# On Weak Solvability of Boundary Value Problems for Elliptic Systems

Save this PDF as:

Size: px
Start display at page:

## Transcription

2 19 FELIPE PONCE, LEONID LEBEDE & LEONARDO RENDÓN 1. Introduction Systems of elliptic partial differential equations arise frequently in problems of continuum mechanics. To formulate boundary value problems BPs, we should supply the equations with some boundary conditions. We will consider the linear partial differential equations derived from an energy type functional. We will get solutions to a BP minimizing the energy type functional. Such a solution will be called weak. As we expect to apply the results to physical problems we use the terminology like energy, displacements, etc. In general case solution of a BP will be reduced to the minimization problem for a functional of the form Eu = 1 Bu, u fu in a Hilbert space H, where B is a symmetric bilinear form and f is a linear functional in H. It is well know that Theorem 1.1 [15, 16]. Let B be a continuous symmetric bilinear form in a Hilbert space H. If f is a continuous linear functional and Bu, u C u for every u H and some constant C > 0, then E attains a unique minimum in H. Furthermore, u 0 is the minimum point if and only if it is a unique solution to equation Bu 0, v = fv 1 for every v H. This simple result allows us to prove existence uniqueness theorems for mechanical problems; it motivates the following definition. Definition 1.. A bilinear form B in a Hilbert space is coercive if Bu, u C u for every u H and some constant C. In the proof of existence and uniqueness of weak solutions of the problems under consideration, the most troublesome point is to establish the coerciveness of B. In linear elasticity such coerciveness is also called Korn s inequality which was first proved in [13, 14]. Subsequent generalizations of Korn s inequality can be found in [9, 5, 17, 11, 6, 1]. One of the most important latest papers on Korn s inequality is due to. A. Kondrat ev and O. A. Oleynik [1], where it is established in a general form. Similar problems frequently arise in equilibrium or stationary linear boundary value physical problems [3, 4, 7, 8, 10]. We extend some known results for linear elasticity to a general case and consider a general coerciveness problem of B, construction of abstract energy spaces related to B as well as the weak setup of corresponding boundary value problems and their solvability. Assume that Ω is a connected bounded open set in R n with piecewise C 1 boundary Ω. Let B be a positive bilinear form, that is, if B u, u = 0 for a smooth vector function u : Ω R n R n, such that u Γ = 0 for some open set Γ Ω, it follows that u = 0. Later we will show when this condition is olumen 47, Número, Año 013

3 SOLABILITY OF ELLIPTIC SYSTEMS 193 valid. The linear space of smooth functions with u Γ = 0 constitutes an inner product space if we take B u, v as an inner product. Completion of this space with respect to the norm induced by the inner product is an energy space. The bilinear form used for the inner product is B u, v = m a ij xl i u, u, x Lj v, v, x dx where a ij L Ω are components of a symmetric positive definite matrix and the linear forms L i are L i u, u, x = n k,l=1 b kl i x k u l + c k i xu k. Let f : Ω R n R n and g : Ω Γ R n be vector functions, these define the energy functional E u = 1 B u, u f u dx g u ds. 3 Ω Ω Γ k=1 By analogy with elasticity equations, we present B in the form B u, v = [ s Ω t k,s l d klst k u l + q kst u k + q tkl k u l + p kt u k ]v t dx + k,l k [ Ω Γ t k,l,s d klst k u l n s + k,s q kst u k n s ]v t ds, where ds is the surface area element, n the normal vector to the surface and the coefficients are d klst = i,j a ij b kl i b st j, q kst = i,j a ij c k i b st j, p kt = i,j a ij c k i c t j. If the functions involved in E are sufficiently smooth and satisfy the boundary condition, functional E can be obtained from integrating by parts. Using standard methods of calculus of variations, assuming existence of a smooth minimizer u of E and minimizing E over the set of smooth functions equal to zero on Γ, we get the corresponding system of partial differential equations s d klst k u l + q kst u k + q tkl k u l + p kt u k = f t 4 k,s l k,l k Revista Colombiana de Matemáticas

4 194 FELIPE PONCE, LEONID LEBEDE & LEONARDO RENDÓN for t = 1,..., n. Moreover, on Ω Γ we obtain the natural condition d klst k u l + q kst u k n s = g t 5 k,s l for t = 1,..., n, that is analogous to Neumann s condition for the Laplace equation, whereas on Γ we have Dirichlet condition u Γ = 0. 6 Definition 1.3. A vector function u is a weak solution of the system in 4 with boundary conditions 5 and 6 if it is a minimum point of the energy functional E u in 3. We will present some sufficient conditions for coerciveness and continuity of B in the Hilbert space H of vector functions in [ W 1, Ω ] n with zero trace value in Γ Ω. In this way we will show that the energy norm B u, u 1 is an equivalent norm in H, which in turn insure existence and uniqueness of a weak solution of the boundary value problem 4 6 in the space H. We apply this result to a particular type of bilinear forms, generalizing a result from the theory of shallow shells [18] and consider an example of boundary value problems for a class of elliptic equations including the equations of linear elasticity. Notation. The norm of a vector field u in [ L q Ω ] n = L q Ω L q Ω and the L Ω norm of u are, respectively, u q = { Ω i=1 u i q } 1 q and u { = i u j dx The norm of [ W 1, Ω ] n = W 1, Ω W 1, Ω is u 1, = { u + u } 1.. Equivalence of Energy and Sobolev s Norms Let us rewrite the symmetric bilinear functional B u, v m = a ij xl i u, u, x Lj v, v, x dx 7 where a ij is a symmetric positive definite matrix almost everywhere a.e. and the linear forms L i are n L i u, u, x = b kl i x k u l + c k i xu k. k,l=1 k=1 } 1. olumen 47, Número, Año 013

5 We can also write SOLABILITY OF ELLIPTIC SYSTEMS 195 L i u, u, x = Mi u, x + Ni u, x, where M i and N i are linear forms. In what follows, we assume that a ij, b kl i L Ω 8 c k i L q Ω for n < q. 9 By the positive definiteness of a ij, there exists a positive bounded a.e. function h such that B u, u m hx L i u, u, x dx. Ω i=1 From this, it is easy to prove that B is positive definite if and only if L i u, u, x = 0 a.e. for every i implies that u = 0, a.e. We will use the following classical properties of Sobolev spaces Theorem.1 The Embedding theorem []. Suppose that Ω is a bounded open set satisfying the cone condition. Then i If n =, W 1, Ω is continuously and compactly embedded in L q Ω, for 1 q < ; that is, for every u W 1, Ω, there exists a constant C such that u q C u 1, and if u n u in W 1, Ω, then u n 0 in L q Ω. ii If n >, W 1, Ω is continuously embedded in L q Ω, for 1 q and compactly embedded in L q Ω, for 1 q < n n. n n, The continuous embedding is known as Sobolev embedding theorem, and the compact embedding as Rellich-Kondrachov theorem. The goal of this section is to show that the terms M i, which include only first order derivatives, determine the equivalence of the energy norm with the Sobolev s norm. Theorem.. If B is the bilinear form given by the Equation 7 and m a ij xm i u, x Mj u, x dx C u, 10 then the energy norm B u, u 1 is equivalent to the norm of H. Corollary.3. Suppose that B satisfies the hypothesis of Theorem. and let f : Ω R n R n and g : Ω Γ R n be vector functions. If Revista Colombiana de Matemáticas

6 196 FELIPE PONCE, LEONID LEBEDE & LEONARDO RENDÓN f [ L p Ω ] n and g [ L p Ω Γ ] n for 1 < p, p, if n =, [ ] n n, f L n n+ Ω and g [L n 1 n Ω Γ] if n >. Then for the corresponding boundary value problem 4 6 there exists a unique weak solution in H. Proof. It is a consequence of Theorem 1.1 and the continuity of the functionals Ω f v dx and Ω Γ g v dx in [ W 1, Ω ] n. Inequality 10 is of Korn s type. The name comes from the classical Korn s inequality in elasticity, which depends strongly on the fact that the vector field is zero on some open subset of the boundary i v j + j v i dx C v. Before proving Theorem., we need the following lemma. Lemma.4. Suppose that Ω is a bounded open set satisfying the cone condition and Z [ W 1, Ω ] n is a closed subspace. Let B be a continuous positive definite bilinear form in Z. If the conditions v k 0 in [ W 1, Ω ] n and B v k, v k 0 imply that vk 0, then for some constant C. B v, v C v 1, 11 Proof. Suppose contrarily that B v, v C v 1, is false for any C. Then we can find a sequence { v k } such that B vk, v k 0 and vk 1, = 1. As Z is a Hilbert space and { v k } is bounded, we can assume that the sequence converges weakly to some vector v 0. As B, v 0 is a continuous functional, by positive definiteness of B, 0 B v k v 0, v k v 0 B vk, v k B vk, v 0 + B v0, v 0. Taking the limit when k we get 0 Bv 0, v 0. So, B v 0, v 0 = 0, which implies that v0 = 0. Thus v k 0 in [ W 1, Ω ] n. As B v k, v k 0. by the assumptions we conclude that vk 0. As v k 0 in [ W 1, Ω ] n, by Rellich Kondrachov s theorem we see that v k 0 in [ L Ω ] n. Thus vk 0 in [ W 1, Ω ] n, which contradicts the equality vk 1, = 1. olumen 47, Número, Año 013

7 SOLABILITY OF ELLIPTIC SYSTEMS 197 Proof of Theorem.. To verify the continuity of the bilinear form B, by the symmetry of B it suffices to show that B u, u C u 1,. Choosing p according to 9 and q such that 1 p + 1 q = 1, for the components of the bilinear form we have a ij b kl i b st j k u l s u t dx 1 a ij b kl i b st j k u l + s u t Ω Ω C 1 u 1,, 1 a ij b kl i c s j k u l u s dx a ij b kl i c s j p k u l u s q Ω C u 1,, 13 a ij c k i c s ju k u s dx 1 a ij c k i p u k q + c s j u p s q So we conclude that B is continuous. C 3 u 1,. 14 By Lemma.4, now we only need to prove that u k 0 in [ W 1, Ω ] n and B u k, u k 0 together imply that uk 0. We write the bilinear form as follows: B m u k, u k = a ij xm i uk, x M j uk, x dx + m a ij xm i uk, x N j uk, x dx + m a ij xn i uk, x N j uk, x dx. 15 By Rellich-Kondrachov s theorem we know that uk q 0. The first Inequality in 14 shows that the third term on the right-hand side of the Inequality 15 tends to zero. In a similar fashion, by the first inequality in 13 and the boundedness of i u k,j, the second term also tends to zero. Thus, by Equation 10, we get uk 0 as k, which concludes the proof. When we try to prove the coerciveness of B, the uniform positive definiteness of a ij allows us to simplify the proof and to extend theorems on equivalence between energy norms and Sobolev s norms. Definition.5. The terms a ij are uniformly positive definite if m m a ij xξ i ξ j θ i,j=1 i=1 ξ i Revista Colombiana de Matemáticas

8 198 FELIPE PONCE, LEONID LEBEDE & LEONARDO RENDÓN for every x Ω and some constant θ > 0. This stronger requirement implies that B u, u θ Ω m Li u, u, x dx. i=1 In general, it is easier to prove the coerciveness of the form in the right hand side, obtaining thus the coerciveness of the original bilinear form. Moreover, once it is proved that B u, v = m L i u, u, x Li v, v, x dx 16 Ω i=1 is coercive, we can extend this result to a wider class of bilinear forms. Theorem.6. Suppose that a ij x is uniformly positive definite and T x : R m R m m v i t ij xv j t=1 17 is a non-singular linear operator at each point, such that T x 1 is bounded. The bilinear form B defined by m m a ij m t it xl t u, u, x t jt xl t v, v, x dx 18 t=1 is coercive if B in 16 is coercive. Proof. By uniform coerciveness of a ij, the theorem follows from the inequality m m m, t ij xl j u, u, x C Li u, u, x i=1 j=1 where C > 0 is a constant, after integrating over Ω. But this inequality is an easy consequence of [ T x ] v t=1 i=1 1 T x 1 v 1 v C 1 where is the euclidean norm in R m and T x 1 C 1 <. olumen 47, Número, Año 013

9 SOLABILITY OF ELLIPTIC SYSTEMS Generalization of a Bilinear Form in Elasticity The following bilinear form is an extension of the body strain energy in theory of shallow shells, B u, v = Ω i,j,k,l=1 a ijkl xl ij u, u, x Lkl v, v, x dx where a ijkl = a klij L Ω are uniformly positive definite, that is, for some θ > 0 and every x Ω we have i,j,k,l=1 The linear components are a ijkl xξ ij ξ kl θ i,j=1 ξ ij. where c k ij L Ω. L ij v = i v j + j v i + c k ijv k By the uniform positive definiteness of a ijkl, instead of B, we prove the coerciveness of Bu, v = L ij u, u, x Lij v, v, x dx. As noted in the preceding section, by means of Theorem.6, we can extend equivalence of the energy norm B u, u 1 to a greater class of energy norms. Linear forms L ij can be considered as the components in the canonical basis of operator Lv = v + v T + C v, where the components of C v are k ck ij v k. So we have the identity B u, v = Ω tr LuLv T dx. 19 An additional advantage of B is its invariance under orthogonal transformations. To see this, suppose {e i } is an orthonormal basis for R n and let e i = i qi i e i define a new orthonormal basis, whose inverse transformation is e i = i qi i e i, where j qj i qk j = δk i. As neither v nor C v depend on a particular basis and the trace of a matrix is invariant under orthogonal transformations, we have from Equation 19 that Revista Colombiana de Matemáticas

10 00 FELIPE PONCE, LEONID LEBEDE & LEONARDO RENDÓN i v j + j v i + c k ijv k dx = k=1 Ω i,j =1 i v j + j v i + k =1 c k i j v k dx where c k i j = i,j,k ck ij qk k qi i qj j and Ω is the domain in the new coordinates. Since the norm is an invariant, it can be seen that c k i j i,j,k=1 c k ij = C. As equation 10 holds by the classical Korn s inequality, it remains to prove that B is positive definite. Theorem 3.1. If Bu, u = 0, then u = 0 a.e. Proof. If Bu, u = 0, then for any basis of R n i u i = k =1 c k i i u k. Let us draw, rotating if it is necessary, a hypercube with side of length L such that Γ passes through adjacent sides of the hypercube Figure 1a. The set of points in Ω enclosed by Γ and the hypercube is called. The faces of the hypercube in Ω are the planes x i = a i. Let n i be the components of the normal vector to the surface. The function x i a i u i n i is zero on. Thus, using Gauss-Green formula, we get i x i a i u i dx = u i + x i a i u i i u i dx = 0. It follows u i dx L = L x i a i u i i u i dx x i a i u i k =1 c k c i i i + k =1 c k i i u k dx i i u i + u k dx k =1 c k i i u i + k i c k i i u k dx olumen 47, Número, Año 013

11 SOLABILITY OF ELLIPTIC SYSTEMS 01 L k =1 c k i i nl C u dx. u dx Taking the sum over all the components of u we have u dx n L C u dx. 0 Choosing L small enough so that n L C < 1, we get u = 0 a.e. in. This inequality makes sense for u [ W 1, Ω ] n. It is important to note that the length L does not depend on the location of the hyperrectangles in Ω. As for every point in Γ there exists a neighborhood where u = 0 a.e., then in a neighborhood Ω of Γ we have u = 0 a.e.. From this neighborhood, using hyperrectangles as in Figure 1b we can extend the equality u = 0 a.e.. The set Ω can be covered by a net of hyperrectangles stemming from Ω, covering a subset Ω 1 Figure. Next, we use smaller hyperrectangles covering a greater set Ω Ω and so on, using at most countable many hyperrectangles, obtaining u = 0 a.e. in Ω. a Area of integration. b Extension by rectangles. Figure 1 The system of equations related to B is l d ijkl e ij + s i,j c s iju s + i,j q ijk e ij + s c s iju s = f k 1 Revista Colombiana de Matemáticas

12 0 FELIPE PONCE, LEONID LEBEDE & LEONARDO RENDÓN Γ Figure. The function u is zero in the region Ω 1, depicted by gray rectangles. where d ijkl = a ijkl + a ijlk, q ijk = s,t aijst c k st and e ij = i u j + j u i. Natural conditions are d ijkl e ij + c s iju s n l = g k. s i,j,l So we have established the following theorem. Theorem 3.. Let f : Ω R n R n and g : Ω Γ R n be vector functions such that f [ L p Ω ] n and g [ L p Ω Γ ] n for 1 < p, p, if n =, [ ] n n, f L n n+ Ω and g [L n 1 n Ω Γ] if n >. Then for the system of partial differential equations 1 with boundary conditions 6 and exists a unique solution in H. Acknowledgements. This work was supported by Universidad Nacional de Colombia, Bogotá, under grant No The first author was supported by a grant from MAZDA Foundation for Art and Science. References [1] G. Acosta, R. G. Durán, and F. L. García, Korn Inequality and Divergence Operator: Counterexamples and Optimality of Weighted Estimates, Proc. Amer. Math. Soc , [] R. A. Adams and J. F. Fournier, Sobolev Spaces, nd ed., Elsevier, Amsterdam, The Netherlands, 003. olumen 47, Número, Año 013

13 SOLABILITY OF ELLIPTIC SYSTEMS 03 [3] H. Altenbach,. A. Eremeyev, and L. P. Lebedev, On the Existence of Solution in the Linear Elasticity with Surface Stresses, ZAMM , [4], On the Spectrum and Stiffness of an Elastic Body with Surface Stresses, ZAMM , [5] P. G. Ciarlet, Mathematical Elasticity. ol. I: Three-Dimensional Elasticity, vol. 1, Elsevier, Amsterdam, The Netherlands, [6] S. Conti, D. Faraco, and F. Maggi, A New Approach to Counterexamples to L 1 Estimates: Korn s Inequality, Geometric Rigidity, and Regularity for Gradients of Separately Convex Functions, Arch. Rational Mech. Anal , [7] H. L. Duan, J. Wang, and B. L. Karihaloo, Theory of Elasticity at the Nanoscale, Advances in Applied Mechanics 4 009, [8] G. Fichera, Handbuch der Physik, ch. Existence Theorems in Elasticity, Springer, Berlin, Germany, 197 de. [9] K. O. Friederichs, On the Boundary-alue Problems of the Theory of Elasticity and Korn s Inequality, Ann. of Math , [10] M. E. Gurtin and A. I. Murdoch, A Continuum Theory of Elastic Material Surfaces, Arch. Rat. Mech. Analysis , [11] C. O. Horgan, Korn s Inequalities and Their Applications in Continuum Mechanics, SIAM rev , no. 4, [1]. A. Kondrat ev and O. A. Oleynik, Boundary-value Problems for the System of Elasticity Theory in Unbounded Domains. Korn s Inequalities, Russ. Math. Surv , no. 5, [13] A. Korn, Solution générale du problème d équilibre dans la théorie de l élasticité dans le cas où les efforts sont donnès à la surface, Ann. Fac. Sci. Toulouse , no., fr, in french. [14], Über einige Ungleichungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen, Bull. Int. Cracovie Akademie Umiejet, Classe des Sci. Math. Nat. 1909, de, in german. [15] L. P. Lebedev, I. I. orovich, and M. J. Cloud, Functional Analysis in Mechanics, nd ed., Springer-erlag, New York, USA, 013. [16] S. G. Mikhlin, The Problem of the Minimum of a Quadratic Functional, Governmental publisher, Moscow, Rusia, 195, in russian. Revista Colombiana de Matemáticas

14 04 FELIPE PONCE, LEONID LEBEDE & LEONARDO RENDÓN [17] J. Necas and I. Hlavácek, Mathematical Theory of Elastic and Elasto- Plastic Bodies: An Introduction, Elsevier, [18] I. I. orovich, Nonlinear Theory of Shallow Shells, Springer-erlag, New York, USA, Recibido en mayo de 013. Aceptado en septiembre de 013 Departamento de Matemáticas Universidad Nacional de Colombia Facultad de Ciencias Carrera 30, calle 45 Bogotá, Colombia olumen 47, Número, Año 013

### Multiplicative Relaxation with respect to Thompson s Metric

Revista Colombiana de Matemáticas Volumen 48(2042, páginas 2-27 Multiplicative Relaxation with respect to Thompson s Metric Relajamiento multiplicativo con respecto a la métrica de Thompson Gerd Herzog

### The Stekloff Problem for Rotationally Invariant Metrics on the Ball

Revista Colombiana de Matemáticas Volumen 47(2032, páginas 8-90 The Steklo Problem or Rotationally Invariant Metrics on the Ball El problema de Steklo para métricas rotacionalmente invariantes en la bola

### A Simple Observation Concerning Contraction Mappings

Revista Colombiana de Matemáticas Volumen 46202)2, páginas 229-233 A Simple Observation Concerning Contraction Mappings Una simple observación acerca de las contracciones German Lozada-Cruz a Universidade

### On the existence of multiple principal eigenvalues for some indefinite linear eigenvalue problems

RACSAM Rev. R. Acad. Cien. Serie A. Mat. VOL. 97 (3), 2003, pp. 461 466 Matemática Aplicada / Applied Mathematics Comunicación Preliminar / Preliminary Communication On the existence of multiple principal

### Regularity of the solutions for a Robin problem and some applications

Revista Colombiana de Matemáticas Volumen 42(2008)2, páginas 127-144 Regularity of the solutions for a Robin problem and some applications Regularidad de las soluciones para un problema de Robin y algunas

### A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of Rou-Huai Wang

A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of Rou-Huai Wang 1. Introduction In this note we consider semistable

### AND A NEW PROOF OF KORN S INEQUALITY

Mathematical Models and Methods in Applied Sciences Vol. 15, No. 2 (2005) 259 271 c World Scientific Publishing Company ANOTHER APPROACH TO LINEARIZED ELASTICITY AND A NEW PROOF OF KORN S INEQUALITY PHILIPPE

### 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is:

CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is: with coefficients given by: a n = 1 π f(x) a 0 2 + a n cos(nx) + b n sin(nx), n 1 f(x) cos(nx)dx

### ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction

ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZ-PÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that

### Galerkin Approximations and Finite Element Methods

Galerkin Approximations and Finite Element Methods Ricardo G. Durán 1 1 Departamento de Matemática, Facultad de Ciencias Exactas, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina. Chapter 1 Galerkin

### A simple application of the implicit function theorem

Boletín de la Asociación Matemática Venezolana, Vol. XIX, No. 1 (2012) 71 DIVULGACIÓN MATEMÁTICA A simple application of the implicit function theorem Germán Lozada-Cruz Abstract. In this note we show

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### Høgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver

Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin e-mail: chechkin@mech.math.msu.su Narvik 6 PART I Task. Consider two-point

### THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

### LECTURE NOTES: FINITE ELEMENT METHOD

LECTURE NOTES: FINITE ELEMENT METHOD AXEL MÅLQVIST. Motivation The finite element method has two main strengths... Geometry. Very complex geometries can be used. This is probably the main reason why finite

### MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.

MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α

### 2.3 Convex Constrained Optimization Problems

42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

### Extracting the roots of septics by polynomial decomposition

Lecturas Matemáticas Volumen 29 (2008), páginas 5 12 ISSN 0120 1980 Extracting the roots of septics by polynomial decomposition Raghavendra G. Kulkarni HMC Division, Bharat Electronics Ltd., Bangalore,

### CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian. Pasquale Candito and Giuseppina D Aguí

Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian Pasquale

### Numerical Solutions to Differential Equations

Numerical Solutions to Differential Equations Lecture Notes The Finite Element Method #2 Peter Blomgren, blomgren.peter@gmail.com Department of Mathematics and Statistics Dynamical Systems Group Computational

### A COMMENT ON LEAST-SQUARES FINITE ELEMENT METHODS WITH MINIMUM REGULARITY ASSUMPTIONS

INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 10, Number 4, Pages 899 903 c 2013 Institute for Scientific Computing and Information A COMMENT ON LEAST-SQUARES FINITE ELEMENT METHODS WITH

### Powers of Two in Generalized Fibonacci Sequences

Revista Colombiana de Matemáticas Volumen 462012)1, páginas 67-79 Powers of Two in Generalized Fibonacci Sequences Potencias de dos en sucesiones generalizadas de Fibonacci Jhon J. Bravo 1,a,B, Florian

### INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES

INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES FERNANDO BOMBAL AND IGNACIO VILLANUEVA Abstract. We study and characterize the integral multilinear operators on a product of C(K) spaces in terms of the

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### Publikationsliste. 1 Referierte Zeitschriftenartikel

Publikationsliste 1 Referierte Zeitschriftenartikel [1 ] An estimate for the maximum of solutions of parabolic equations with the Venttsel condition, Vestnik Leningrad. Univ. (Ser. Mat. Mekh. Astronom.,

### Extremal equilibria for reaction diffusion equations in bounded domains and applications.

Extremal equilibria for reaction diffusion equations in bounded domains and applications. Aníbal Rodríguez-Bernal Alejandro Vidal-López Departamento de Matemática Aplicada Universidad Complutense de Madrid,

### AFED - Fourier Analysis and Differencial Equations

Coordinating unit: Teaching unit: Academic year: Degree: ECTS credits: 2016 205 - ESEIAAT - Terrassa School of Industrial, Aerospace and Audiovisual Engineering 749 - MAT - Department of Mathematics BACHELOR'S

### ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING. 0. Introduction

ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING Abstract. In [1] there was proved a theorem concerning the continuity of the composition mapping, and there was announced a theorem on sequential continuity

### LEVERRIER-FADEEV ALGORITHM AND CLASSICAL ORTHOGONAL POLYNOMIALS

This is a reprint from Revista de la Academia Colombiana de Ciencias Vol 8 (106 (004, 39-47 LEVERRIER-FADEEV ALGORITHM AND CLASSICAL ORTHOGONAL POLYNOMIALS by J Hernández, F Marcellán & C Rodríguez LEVERRIER-FADEEV

### Shape Optimization Problems over Classes of Convex Domains

Shape Optimization Problems over Classes of Convex Domains Giuseppe BUTTAZZO Dipartimento di Matematica Via Buonarroti, 2 56127 PISA ITALY e-mail: buttazzo@sab.sns.it Paolo GUASONI Scuola Normale Superiore

### Theory of Sobolev Multipliers

Vladimir G. Maz'ya Tatyana O. Shaposhnikova Theory of Sobolev Multipliers With Applications to Differential and Integral Operators ^ Springer Introduction Part I Description and Properties of Multipliers

### Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

### 1 Introduction, Notation and Statement of the main results

XX Congreso de Ecuaciones Diferenciales y Aplicaciones X Congreso de Matemática Aplicada Sevilla, 24-28 septiembre 2007 (pp. 1 6) Skew-product maps with base having closed set of periodic points. Juan

### OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN. E.V. Grigorieva. E.N. Khailov

DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Supplement Volume 2005 pp. 345 354 OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN E.V. Grigorieva Department of Mathematics

### F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)

Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p -CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p

### CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation

Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in

### The Positive Supercyclicity Theorem

E extracta mathematicae Vol. 19, Núm. 1, 145 149 (2004) V Curso Espacios de Banach y Operadores. Laredo, Agosto de 2003. The Positive Supercyclicity Theorem F. León Saavedra Departamento de Matemáticas,

### Duality of linear conic problems

Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

### Error estimates for nearly degenerate finite elements

Error estimates for nearly degenerate finite elements Pierre Jamet In RAIRO: Analyse Numérique, Vol 10, No 3, March 1976, p. 43 61 Abstract We study a property which is satisfied by most commonly used

### PARTIAL DIFFERENTIAL EQUATIONS

CHAPTER 1 PARTIAL DIFFERENTIAL EQUATIONS Many natural processes can be sufficiently well described on the macroscopic level, without taking into account the individual behavior of molecules, atoms, electrons,

### 17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function

17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function, : V V R, which is symmetric, that is u, v = v, u. bilinear, that is linear (in both factors):

### Høgskolen i Narvik Sivilingeniørutdanningen

Høgskolen i Narvik Sivilingeniørutdanningen Eksamen i Faget STE6237 ELEMENTMETODEN Klassen: 4.ID 4.IT Dato: 8.8.25 Tid: Kl. 9. 2. Tillatte hjelpemidler under eksamen: Kalkulator. Bok Numerical solution

Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650

### Systems with Persistent Memory: the Observation Inequality Problems and Solutions

Chapter 6 Systems with Persistent Memory: the Observation Inequality Problems and Solutions Facts that are recalled in the problems wt) = ut) + 1 c A 1 s ] R c t s)) hws) + Ks r)wr)dr ds. 6.1) w = w +

### We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.

Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to

### IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

### Computing divisors and common multiples of quasi-linear ordinary differential equations

Computing divisors and common multiples of quasi-linear ordinary differential equations Dima Grigoriev CNRS, Mathématiques, Université de Lille Villeneuve d Ascq, 59655, France Dmitry.Grigoryev@math.univ-lille1.fr

### Row Ideals and Fibers of Morphisms

Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Vectors, Gradient, Divergence and Curl.

Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use

### A RIGOROUS AND COMPLETED STATEMENT ON HELMHOLTZ THEOREM

Progress In Electromagnetics Research, PIER 69, 287 304, 2007 A RIGOROU AND COMPLETED TATEMENT ON HELMHOLTZ THEOREM Y. F. Gui and W. B. Dou tate Key Lab of Millimeter Waves outheast University Nanjing,

### EXISTENCE OF SOLUTIONS TO NONLINEAR, SUBCRITICAL HIGHER-ORDER ELLIPTIC DIRICHLET PROBLEMS WOLFGANG REICHEL AND TOBIAS WETH

EXISTENCE OF SOLUTIONS TO NONLINEAR, SUBCRITICAL HIGHER-ORDER ELLIPTIC DIRICHLET PROBLEMS WOLFGANG REICHEL AND TOBIAS WETH Abstract. We consider the -th order elliptic boundary value problem Lu = f(x,

### Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

### Chapter 5: Application: Fourier Series

321 28 9 Chapter 5: Application: Fourier Series For lack of time, this chapter is only an outline of some applications of Functional Analysis and some proofs are not complete. 5.1 Definition. If f L 1

### Inner product. Definition of inner product

Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product

### A MIXED TYPE IDENTIFICATION PROBLEM RELATED TO A PHASE-FIELD MODEL WITH MEMORY

Guidetti, D. and Lorenzi, A. Osaka J. Math. 44 (27), 579 613 A MIXED TYPE IDENTIFICATION PROBLEM RELATED TO A PHASE-FIELD MODEL WITH MEMORY DAVIDE GUIDETTI and ALFREDO LORENZI (Received January 23, 26,

### On Spectral Compactness of Von Neumann Regular Rings

Revista Colombiana de Matemáticas Volumen 4620121, páginas 81-95 On Spectral Compactness of Von Neumann Regular Rings Sobre la compacidad espectral de los anillos regulares de von Neumann Ibeth Marcela

### PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION

STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume L, Number 3, September 2005 PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION YAVUZ ALTIN AYŞEGÜL GÖKHAN HIFSI ALTINOK Abstract.

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

### MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

### UNIVERSITETET I OSLO

NIVERSITETET I OSLO Det matematisk-naturvitenskapelige fakultet Examination in: Trial exam Partial differential equations and Sobolev spaces I. Day of examination: November 18. 2009. Examination hours:

### m = 4 m = 5 x x x t = 0 u = x 2 + 2t t < 0 t > 1 t = 0 u = x 3 + 6tx t > 1 t < 0

ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 21, Number 2, Spring 1991 NODAL PROPERTIES OF SOLUTIONS OF PARABOLIC EQUATIONS SIGURD ANGENENT * 1. Introduction. In this note we review the known facts about

### Mathematical Methods of Engineering Analysis

Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Monotone maps of R n are quasiconformal

Monotone maps of R n are quasiconformal K. Astala, T. Iwaniec and G. Martin For Neil Trudinger Abstract We give a new and completely elementary proof showing that a δ monotone mapping of R n, n is K quasiconformal

### Stiffness Matrices of Isoparametric Four-node Finite Elements by Exact Analytical Integration

Stiffness Matrices of Isoparametric Four-node Finite Elements by Exact Analytical Integration Key words: Gautam Dasgupta, Member ASCE Columbia University, New York, NY C ++ code, convex quadrilateral element,

### Absolute Value Programming

Computational Optimization and Aplications,, 1 11 (2006) c 2006 Springer Verlag, Boston. Manufactured in The Netherlands. Absolute Value Programming O. L. MANGASARIAN olvi@cs.wisc.edu Computer Sciences

### Labeling outerplanar graphs with maximum degree three

Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics

### CONTRIBUTIONS TO ZERO SUM PROBLEMS

CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg

### Elasticity Theory Basics

G22.3033-002: Topics in Computer Graphics: Lecture #7 Geometric Modeling New York University Elasticity Theory Basics Lecture #7: 20 October 2003 Lecturer: Denis Zorin Scribe: Adrian Secord, Yotam Gingold

### Class Meeting # 1: Introduction to PDEs

MATH 18.152 COURSE NOTES - CLASS MEETING # 1 18.152 Introduction to PDEs, Fall 2011 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u = u(x

### Properties of BMO functions whose reciprocals are also BMO

Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,and

### Notes on weak convergence (MAT Spring 2006)

Notes on weak convergence (MAT4380 - Spring 2006) Kenneth H. Karlsen (CMA) February 2, 2006 1 Weak convergence In what follows, let denote an open, bounded, smooth subset of R N with N 2. We assume 1 p

### PIECEWISE LINEAR FINITE ELEMENT METHODS ARE NOT LOCALIZED

MATHEMATICS OF COMPUTATION Volume 73, Number 247, Pages 1195 1 S 25-5718(3)1584-9 Article electronically published on July 14, 23 PIECEWISE LINEAR FINITE ELEMENT METHODS ARE NOT LOCALIZED ALAN DEMLOW Abstract.

### 1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

### Remark on Rotation of Bilinear Vector Fields

Remark on Rotation of Bilinear Vector Fields A.M. Krasnosel skii Institute for Information Transmission Problems, Moscow, Russia E-mail: Sashaamk@iitp.ru A.V. Pokrovskii National University of Ireland

### Introduction to Flocking {Stochastic Matrices}

Supelec EECI Graduate School in Control Introduction to Flocking {Stochastic Matrices} A. S. Morse Yale University Gif sur - Yvette May 21, 2012 CRAIG REYNOLDS - 1987 BOIDS The Lion King CRAIG REYNOLDS

### On a tree graph dened by a set of cycles

Discrete Mathematics 271 (2003) 303 310 www.elsevier.com/locate/disc Note On a tree graph dened by a set of cycles Xueliang Li a,vctor Neumann-Lara b, Eduardo Rivera-Campo c;1 a Center for Combinatorics,

### SOME METHODS FOR CALCULATING STIFFNESS PROPERTIES OF PERIODIC STRUCTURES

48 (2003) APPLICATIONS OF MATHEMATICS No. 2, 97 110 SOME METHODS FOR CALCULATING STIFFNESS PROPERTIES OF PERIODIC STRUCTURES Stein A. Berggren, Dag Lukkassen, Annette Meidell, and Leon Simula, Narvik (Received

### An optimal transportation problem with import/export taxes on the boundary

An optimal transportation problem with import/export taxes on the boundary Julián Toledo Workshop International sur les Mathématiques et l Environnement Essaouira, November 2012..................... Joint

### Introduction to the Finite Element Method

Introduction to the Finite Element Method 09.06.2009 Outline Motivation Partial Differential Equations (PDEs) Finite Difference Method (FDM) Finite Element Method (FEM) References Motivation Figure: cross

### Some remarks on Phragmén-Lindelöf theorems for weak solutions of the stationary Schrödinger operator

Wan Boundary Value Problems (2015) 2015:239 DOI 10.1186/s13661-015-0508-0 R E S E A R C H Open Access Some remarks on Phragmén-Lindelöf theorems for weak solutions of the stationary Schrödinger operator

### A mixed FEM for the quad-curl eigenvalue problem

Noname manuscript No. (will be inserted by the editor) A mixed FEM for the quad-curl eigenvalue problem Jiguang Sun Received: date / Accepted: date Abstract The quad-curl problem arises in the study of

### Example 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x

Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be written

### Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula:

Chapter 7 Div, grad, and curl 7.1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: ( ϕ ϕ =, ϕ,..., ϕ. (7.1 x 1

### DOCUMENTS DE TREBALL DE LA FACULTAT D ECONOMIA I EMPRESA. Col.lecció d Economia

DOCUMENTS DE TREBALL DE LA FACULTAT D ECONOMIA I EMPRESA Col.lecció d Economia E13/89 On the nucleolus of x assignment games F. Javier Martínez de Albéniz Carles Rafels Neus Ybern F. Javier Martínez de

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### ON INTEGERS EXPRESSIBLE BY SOME SPECIAL LINEAR FORM

Acta Math. Univ. Comenianae Vol. LXXXI, (01), pp. 03 09 03 ON INTEGERS EXPRESSIBLE BY SOME SPECIAL LINEAR FORM A. DUBICKAS and A. NOVIKAS Abstract. Let E(4) be the set of positive integers expressible

### Math Department Student Learning Objectives Updated April, 2014

Math Department Student Learning Objectives Updated April, 2014 Institutional Level Outcomes: Victor Valley College has adopted the following institutional outcomes to define the learning that all students

### Solutions series for some non-harmonic motion equations

Fifth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 10, 2003, pp 115 122. http://ejde.math.swt.edu or http://ejde.math.unt.edu

### Summary of week 8 (Lectures 22, 23 and 24)

WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry

### 9 More on differentiation

Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......

### NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

### Metric Spaces. Chapter 1

Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

### New Results on the Burgers and the Linear Heat Equations in Unbounded Domains. J. I. Díaz and S. González

RACSAM Rev. R. Acad. Cien. Serie A. Mat. VOL. 99 (2), 25, pp. 219 225 Matemática Aplicada / Applied Mathematics Comunicación Preliminar / Preliminary Communication New Results on the Burgers and the Linear

### ON A MIXED SUM-DIFFERENCE EQUATION OF VOLTERRA-FREDHOLM TYPE. 1. Introduction

SARAJEVO JOURNAL OF MATHEMATICS Vol.5 (17) (2009), 55 62 ON A MIXED SUM-DIFFERENCE EQUATION OF VOLTERRA-FREDHOLM TYPE B.. PACHPATTE Abstract. The main objective of this paper is to study some basic properties

### Remark on Rotation of Bilinear Vector Fields

Remark on Rotation of Bilinear Vector Fields A.M. Krasnosel skii Institute for Information Transmission Problems, Moscow, Russia E-mail: amk@iitp.ru A.V. Pokrovskii Department of Applied Mathematics and