(x) dσ(x), { H u(x) τ} where τ is given in (3.5.5). Now thanks to ρ k v k L () = u k ū L () < 1/k, k 0 > 0 such that H uu L ()ρ k v k L () τ < H uu L () kτ < 1 k k 0, therefore τ ρ k v k (x) H uu L ()v 2 k(x) for a.a. x and k k 0. Then we have lim sup k lim sup k (3.5.14) { ( 1 { H u >τ} ρ H u v k + 1 ) k 2 H uu vk 2 dσ(x) + 1 } H uu v 2 2 { H k dσ(x) u τ} { ( H uu L () + 1 { H u >τ} 2 H ) uu vk 2 dσ(x) + 1 } H uu v 2 2 { H k dσ(x) u τ} ( H uu L () + 1 { H u >τ} 2 H ) uu v 2 dσ(x) + 1 H uu v 2 dσ(x) 2 { H u τ} 1 H uu v 2 dσ(x). 2 Combining (3.5.13) and (3.5.14), we obtain the desired lower estimate. Step 4: Final contradiction. Since v = 0, there holds z = 0. This fact, along with v k L 2 () = 1, (3.5.12)(3.5.14) and the second condition in (3.5.5), leads to { 0 lim sup ( H uu ()+ 1 L k { H u >τ} 2 H ) uu vk 2 dσ(x) + 1 } H uu v 2 2 { H k dσ(x) u τ} { Huu L lim sup () v k 2 { H k 2 dσ(x) + µ } v 2 u >τ} 2 { H k dσ(x) u τ} min{ H { } uu L (), µ} lim sup vk 2 dσ(x) 2 k = min{ H uu L (), µ} > 0, 2
105 3.5. SECONDORDER OPTIMALITY CONDITIONS 97 yielding the contradiction we were looking for. By virtue of the identity (3.5.4), the secondorder sufficient optimality condition (3.5.5) can be formulated in terms of the Lagrange function as follows D(y,u) 2 L(ȳ, ū, ϕ)(z, v)2 > 0 (z, v) (H 1 () Cū)\{(0, 0)} verifying z = G (ū)v, H uu (x) µ if H u (x) τ for a.a. x. Remark A comparison of the first inequality of (3.5.5) with the analogous of (3.5.1) shows that the gap between the necessary and sufficient conditions is minimal. On the other hand, the second inequality of (3.5.5) is stronger than the corresponding one of (3.5.1). In general, we cannot take τ = 0 in (3.5.5). The reader is referred to Dunn [52], see also Casas et al. [30, page 24], for a simple example showing the impossibility of taking τ = 0. Remark The statement of Theorem 3.18 involves both the L 2 () and L () norms. This phenomenon is called twonorm discrepancy: the differentiability of the cost function J requires the L () norm (see Theorem 3.7), while the L 2 () norm is the natural one to deduce sufficient optimality condition for strict local minima. Remark Let us note that H uu (x) = ( 2 l/ u 2 )(x, ȳ(x), ū(x)). Therefore, if the second derivative of l w.r.t. the third variable has a positive lower bound for a.a. x then the second condition of (3.5.5) is satisfied automatically. A standard example is given by the function l(x, y, u) = l 0 (x, y) + λ 2 u2 with λ > 0. In this case, controls satisfying the assumptions of Theorem 3.18 are locally optimal even in the sense of L 2 (), i.e. the problem of the twonorm discrepancy is resolved. To prove this we follow the lines of the proof of Theorem 3.18 with the following differences. We assume that there exists a sequence {u k } k=1 U ad with u k ū L 2 () < 1 k and J(ū) + 1 k u k ū 2 L 2 () > J(u k ) k N, instead of (3.5.6). Then, by using the identity H uu (x) λ and v k L 2 () = 1, we can considerably shorten the previous proof in the following way { 1 0 lim sup H u (x)v k (x) dσ(x) + 1 } H k k ρ k 2 uu(x)vk(x) 2 dσ(x) { 1 = lim sup k ρ H u (x) v k (x) dσ(x) + 1 } H uu (x)v 2 k 2 k(x) dσ(x) { } λ lim sup v 2 k 2 k(x) dσ(x) = λ 2. This yields the desired contradiction.
106 98 3. THE OPTIMAL CONTROL PROBLEM We finish this section by giving another sufficient optimality condition equivalent to (3.5.5). The proof of this equivalence is carried out in Casas and Mateos [25, Theorem 4.4]. Theorem Let ū be an admissible control for problem (P) and ϕ W 1,r () satisfying (3.3.5) and (3.3.6). Then (3.5.5) holds if and only if there exist δ > 0 and ρ > 0 such that (3.5.15) J (ū)h 2 δ h 2 L 2 () h C ρ ū, where 0 if ū(x) = u a (x) C ρ ū = h L 2 () h(x) 0 if ū(x) = u b (x) = 0 if d(x) > ρ for a.a. x. Remark Since C ρ ū is strictly contained in Cū, the reader might expect that the condition (3.5.15) is stronger than (3.5.5). The fact is that they are equivalent. Remark There are at least two advantages for studying sufficient conditions of the form (3.5.5). First, we can compare it with the necessary one given by (3.5.1). Second, in contrast to the equivalent condition (3.5.15), the proof of its sufficiency is close to the method known from the theory of nonlinear optimization in finitedimensional spaces. Nevertheless, condition (3.5.15) is the one used for numerical purposes; in order to deduce error estimates for the control problem (P), we will make explicit use of (3.5.15); see Section 4.3.
107 CHAPTER 4 Numerical approximation of the control problem 4.1. Introduction In this chapter, we focus on the numerical analysis of the following optimal boundary control problem min J(u) = L(x, y u (x)) dx + l(x, y u (x), u(x)) dσ(x), (P) subject to u U ad = {u L () α u(x) β a.e. x }, (y u, u) satisfying the state equation (1.1.1). To simplify the discussion below the box constraints α and β in the problem under consideration are now real numbers and α < β. Based on a standard finite element approximation, we will introduce a finite dimensional control problem (P h ) with h > 0. The state functions are discretized by linear finite element ansatz functions, i.e. by functions belonging to the set Y h defined on page 37. Concerning the discretization of control functions, we consider piecewise constant controls. At this point, we mention two other approaches for the approximation of (P). The continuous piecewise linear approximation of Neumann controls of semilinear equations was studied carefully by Casas and Mateos [27]. Other contributions to this kind of approximation were made by Casas [16], concerning distributed problems, and by Casas and Raymond [31] for Dirichlet boundary control problems. Another approach is the variational discretization concept suggested by Hinze in [63] that was applied to linear quadratic distributed problems and extended in [27] to Neumann controls of semilinear equations. The idea is to discretize the state but not the control, thus an infinite dimensional optimization problem has to be solved. Theoretical results for both techniques in the context of distributed quasilinear control problems were recently obtained by Casas and Tröltzsch [38]. For a particular Neumann control problem in a convex domain some numerical results are presented in Section 4.4. In our tests, we observe the same rate of convergence of the approximations as related numerical tests for semilinear problems; cf. [27] and [65]. Our main aims in this chapter are twofold. First, we are going to study the convergence of a sequence {ū h } h>0 of local optimal controls for the discretized problem (P h ) to a local solution ū of the continuous problem (P); see Section 4.2. Second, 99
108 NUMERICAL APPROXIMATION OF THE CONTROL PROBLEM we prove estimates for the error ū ū h in the L 2 () and L () norms; see Section 4.3. The chapter ends with some numerical experiments shown in Section 4.4. The preceding chapters have revealed that the analysis of control problems governed by quasilinear equations of the type (1.1.1) is much more complicated than expected at first sight. In fact, they are much more difficult than the corresponding control problems governed by semilinear and monotone equations. Indeed, although we were able to prove that, under quite general assumptions, (1.1.1) has a unique solution (see Chapter 1), the nonmonotone character of the equation introduced many difficulties which range from the linearization of (1.1.1) to the analysis of the discrete equation. To deal with the fact that solutions of the discrete equation are probably not unique we proved only local uniqueness as well as the differentiability of the mapping associating to each control its corresponding locally unique discrete state; see Section 2.6. These results, along with first and secondorder optimality conditions and some extra regularity of solutions of (P) to be approximated, are essential ingredients in the proof of error estimates for optimal controls. The case of a distributed control problem associated with a quasilinear equation similar to (1.1.1) was studied by Casas and Tröltzsch in [36, 37, 38]. However, as it is well known, the analysis for boundary controls is often more complicated than for the distributed ones because of the lower regularity of the states corresponding to boundary controls. Therefore, it has required a different approach to the one developed in [37] to carry out the numerical analysis of the control problem (P). Our study is limited to polygonal domains in the plane, since the case of curved domains introduces some additional difficulties that are beyond the scope of this thesis. The reader is referred to Casas et al. [24], Casas and Sokolowski [33] and Deckelnick et al. [47] for the analysis of boundary control problems in curved domains. However, we do not assume the convexity of the domain as required in most of the previous papers dealing with error estimates for boundary control problems; see Casas and Mateos [27], Casas et al. [29], Casas and Raymond [31] and Krumbiegel et al. [74], all of them devoted to linear or semilinear equations. Because of the higher regularity of the state and adjoint state in convex domains, we obtain better error estimates for optimal controls under the convexity assumption; see Theorems 4.10 and Though for the sake of brevity, we have considered only the approximation of the controls by using piecewise constant functions, other possibilities as described above are open; see Casas [16], Hinze [63] and Meyer and Rösch [79]. The control ū will stand for a strict local minimum of the control problem (P), i.e. there exists ε > 0 such that (4.1.1) J(u) > J(ū) u U ad B L ()(ū, ε) with u ū. Let us denote by ȳ the state associated with ū.
109 4.2. APPROXIMATION BY PIECEWISE CONSTANT CONTROLS 101 Throughout this chapter, we suppose that the Assumptions , 1.17, 1.24 and 3.6 hold, with β p > 4/3 and γ > 2 (see Assumption 3.6). Moreover, we assume that (3.3.9)(3.3.10) on pages are satisfied. Let us remark that the assumption β p > 4/3 is only made to avoid heavy notations and can be relaxed to min{β, p} > 4/3. In the sequel, we will often consider the case when p > 2. Hence, β p > 2 but, in fact, it is sufficient to require min{β, p} > Approximation of the controls by piecewise constant functions Let {T h } h>0 be a family of triangulations of, as defined on page 37, with maximum mesh size h > 0. For fixed h we denote by {T j } N(h) j=1 the family of triangles of T h with one side on. If the vertices of T j are x j and x j+1 then [x j, x j+1 ] := T j, 1 j N(h), with x N(h)+1 := x 1. Associated with the previous notation, we denote by U h L () the set of piecewise constant control functions, i.e. N(h) U h = u h L () u h = u j χ (xj,x j+1 ), u j R. Let us take q (2, 4) arbitrary but fixed and h 0, ρ and κ ρ, as in Theorem 2.23 on page 61. Further, h 1 h 0 is as in Theorem 2.25 on page 63. Let ε > 0 be chosen small enough such that (4.1.1) and 0 < ε < hold. Then we have j=1 ρ σ() 1 /2 u B L ()(ū, ε) = u ū L 2 () εσ() 1 /2 < ρ = u B L 2 ()(ū, ρ). Hence, according to Theorem 2.23, if u B L ()(ū, ε) the discrete state equation (2.3.1) on page 39 has, for every h < h 0, a unique solution y h (u) B W 1, q ()(ȳ, κ ρ ) Y h. Let us set U ad,h = U h U ad = {u h U h α u h β a.e. on } and consider for h < h 1 the auxiliary discrete control problem min J h (u h ) = L(x, y h (u h )) dx + l(x, y h (u h ), u h ) dσ(x), (Ph) ε s.t. u h U ad,h B L ()(ū, ε), (y h (u h ), u h ) satisfying the discrete state equation (2.3.1) with u = u h. We should underline that B L ()(ū, ε) is an additional constraint on the controls. However, we will see later that this constraint is not active if h is small enough. Consequently, we will remove it and finally we will introduce below the discrete problem in a standard way. By defining α h,j = max{α, max ū(x) ε} and β h,j = min{β, min x [x j,x j+1 ] x [x j,x j+1 ] ū(x) + ε},
110 NUMERICAL APPROXIMATION OF THE CONTROL PROBLEM we have the following equivalence for u h U ad,h : u h B L ()(ū, ε) α h,j u h (xj,x j+1 ) β h,j 1 j N(h). Theorem 4.1. The discrete problem (P ε h) admits at least one optimal solution. Proof. Since we are in finite dimension, the set of admissible controls is compact and nonempty in U h. This fact, along with the continuity of the functional J h, yields the existence of a minimum of (Ph). ε The next two theorems are the counterparts of Theorems 3.7 and 3.9 and their proofs are analogous. Theorem 4.2. For every h < h 1 the functional J h : B L ()(ū, ε) R is of class C 2 and its derivative is given by ( ) l (4.2.1) J h(u)v = u (x, y h(u), u) + ϕ h (u) v dσ(x), where ϕ h (u) Y h is the unique solution of the problem { (4.2.2) a(x, y h (u)) ϕ h (u) φ h + a y (x, y h(u))φ h y h (u) ϕ h (u) + f } y (x, y L h(u))ϕ h (u)φ h dx = y (x, y l h(u))φ h dx + y (x, y h(u), u)φ h dσ(x) for every φ h Y h. The existence and uniqueness of ϕ h (u) Y h satisfying (4.2.2) is an immediate consequence of Theorem 2.27 on page 67. Theorem 4.3. For every h < h 1 let ū h be a local minimum of the problem (Ph) ε and ȳ h := y h (ū h ). Then there exists a unique solution ϕ h Y h of the problem { ( a a(x, ȳ h ) ϕ h φ h + y (x, ȳ h) ȳ h ϕ h + f ) } y (x, ȳ h) ϕ h φ h dx L = y (x, ȳ l h)φ h dx + y (x, ȳ h, ū h )φ h dσ(x) φ h Y h and (4.2.3) N(h) j=1 xj+1 x j { } l u (x, ȳ h, ū j ) + ϕ h dσ(x) (u j ū j ) 0 u j [α h,j, β h,j ], where ū j = ū h (xj,x j+1 ). Moreover, there exist unique real numbers { s j } N(h) j=1 such that xj+1 { } l (4.2.4) u (x, ȳ h(x), s j ) + ϕ h (x) dσ(x) = 0 1 j N(h). x j
111 4.2. APPROXIMATION BY PIECEWISE CONSTANT CONTROLS 103 Finally, the function ū h satisfies the following projection formula (4.2.5) ū j = Proj [αh,j,β h,j] { s j} = max {min {β h,j, s j }, α h,j } 1 j N(h). The existence and uniqueness of the numbers s j follow easily from inequality (3.3.9) on page 80. Our next goal is to prove the convergence of solutions of (P ε h) to ū in L (). For this purpose, we need some preparatory lemmas. Lemma 4.4. Let 0 < τ < 1/2 be fixed. Suppose that h < h 2, where h 2 is given as in Proposition 2.26, and v, u U ad B L 2 ()(ū, ρ). Then there holds the estimate (4.2.6) y u y h (v) X + ϕ u ϕ h (v) X C ( h σ + u v L 2 ()), with C > 0 being dependent on τ, and σ takes the following values nonconvex convex X = L 2 () 3/4 1 X = H 1 () 1/2 1/2 X = L 2 () 5/8 3/4 X = L () 1/2 τ 1/2 τ Finally, if is convex, p > 2 and u W 1 1/ϱ,ϱ () with ϱ > 2, then (4.2.7) y u y h (v) H 1 () + ϕ u ϕ h (v) H 1 () C ( h + u v L 2 ()), and y u y h (v) L () + ϕ u ϕ h (v) L () C ( h 1 τ + h τ u v L 2 ()). Proof. We split the proof in two parts. Part 1: Estimates for the states. Let us consider the first two cases when is convex or nonconvex and deduce estimates for y u y h (v) in the norms of L 2 () and H 1 (). Using the estimate (2.3.37) on page 52 we get y u y h (v) L 2 () y u y h (u) L 2 () + y h (u) y h (v) L 2 () C ( h + u v L 2 ()). The estimation of the last term follows from the mean value theorem, Theorem 2.25 on page 63 as well as inequality (2.6.14) on page 66: (4.2.8) y h (u) y h (v) W 1, q () sup G h(v + t(u v)) L(L 2 (),W 1, q ()) u v L 2 (). t [0,1]
112 NUMERICAL APPROXIMATION OF THE CONTROL PROBLEM Analogously, we prove the estimate in the H 1 () norm. Let us focus now on the estimates in the norms defined on. Taking in inequality (2.7.3) on page 67 ε = h if is nonconvex and ε = h 2 if is convex and arguing as above, it is a immediate consequence of Corollary 2.15 on page 52 that y u y h (v) L 2 () C ( h σ + u v L 2 ()), with σ = 3/4 if is nonconvex and σ = 1 if is convex. This implies the estimates for the states in the L 2 () norm as given in the lemma. To derive the estimate for y u y h (v) L () we make use of the embedding W 1,q () C( ) q > 2. For q = min{ q, 2/(1 τ)} it follows q (2, 4), (2/q) (1/2) (1/2) τ and there exists C > 0, dependent on q and consequently on τ, such that Using the triangle inequality, we obtain y u y h (v) L () C y u y h (v) W 1,q (). y u y h (v) W 1,q () y u y h (u) W 1,q () + y h (u) y h (v) W 1,q (). Again, for the second term we can apply (4.2.8). For the first one, we deduce from the estimate (2.3.38) on page 52 y u y h (u) W 1,q () Ch 2 /q 1/2 Ch 1 /2 τ. It remains to consider the case when is convex, p > 2 and u W 1 1/ϱ,ϱ () (ϱ > 2). Now y u W 2,q () for some 2 < q min{p, ϱ, q, 2/(1 τ)}; see Theorem 1.21 on page 18. Obviously, the estimate (4.2.7) for the states follows from (2.3.40) and the above argumentation. Finally, the W 2,q () regularity of y u and an obvious modification of the proof of inequality (2.3.23) on page 47 yield in the same way as in the nonconvex case that y u y h (v) L () C ( h 1 τ + u v L 2 ()). Part 2: Estimates for the adjoint states. Once again we start by deducing the estimates for the adjoint states in the norms of L 2 () and H 1 () when is convex or nonconvex. From the inequalities (2.7.24)(2.7.25) on page 71, Corollary 1.37 on page 33 and taking into account the estimates for the states proved above, we obtain ϕ u ϕ h (v) L 2 () ϕ u ϕ v L 2 () + ϕ v ϕ h (v) L 2 () C ( u v L 2 () + h σ), where σ attains the values given in the statement of the lemma. The estimate in the H 1 () norm is obtained in the same way. For the L 2 () error we can proceed as for the states to deduce the desired estimates. Finally, let us consider the error in the L () norm. Taking once again q = min{ q, 2/(1 τ)} > 2, along with the triangle inequality, ϕ u ϕ h (v) L () C ϕ u ϕ h (v) W 1,q () C ( ϕ u ϕ v W 1,q () + ϕ v ϕ h (v) W 1,q ()).
113 4.2. APPROXIMATION BY PIECEWISE CONSTANT CONTROLS 105 Following the proof of (2.3.23), we observe that the order of convergence of the second summand in the previous inequality is O(h 1 /2 τ ); notice that ϕ v H 3 /2 (). This fact, along with Corollary 1.37, yields the estimates we were looking for. Let us study now the more delicate case when is convex, u W 1 1/ϱ,ϱ () (ϱ > 2) and p > 2. The W 2,q () regularity of ϕ u, with some q > 2, can be shown as in the proof of Theorem 3.10(2); see pages However, in contrast to the states, we do not have an inequality analogous to (4.2.8) for the adjoint states. In order to benefit from the higher regularity of ϕ u, we will argue in a different way. Let us consider first the error in the H 1 () norm. The triangle inequality ϕ u ϕ h (v) H 1 () ϕ u ϕ h (u) H 1 () + ϕ h (u) ϕ h (v) H 1 () delivers the order O(h) of convergence for the first summand; compare also the estimate (2.7.26) on page 72. For the second one we are going to prove that (4.2.9) ϕ h (u) ϕ h (v) H 1 () C ( h + u v L 2 ()). Once this is shown, we conclude (4.2.7). For the estimation of ϕ h (u) ϕ h (v) H 1 () we subtract the equations satisfied by both functions, insert ϕ h (u) ϕ h (v) as test function and arrive at ϕ h (u) ϕ h (v) 2 H 1 () ( C a(x, y h (u)) ϕ h (u) ϕ h (v) 2 + f ) y (x, y h(u)) (ϕ h (u) ϕ h (v)) 2 dx ( = C (a(x, y h (v)) a(x, y h (u))) ϕ h (v) ( ϕ h (u) ϕ h (v)) dx ( a + y (x, y h(v)) y h (v) ϕ h (v) a ) y (x, y h(u)) y h (u) ϕ h (u) ( f (ϕ h (u) ϕ h (v)) dx + y (x, y h(v)) f ) y (x, y h(u)) ϕ h (v) (ϕ h (u) ϕ h (v)) dx ( L + y (x, y h(u)) L ) y (x, y h(v)) (ϕ h (u) ϕ h (v)) dx ( l + y (x, y h(u), u) l ) ) y (x, y h(v), v) (ϕ h (u) ϕ h (v)) dσ(x) C ( y h (v) y h (u) W 1, q () + u v L 2 () hence + ϕ h (u) ϕ h (v) L 4 ()) ϕh (u) ϕ h (v) H 1 (), (4.2.10) ϕ h (u) ϕ h (v) H 1 () C ( u v L 2 () + ϕ h (u) ϕ h (v) L 4 ()).
114 NUMERICAL APPROXIMATION OF THE CONTROL PROBLEM At this point, let us demonstrate only the estimation of the most difficult term a y (x, y h(v)) y h (v) ϕ h (v) a y (x, y h(u)) y h (u) ϕ h (u) ϕ h(u) ϕ h (v) dx C ( y h (v) y h (u) W 1, q () + ϕ h (u) ϕ h (v) H 1 ()) ϕh (u) ϕ h (v) L 4 (), having used the fact that {y h (u)} h<h2, {y h (v)} h<h2 and the sequences of the corresponding adjoint states are bounded in W 1,4 (); see Corollary 2.31 on page 74. From Lemma 2.3 on page 39 and inequality (4.2.10) we infer ϕ h (u) ϕ h (v) H 1 () C ( u v L 2 ()+ ϕ h (u) ϕ h (v) 1 /2 L 2 () ϕ ) h(u) ϕ h (v) 1 /2 H 1 () and by Young s inequality, ϕ h (u) ϕ h (v) H 1 () C u v L 2 () + C2 2 ϕ h(u) ϕ h (v) L 2 () ϕ h(u) ϕ h (v) H 1 (). Next let us get an estimate for ϕ h (u) ϕ h (v) L 2 (). Since is convex, it follows from inequality (2.7.25) and Corollary 1.37 that ϕ h (u) ϕ h (v) L 2 () ϕ u ϕ v L 2 () + ϕ h (u) ϕ u L 2 () + ϕ v ϕ h (v) L 2 () C ( h + u v L 2 ()). Collecting all these results, we conclude (4.2.9). It remains to estimate the error ϕ u ϕ h (v) in the L () norm. Using the same triangle inequality as for the H 1 () norm and taking 2 < q = min{q, q, 2/(1 τ)}, we obtain ϕ u ϕ h (v) L () C ϕ u ϕ h (v) W 1,q () C ( ϕ u ϕ h (u) W 1,q () + ϕ h (u) ϕ h (v) W 1,q ()). For the first term we deduce as for the state functions ϕ u ϕ h (u) W 1,q () Ch 2 /q Ch 1 τ. For the second term we use the inverse inequality (2.2.6) on page 38 ϕ h (u) ϕ h (v) W 1,q () Ch 2 /q 1 ϕ h (u) ϕ h (v) H 1 () Inserting (4.2.9) in the last inequality, we get Ch τ ϕ h (u) ϕ h (v) H 1 (). ϕ h (u) ϕ h (v) W 1,q () Ch τ ( h + u v L 2 ()) = C ( h 1 τ + h τ u v L 2 ()), hence ϕ u ϕ h (v) L () C ( h 1 τ + h τ u v L 2 ()).
115 4.2. APPROXIMATION BY PIECEWISE CONSTANT CONTROLS 107 Lemma 4.5. For every h < h 1 let u h U ad,h B L 2 ()(ū, ρ) and u U ad be given. (1) Under the weak convergence u h u in L 1 () as h 0, there holds (4.2.11) lim y h (u h ) y u C( ) H 1 () = 0 and lim inf J h (u h ) J(u). h 0 h 0 (2) If u h u strongly in L 1 () as h 0 then lim h 0 J h (u h ) = J(u). Proof. Thanks to the weak convergence u h u in L 1 (), {u h } h<h1 converges weakly to u in L 2 () and u B L 2 ()(ū, ρ). In view of the definition of G h, we have for y h (u h ) = G h (u h ) that {y h (u h )} h<h1 B W 1, q ()(ȳ, κ ρ ) Y h, therefore there exists a subsequence {y hk (u hk )} k=1 converging weakly to some element ỹ B W 1, q ()(ȳ, κ ρ ). Since the embedding W 1, q () C( ) is compact, this convergence is strong in C( ). To show that ỹ = y u we consider the equation satisfied by y hk (u hk ): (4.2.12) {a(x, y hk (u hk )) y hk (u hk ) φ hk +f(x, y hk (u hk ))φ hk } dx= u hk φ hk dσ(x) φ hk Y hk. Let us take in (4.2.12) φ hk = Π hk φ, with φ H 2 () arbitrarily chosen and Π h introduced on page 37. Then by passing to the limit, we obtain, along with the strong convergences y hk (u hk ) ỹ in C( ) and Π hk φ φ in H 1 (), the weak convergences y hk (u hk ) ỹ in W 1, q () and u hk u in L 2 (), and the assumptions on a and f, that {a(x, ỹ) ỹ φ + f(x, ỹ)φ} dx = uφ dσ(x). Finally, the density of H 2 () in H 1 () and the uniqueness of the solution of (1.1.1) imply ỹ = y u. Since all the subsequences have the same limit, we conclude that lim h 0 y h (u h ) y u C( ) = 0. Let us prove now the convergence y h (u h ) y u in the H 1 () norm. To this end, we write y h (u h ) y u H 1 () y h (u h ) y uh H 1 () + y uh y u H 1 (). Now since y h (u h ) and y uh are the discrete and continuous states associated with the same control and {u h } h<h1 is bounded in L 2 (), we can use the error estimate (2.3.37) on page 52 to deduce that the first summand tends to zero as h 0. Next we show that the second summand converges to zero, too. At first, similar to the proof of Theorem 3.3 on page 77 it can be shown that y uh y u weakly in H 3 /2 (); notice that {y uh } h<h1 is bounded in H 3 /2 () (Theorem 1.18 on page 16). Then the proof of the convergence of the states in H 1 () is completed due to the compactness of the embedding H 3 /2 () H 1 (). For the proof of the inequality in (4.2.11) the reader is referred to Casas and Mateos [26, Lemma 11]. Finally, let us prove the last assertion of the lemma. From the dominated convergence theorem, along with the fact that {u h } h<h1 is bounded in U ad,h B L 2 ()(ū, ρ) and u h u strongly in L 1 (), we deduce the strong convergence u h u in L 2 (). The
116 NUMERICAL APPROXIMATION OF THE CONTROL PROBLEM convergence of the cost functions is then an immediate consequence of the continuity of the controltostate mapping G : L 2 () H 3 /2 (), u y u, and (4.2.11). The next theorem, which is the main result of this section, establishes the convergence in L () of solutions of (P ε h) to ū. Theorem 4.6. Let {ū h } h<h1 there holds the convergence be a family of solutions of (P ε h) and ȳ h := y h (ū h ). Then lim h 0 ( ūh ū L () + ȳ h ȳ L () + ȳ h ȳ H 1 () ) = 0. Proof. Since {ū h } h<h1 is bounded in L (), there exists a weakly convergent subsequence in L (), still indexed by h, to some ũ U ad B L ()(ū, ε). Although the proof of the identity ū = ũ is standard, see for instance Casas et al. [29, Theorem 4.4], we will briefly sketch it for convenience of the reader. From Theorem 3.10(1) on page 81 we know that ū C 0, 1/2 (). Now consider the projection operator P h : L 1 () U h defined by 1 xj+1 (P h u) (xj,x j+1 ) = u(x) dσ(x) x j+1 x j x j which satisfies the inequality ū P h ū L () Ch 1 /2 ū C 0,1/2 (), with some positive constant C independent of ū and h. Hence, for all h small enough P h ū is admissible for (P ε h) and from Lemma 4.5(1) (in particular, ū h ũ weakly in L 1 ()) we get J(ũ) lim inf h 0 J h (ū h ) lim sup h 0 J h (ū h ) lim sup J h (P h ū) = J(ū). h 0 Finally, this inequality, along with (4.1.1), yields ũ = ū and lim h 0 J h (ū h ) = J(ū). In view of Lemma 4.5, this implies the desired convergence properties for the states. The rest of the proof is split into two steps. Step 1: L 2 () convergence of the controls. We will follow here Arada et al. [5]. By the definitions of J h and J, the convergence lim h 0 J h (ū h ) = J(ū) implies that {l(x, ȳ h, ū h ) l(x, ȳ, ū)} dσ(x) = 0. lim h 0 On the other hand, since ū h ū weakly in L (), there holds l u (x, ȳ, ū)(ū h ū) dσ(x) 0 as h 0.
117 4.2. APPROXIMATION BY PIECEWISE CONSTANT CONTROLS 109 Then invoking inequality (3.3.9) on page 80, Assumption 3.6 and (4.2.11), we have for some measurable function θ h (x) [0, 1] and v h (x) := ū(x) + θ h (x)(ū h (x) ū(x)) that Λ l 2 ū h ū 2 L 2 () 1 2 l 2 u (x, ȳ, v h)(ū 2 h ū) 2 dσ(x) l = {l(x, ȳ, ū h ) l(x, ȳ, ū)} dσ(x) u (x, ȳ, ū)(ū h ū) dσ(x) = {l(x, ȳ, ū h ) l(x, ȳ h, ū h )} dσ(x) l + {l(x, ȳ h, ū h ) l(x, ȳ, ū)} dσ(x) u (x, ȳ, ū)(ū h ū) dσ(x) C ȳ h ȳ L 2 () + {l(x, ȳ h, ū h ) l(x, ȳ, ū)} dσ(x) l u (x, ȳ, ū)(ū h ū) dσ(x) 0 when h 0. Step 2: L () convergence of the controls. Given x, let 1 j N(h) be such that x [x j, x j+1 ]. From the projection formulas (3.3.12) on page 81 and (4.2.5) for the optimal controls and the contractivity of the projection we obtain (4.2.13) ū(x) ū j = Proj[α,β] { s(x)} Proj [αh,j,β h,j ]{ s j } Proj[α,β] { s(x)} Proj [αh,j,β h,j ]{ s(x)} + Proj[αh,j,β h,j ]{ s(x)} Proj [αh,j,β h,j ]{ s j } Proj [α,β] { s(x)} Proj [αh,j,β h,j ]{ s(x)} + s(x) sj. Next we demonstrate that the first summand in (4.2.13) is zero for sufficiently small h. Assume that h 1 /2 Λū < ε/2, where Λū is the Hölder constant of ū. Because α α h,j β h,j β, we have that (4.2.14) Proj [α,β] { s(x)} Proj [αh,j,β h,j ]{ s(x)} = 0 if α h,j s(x) β h,j. Let us now assume that α h,j > s(x). Under this assumption, we will show that α = α h,j and consequently (4.2.14) holds true. We will argue by contradiction. If α < α h,j then ū(x) < α h,j = max x [xj,x j+1 ] ū( x) ε, hence there exists x [x j, x j+1 ] such that ū(x) < ū(x) ε. But this leads to the following contradiction ε < ū(x) ū(x) Λū x x 1 /2 Λūh 1 /2 < ε 2. Analogously, we argue if β h,j < s(x). Now from (4.2.13) it follows (4.2.15) ū ū h L () s s h L (), where s h := N(h) j=1 s j χ (xj,x j+1 ).
118 NUMERICAL APPROXIMATION OF THE CONTROL PROBLEM For the proof of the uniform convergence s h s as h 0 we refer to Casas et al. [29, pages ]. We should remark that the proof of the latter result relies on the C 0, 1/2 () regularity of s and the convergences ȳ h ȳ and ϕ h (ū h ) ϕ in L () as a consequence of the convergence ū h ū in L 2 () and Lemma 4.4. Because of the uniform convergence ū h ū as h 0, ū h is a local solution of the problem min J h (u h ) = L(x, y h (u h )) dx + l(x, y h (u h ), u h ) dσ(x), (P h ) s.t. u h U ad,h, (y h (u h ), u h ) satisfying the discrete state equation (2.3.1) with u = u h, for every sufficiently small h. Therefore, (4.2.3) can be rewritten as N(h) xj+1 { } l u (x, ȳ h, ū j ) + ϕ h dσ(x) (u j ū j ) 0 u j [α, β] j=1 x j and the expression (4.2.5) can be formulated as (4.2.16) ū j = Proj [α,β] { s j } 1 j N(h) Error estimates for optimal controls In this section, ū is supposed to be a local minimum of (P) satisfying the secondorder sufficient condition for optimality (3.5.15) on page 98 and {ū h } h<h1 (h 1 > 0 is given in Theorem 2.25 on page 63) is a sequence of local solutions of (P h ) converging uniformly to ū; see Theorem 4.6. As usual, ȳ, ȳ h := y h (ū h ), and ϕ, ϕ h := ϕ h (ū h ), stand for the states and adjoint states corresponding to ū and ū h. Our goal is to estimate ū h ū L 2 () and ū h ū L (). To this aim, we will state and prove three auxiliary lemmas. Lemma 4.7. Let δ > 0 be given as in Theorem Then there exists h 3 h 1 such that δ (4.3.1) 2 ū h ū 2 L 2 () (J (ū h ) J (ū))(ū h ū) h < h 3. Proof. The proof is exactly the same as in Casas et al. [29, Lemma 4.6], therefore we will only sketch the main steps. Setting d h (x) = l u (x, ȳ h(x), ū h (x)) + ϕ h (x) for x, we deduce from Lemma 4.4 that d h d uniformly on. With the help of this convergence and taking ρ as in Theorem 3.22 on page 98, it can be shown that there exists 0 < h h 1 such that (ū h ū) C ρ ū h < h. Hence, according to Theorem
119 4.3. ERROR ESTIMATES FOR OPTIMAL CONTROLS , it follows J (ū)(ū h ū) 2 δ ū h ū 2 L 2 (). Now by applying the mean value theorem, we get for some measurable function θ h with values in [0, 1] that (J (ū h ) J (ū))(ū h ū) = J (ū + θ h (ū h ū))(ū h ū) 2 = (J (ū + θ h (ū h ū)) J (ū)) (ū h ū) 2 + J (ū)(ū h ū) 2 δ ū h ū 2 L 2 () (J (ū + θ h (ū h ū)) J (ū)) (ū h ū) 2. Finally, it is enough to choose 0 < h 3 h such that (J (ū + θ h (ū h ū)) J (ū)) (ū h ū) 2 δ 2 ū h ū 2 L 2 () h < h 3 to prove (4.3.1). The last inequality is obtained from the representation of J (Eq. (3.3.3) on page 79) after long but easy computations. Let us demonstrate exemplarily the estimation of one of the terms involved in (3.3.3). Setting û h = ū + θ h (ū h ū), v h = ū h ū and exploiting inequality (1.6.11) on page 26, we deduce ϕ 2 a û yûh h y (x, y 2 û h ) (G (û h )v h ) 2 2 a ϕū yū y (x, y ū) (G (ū)v 2 h ) 2 dx C (( ϕûh ϕū W 1,4 () + yûh yū W 1,4 ()) vh 2 L 2 () + (G (û h ) G (ū)) v h L 4 () v h L 2 ()). Moreover, by subtracting the equations satisfied by G (û h )v h and G (ū)v h and using again (1.6.11), we get (G (û h ) G (ū)) v h H 3/2 () C y û h yū W 1,4 () v h L 2 (). Hence, given γ > 0, these inequalities, together with the uniform convergence ū h ū (and consequently û h ū), Corollary 1.32 on page 27 and the estimate (1.7.11) on page 33, yield for h small enough that ϕ 2 a û yûh h y (x, y 2 û h ) (G (û h )v h ) 2 2 a ϕū yū y (x, y ū) (G (ū)v 2 h ) 2 In the next lemma, we estimate the convergence of J h to J. dx γ v h 2 L 2 (). Lemma 4.8. Let h 2 > 0 be given as in Proposition If is nonconvex there exists C 0 > 0 such that, for all v L 2 () and h < h 2, (4.3.2) (J h(ū h ) J (ū h )) v C 0 h 5 /8 v L 2 ().
120 NUMERICAL APPROXIMATION OF THE CONTROL PROBLEM If is convex and p > 2 then for every γ > 0 there exists C γ > 0 such that, for all v L 2 () and all h < h 2, (4.3.3) (J h(ū h ) J (ū h )) v ( C γ h + γ ū h ū L 2 ()) v L 2 (). Proof. Invoking the expressions of the derivatives J h and J given by the formula (3.3.2) on page 79 and (4.2.1), respectively, it follows (J h(ū h ) J l (ū h ))v u (x, ȳ h, ū h ) l u (x, y ū h, ū h ) + ϕ h ϕūh v dσ(x) (4.3.4) C 1 ( ȳh yūh L 2 () + ϕ h ϕūh L 2 ()) v L 2 (). The result in the nonconvex case is then an easy application of Lemma 4.4: ȳ h yūh L 2 () + ϕ h ϕūh L 2 () Ch 5 /8. To show (4.3.3) we use the following well known property, compare also Grisvard [60, Theorem ]: For every ε > 0 there exists C ε > 0 such that thus we get with the aid of (4.2.6) z L 2 () ε z H 1 () + C ε z L 2 () z H 1 (), ȳ h yūh L 2 () = y h (ū h ) yūh L 2 () ε y h (ū h ) yūh H 1 () + C ε y h (ū h ) yūh L 2 () ε y h (ū h ) yūh H 1 () + C ε C 2 h. By virtue of Corollary 1.32 on page 27, along with the embedding W 1,4 () H 1 () and the boundedness of the sequence {ū h } h<h2 in L 2 (), we have ȳ yūh H 1 () C 3 ū ū h L 2 (). Taking into account that ū C 0,1 () W 1 1/ϱ,ϱ () with ϱ > 2 (Theorem 3.10(2) on page 81), (4.2.7) leads to ȳ y h (ū h ) H 1 () C 4 ( h + ū ūh L 2 ()), Combining the last two inequalities, we deduce y h (ū h ) yūh H 1 () C 5 ( h + ū ūh L 2 ()). The same arguments can be applied to the adjoint state. Hence, by setting γ = εc 5 C 1 and C γ = γ + C ε C 2 C 1, (4.3.4) implies (4.3.3). One key point in the proof of the error estimates for the controls is to find a function u h U ad,h that approximates ū and satisfies J (ū)u h = J (ū)ū. Making use of the
121 4.3. ERROR ESTIMATES FOR OPTIMAL CONTROLS 113 definition of d given on page 80, we define u h = N(h) j=1 u j χ (xj,x j+1 ) U h by 1 xj+1 d(x)ū(x) dσ(x) if Ij 0, I xj+1 j x (4.3.5) u j = j 1 xj+1 I j = d(x) dσ(x). x ū(x) dσ(x) if I j = 0, j x j+1 x j x j The next lemma shows that u h fulfills the required conditions. Lemma 4.9. There exists h 4 > 0 such that, for every h < h 4, the following properties hold: (1) u h U ad,h. (2) J (ū)u h = J (ū)ū. (3) There exists C > 0 independent of h such that ū u h L () Ch σ, where σ = 1/2 if is nonconvex and σ = 1 if is convex and p > 2. Proof. The proof is almost identical to the one of Casas et al. [29, Lemma 4.8]. The only difference is that ū C 0, 1/2 () if is nonconvex. Finally, we derive the main error estimate. Theorem There exists a constant C > 0 independent of h such that (4.3.6) ū ū h L 2 () Ch σ with σ = 1/2 if is nonconvex and σ = 1 if is convex and p > 2. Proof. Since ū h is a local minimum of (P h ), we know that J h(ū h )(u h ū h ) 0, where u h is defined by (4.3.5). The last inequality is equivalent to J (ū h )(ū ū h ) + (J h(ū h ) J (ū h )) (ū ū h ) + J h(ū h )(u h ū) 0. On the other hand, J (ū)(ū h ū) 0. Adding these inequalities, we get (J (ū h ) J (ū)) (ū h ū) (J h(ū h ) J (ū h )) (ū ū h ) + J h(ū h )(u h ū) (4.3.7) = (J h(ū h ) J (ū h )) (ū ū h ) + (J h(ū h ) J (ū h )) (u h ū) + (J (ū h ) J (ū)) (u h ū) + J (ū)(u h ū). From (4.3.1), (4.3.2), Corollaries 1.32 and 1.37 on page 27 and 33, respectively, as well as Lemmas 4.4 and 4.9, we deduce in the nonconvex case δ 2 ū h ū 2 L 2 () C ( ( ) ) h 5 /8 ū ūh L 2 () + h 1 /2 + h 1/2 ū h ū L 2 (),
122 NUMERICAL APPROXIMATION OF THE CONTROL PROBLEM which leads to (4.3.6). If is convex and p > 2 we use (4.3.3) instead of (4.3.2). Similarly, by taking γ = δ/4, it follows from (4.3.7) that δ 4 ū h ū 2 L 2 () C γ h ( ū h ū L 2 () + Ch ) ( ) δ + C h ū h ū L 2 (). Finally, the latter result and Young s inequality imply (4.3.6) with σ = 1. Remark The error estimates deduced in Theorem 4.10 seem to be optimal. This opinion is based on the fact that the interpolation error of the approximation of functions in C 0,σ () by piecewise constant functions is of order O(h σ ). Taking σ = 1/2 if is nonconvex and σ = 1 if is convex and p > 2, this is the same order that we have obtained in the previous theorem. At the end, we prove the error estimate in L (). Theorem Let 0 < τ < 1/2 be fixed. There exists a constant C τ dependent on τ but not on h, such that ū ū h L () C τ h σ τ with σ = 1/2 if is nonconvex and σ = 1 if is convex and p > 2. > 0, being Proof. From (4.2.4) and the continuity of the integrand with respect to x (see Eq. (3.3.10) on page 81) we deduce, for every 1 j N(h), the existence of a point ξ j (x j, x j+1 ) such that l (4.3.8) u (ξ j, ȳ h (ξ j ), s j ) + ϕ h (ξ j ) = 0. Moreover, denoting by Λ s the Hölder constant of s (Theorem 3.10 on pages 8081), we get for x (x j, x j+1 ) s(x) s j s(x) s(ξ j ) + s(ξ j ) s j Λ s x ξ j σ + s(ξ j ) s j Λ s h σ + s(ξ j ) s j. Employing this inequality in (4.2.15), along with the properties (3.3.9) and (3.3.11) given on pages and (4.3.8), yields ū ū h L () max s s j L (x j,x j+1 ) Λ s h σ + max s(ξ j) s j 1 j N(h) 1 j N(h) Λ s h σ + 1 l max Λ l 1 j N(h) u (ξ j, ȳ h (ξ j ), s(ξ j )) l u (ξ j, ȳ h (ξ j ), s j ) Λ s h σ + 1 { l max Λ l 1 j N(h) u (ξ j, ȳ h (ξ j ), s(ξ j )) l u (ξ j, ȳ(ξ j ), s(ξ j )) } + ϕ(ξ j ) ϕ h (ξ j ).
123 4.4. NUMERICAL EXPERIMENTS 115 Finally, taking into account Assumption 3.6, Lemma 4.4 and (4.3.6), we conclude ū ū h L () Λ s h σ + C ( ȳ ȳ h L () + ϕ ϕ h L () Λ s h σ + C ( h σ τ + h τ ū ū h L 2 ()) Cτ h σ τ. Remark The error estimates for the discrete controls given in Theorems 4.10 and 4.12 can also be derived by using an abstract result proved recently by Casas and Tröltzsch [38]. ) 4.4. Numerical experiments In this section, we present some numerical experiments for the discrete approach of (P) and compare the order of convergence proved in Theorems 4.10 and 4.12 with the experimental ones. We have tested the convergence theory by two examples. In both cases, we consider the following optimal control problem min J(u) = 1 (E) 2 y u y d 2 L 2 () u u d 2 L 2 () + η(x)u(x) dσ(x) subject to u U ad := {u L () α u(x) β a.e. x }, { div [a(x, yu ) y u ] + f(x, y u ) = 0 in, a(x, y u ) ν y u = u(x) + g(x) on. In the first example, is supposed to be convex and we have specified the functions contained in (E) such that we know a local solution of it. The second example is constructed in such a way that we do not know any local minimum of (E). For the generation of the mesh for we use the MATLAB PDE Toolbox. For the optimization a standard SQP method is implemented; see for instance Heinkenschloss and Tröltzsch [62], Kunisch and Sachs [76], Hinze and Kunisch [64] and Tröltzsch [90]. Given (y k, u k, ϕ k ), k N, in step k + 1 we have to solve the following linearquadratic problem to find (y k+1, u k+1, ϕ k+1 ): (QE) k+1 min J(u k+1 ) = J (y k, u k )(y k+1 y k, u k+1 u k ) D2 (y,u)l(y k, u k, ϕ k )(y k+1 y k, u k+1 u k ) 2 subject to u k+1 U ad, (y k+1, u k+1 ) satisfying the linearized equation
124 NUMERICAL APPROXIMATION OF THE CONTROL PROBLEM {( a(x, y k ) y k+1 + a ) y (x, y k)y k+1 y k φ + f } y (x, y k)y k+1 φ dx {( ) f = y (x, y k)y k f(x, y k ) φ + a } y (x, y k)y k y k φ dx + (u k+1 + g) φ dσ(x) φ H 1 (), where J (y, u) := 1 2 y y d 2 L 2 () u u d 2 L 2 () + L(y, u, ϕ) := J (y, u) {a(x, y) y ϕ + ϕf(x, y)} dx + η(x)u(x) dσ(x) and ϕ (u + g) dσ(x). The new iterate ϕ k+1 is the solution of the associated adjoint equation { a(x, y k ) ϕ k+1 φ + a y (x, y k)φ y k ϕ k+1 + f } y (x, y k)ϕ k+1 φ dx { = (y k+1 y ) φ a y (x, y 2 f k)(y k+1 y k ) ϕ k φ ϕ k y (x, y k)(y 2 k+1 y k )φ [ 2 a ϕ k y (x, y k)(y 2 k+1 y k ) y k + a ] } y (x, y k) (y k+1 y k ) φ dx φ H 1 (). The choice of the initial data (y 1, u 1, ϕ 1 ) is indicated in the examples below. To solve each of the linearquadratic problems (QE) k numerically we have applied a primaldual active set strategy, according to Kunisch and Rösch [75]; see also Bergounioux and Kunisch [8]. The linear equations for y k+1 and ϕ k+1 are solved in the same way as equation (2.5.2) on page 56. In order to deal with the convection term ( a/ y)(, y k ) y k ϕ k+1 in the adjoint state equation, we have additionally written our own routine. As an alternative to the above SQP method a semismooth Newton method can be used; see Ito and Kunisch [69] for more details. We have solved the problem (E) using different mesh sizes. Similarly to Section 2.5, the experimental order of convergence for z is given by EOC X (z) = log( z z h 1 X ) log( z z h2 X ) log(h 1 ) log(h 2 ) In the previous formula z, stands for the state, the adjoint state or the control function. We will replace z by z if a locally optimal solution is known (Example 4.14) and by z fine if not (Example 4.15); z fine is the numerically computed optimal (state, adjoint state or control) function on the finest mesh with mesh size h fine = 2 8. Example We fix the following data: = (0, π) 2, α = 20, β = 2, a(x, y(x)) = 1 + (x 1 + x 2 ) 2 + y 2 (x), f(x, y(x)) = 2y(x)(sin 2 (x 1 ) + sin 2 (x 2 )) ξ(x),
125 4.4. NUMERICAL EXPERIMENTS 117 where x = (x 1, x 2 ) and ξ(x) = 2 sin(x 1 ) sin(x 2 )(1 + (x 1 + x 2 ) 2 ) + 6 sin 3 (x 1 ) sin 3 (x 2 ) (x 1 + x 2 )(sin(x 1 ) cos(x 2 ) + cos(x 1 ) sin(x 2 )). Further, we set y d (x) = sin(x 1 ) sin(x 2 ) 2(sin 2 (x 1 ) + sin 2 (x 2 )) in and e(x) = (1 + (x 1 + x 2 ) 2 ) sin(x 1 + x 2 ), g(x) = min{0, e(x) α} + max{0, e(x) β}, η(x) = 1 Proj [α,β] {e(x)} and u d = 0 on. A strict local minimum of (E) is given by ū(x) = Proj [α,β] {e(x)}. The associated state and adjoint state are ȳ(x) = sin(x 1 ) sin(x 2 ) and ϕ 1 in, respectively. Indeed, it can be easily checked that d(x) = ϕ(x)+η(x)+ū(x) = 0 on, hence the firstorder optimality condition (3.3.6) on page 80 is satisfied. Moreover, in view of equality (3.3.3) on page 79, we have for u L () J (u)v 2 = v 2 dσ(x) + zv 2 dx v L 2 (), where z v H 1 () is the solution of equation (1.6.9) on page 25. Consequently, J (ū)v 2 > 0 for every v Cū 0 \ {0}, hence the secondorder sufficient condition (3.5.5) on page 91 is satisfied. Figure 4.1. Optimal control ū computed with COMSOL Multiphysics (left frame); Solid line: ū, dashed line: e (right frame). The finest grid on which we can run the SQP method has a mesh size of order 2 8 ; the range of the degrees of freedom for the state and adjoint state function is 10 5, the corresponding one for the controls is We have taken y 1 1, ϕ 1 2 and u 1 β as initial data. We have used finer grids for a more accurate numerical evaluation of the errors. The results are collected in the table below. For the controls we observe linear convergence for the error in L 2 () and L (). However, we cannot expect to obtain numerically the convergence order O(h 1 τ ), for some 0 < τ < 1/2, in the
126 NUMERICAL APPROXIMATION OF THE CONTROL PROBLEM h EOC L 2 ()(ȳ) EOC H 1 ()(ȳ) EOC L 2 ()( ϕ) EOC L 2 ()(ū) EOC L ()(ū) Table 4.1. Experimental order of convergence for the optimal values. L () norm. Moreover, the experimental order of convergence is O(h 2 ) for both, ȳ ȳ h L 2 () and ϕ ϕ h L 2 (), and O(h) for ȳ ȳ h H 1 (). Next we report on two other discretization techniques known from the literature. In the first approach, the finite dimensional set U h of discrete controls consists of piecewise linear continuous functions, i.e. U h = U lin h := {u L () u C() and u P 1 ((x j, x j+1 )) for 1 j N(h)}. In the context of semilinear elliptic control problems, this kind of discretization was studied already by Casas and Mateos [27] for Neumann controls, and Casas [16] for distributed controls. We mention also the paper by Rösch [84]. In view of the analysis, this approach is more delicate, since there is no pointwise projection formula for the optimal control and its approximation such as (3.3.12) on page 81 and (4.2.16). In both cases, Neumann and distributed controls, superlinear convergence was proved for the optimal controls in the L 2 norm. Moreover, if the cost function J is quadratic w.r.t. the control variable then convergence of order O(h 3 /2 ) and O(h) for the controls in L 2  and L norms were obtained, cf. [27]. For the numerical realization of this approach we have used the same optimization code as for the piecewise constant approximation, with obvious modifications. The second approach is the socalled variational discretization, where only the state and the adjoint state are discretized, i.e. U h = L (), cf. Hinze [63]. Nevertheless, the optimal control is obtained by a projection of the adjoint state to the set of admissible controls, therefore it is piecewise linear. Replacing the optimal control by this projection, the optimality system reduces to the system containing the state and adjoint state equation. For linearquadratic distributed or Neumann control problems numerous numerical tests have shown a quadratic convergence for the optimal controls in the L 2 norm, cf. Hinze [63], Hinze and Matthes [65]. To solve numerically the optimality system we have used the commercial finite element solver of
127 4.4. NUMERICAL EXPERIMENTS 119 the equationbased modeling and simulation environment COMSOL Multiphysics 1. We had two reasons for choosing this solver for our purposes. First, it allows a very simple implementation of the coupled system. Secondly, it is much faster than our own optimization routines, since it follows the socalled allatonce approach, where the optimality system is solved at once. Finally, we refer to Casas and Tröltzsch [38], where both strategies are treated for the distributed control of (1.1.1) with homogeneous Dirichlet boundary conditions. The authors have derived the order O(h σ ) of the L 2 error for the optimal controls, with σ = 3/2 when J is quadratic w.r.t u and the controls are approximated by piecewise linear functions and σ = 2 for the case of variational discretization. By applying these discretization methods to Example 4.14, we want to check if we can observe convergence rates for the optimal controls similar to those for semilinear problems in convex domains. Table 4.2 illustrates the results obtained from the tests of the approaches described above. Obviously, these results reflect exactly the U h = U lin h h EOC L 2 ()(ū) EOC L ()(ū) h U h = L () EOC L 2 ()(ū) Table 4.2. Experimental convergence rates for the optimal controls. order of convergence observed in the case of semilinear Neumann control problems in convex domains, cf. [27] and [65]. Example We fix = (0, 1) (0, 0.5) (0.5, 1) (0, 1) (compare also Example 2.21 on page 58), α = 0, β = 1.8, η 0, g 0 on, a(x, y(x)) = 1+(x 1 +x 2 ) 2 +y 2 (x) and f(x, y(x)) = y + y 3 (x). Furthermore, we set u d (x) = u d (x 1, x 2 ) = Proj [α,β] { x1 x 2 1 /6 + x 1 } and y d is equal to the numerical approximation of y ud on the same mesh on which we have made the computations, i.e. y d := y h (u d ). We have taken y 1 1, ϕ 1 1 and u 1 (α + β)/2 as initial data to run the SQP method. Figure 4.2 shows u d and y h (u d ) with h = 2 7. Obviously, y d y ud as 1 COMSOL Multiphysics,
128 NUMERICAL APPROXIMATION OF THE CONTROL PROBLEM Figure 4.2. Desired control u d (left frame) and associated numerical solution y h (u d ) for h = 2 7 (right frame). h tends to zero and the rate of this convergence is O(h); see Theorem 2.6 on page 41. On the other hand, since we do not know any solution ū of the control problem (E), we have taken as reference solution the numerical solution ū hfine computed on a very fine grid with mesh size h fine = 2 8. Hence, instead of studying the behavior of the error ū ū h X when h 0, we have considered the error ū hfine ū h X, where X = L 2 () or X = L (). We should remark that the order of convergence of the optimal controls in both norms is not affected by the fact that y d is not fixed but depends on h. This is due to the fact that the order of convergence of y d to y ud in the L 2 () norm is better than the expected ones for the optimal controls. More precisely, according to Theorems 4.10 and 4.12, we expect to see numerically the orders O(h 1 /2 ) and O(h 1 /2 τ ) for the optimal controls in the L 2 () and L () norms, respectively, while y d y ud L 2 () = O(h). Table 4.3 shows the convergence speed of ū hfine ū h in the L 2 () and L () norms. We do not report on the convergence rates for the optimal states and adjoint states because of their dependence on the order of the convergence y d y ud as h tends to zero. h EOC L 2 ()(ū hfine ) EOC L ()(ū hfine ) Table 4.3. Experimental convergence rates for the optimal controls. A close look at Table 4.3 reveals that the convergence behavior for the optimal controls is comparable to the one predicted in Theorem 4.12, while the convergence rate in the L 2 () norm is slightly better than the one obtained in Theorem 4.10.
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