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

Save this PDF as:

Size: px
Start display at page:

Download "Regularity of the solutions for a Robin problem and some applications"

## Transcription

1 Revista Colombiana de Matemáticas Volumen 42(2008)2, páginas Regularity of the solutions for a Robin problem and some applications Regularidad de las soluciones para un problema de Robin y algunas aplicaciones Rafael A. Castro T. 1,a 1 Universidad Industrial de Santander, Bucaramanga, Colombia Abstract. In this paper we study the regularity of the solutions for a Robin problem, with a nonlinear term with sub-critical growth respect to a variable. We establish the Sobolev space H 1 () as the orthogonal sum of two subspaces, and we give the first step to demonstrate the existence of solutions of our problem. Key words and phrases. Robin problems, trace operators, variational formulation, weak solutions, Sobolev spaces, bootstrapping, Green formula, orthogonal sum of subspace Mathematics Subject Classification. 35B65, 35J25, 35J60, 35P99. Resumen. En este artículo estudiamos la regularidad de las soluciones de un problema de Robin, con término no lineal con crecimiento subcrítico respecto a una variable. Expresamos el espacio de Sobolev H 1 () como la suma de dos subespacios dando el primer paso para la demostración de existencia de soluciones de nuestro problema. Palabras y frases clave. Problemas de Robin, operador trazo, formulación variacional, soluciones débiles, espacios de Sobolev, argumento iterativo, formula de Green, suma ortogonal de subespacios. a Supported by CAPES (Brazil) and DIF de Ciencias (UIS) Código CB

2 128 RAFAEL A. CASTRO T. 1. Introduction We examine the regularity of the solutions to the problem u H 1 (, ), (P) u = f(x, u(x)), in, γ 1 u + αγ 0 u = 0, on, here is the Laplacian operator, is a bounded domain in R n (n 2) simply connected and with smooth boundary. The function f : R R, satisfies the conditions: f 0 ) The function f is continuous. f 1 ) There exists a constant c > 0 such that f(x, s) c (1 + s σ ), x and s R, where the exponent σ is a constant such that 1 < σ < n+2 n 2 if n 3, 1 < σ < if n = 2. The boundary condition γ 1 u + αγ 0 u = 0 involves the trace operators: γ 0 : H 1 () H 1/2 ( ), and γ 1 : H 1 (, ) H 1/2 ( ), where H 1 (, ) = {u H 1 () : u L 2 ()} with the norm ( 1/2 u H 1 (, ) = u 2 H 1 () + u 2 L ()), 2 for each u H 1 (, ), γ 1 u H 1/2 ( ), and γ 0 u H 1/2 ( ). Identifying the element γ 0 u with the functional γ0u H 1/2 ( ) defined by γ0 u, w = (γ 0 u)wds, w H 1/2 ( ), the boundary condition makes sense in H 1/2 ( ) (Theorem 2.3), and it appears in the functional of the problem (P), see Theorems 3.1, 3.2 and 3.3 in section 3. This work was motivated by Arcoya-Villegas [1], who proved the existence of nontrivial solution of the nonlinear Neumann s problem u = f(x, u), in, u η = 0, on. We study the existence of solutions of the problem (P) in [4]. We write this paper to separate the regularity conditions of the additional existence conditions. Volumen 42, Número 2, Año 2008

3 REGULARITY OF THE SOLUTIONS FOR A ROBIN PROBLEM Preliminary results In this section, we introduce some notations and known theorems to familiarize the reader with the boundary condition γ 1 u+αγ 0 u = 0. We also remember that is an isomorphism between some spaces, properly enunciated in Theorem 2.5. Using the Trace Theorem, Theorem 2.5 and the bootstrapping we prove that the weak solutions of Problem (P) belong to the space H 2 (). (Theorem 3.2). The norm u 2 = u 2 + u 2, (2.1) is the usual norm of the Sobolev space H 1 (). Theorem 2.1. Suppose bounded with boundary of class C 1. Then, there exists a continuous and linear function γ 0 : H 1 () L 2 ( ) such that: a) γ 0 u = u, if u W 1,2 () C ( ), b) γ 0 u L 2 ( ) c 1 u, u H 1 (), here the constant c 1 depends on. The value γ 0 u is called trace of u on. Proof. See [7, Theorem 1, p. 258]. Theorem 2.2 (Trace Theorem). Let R n be open, bounded, and with smooth boundary. Then the trace operators γ j can be extended to continuous linear operators, mapping H m () onto H m j 1/2 ( ), for 0 j m 1. Moreover, the operator γ = (γ 0, γ 1,..., γ j,..., γ m 1) maps H m () onto the product space m 1 j=0 Hm j 1/2 ( ). Finally, the space H0 m () is the Kernel of the operator γ. Proof. See [2, Theorem 3.1, p. 189] or [5, Theorem 5, p. 905]. Theorem 2.3 (Green Formula). There exists a unique operator γ 1 mapping H 1 (, ) into H 1/2 ( ) such that Green Formula holds: γ 1 u, γ 0 v = v u + u v, u H 1 (, ), v H 1 (). (2.2) Remark. The space C ( ) is dense in H 1 (, ). Proof. See [2, Theorem 2.1, p. 174]. Theorem 2.4 (Compactness of the trace operator). Let R n be an open set, bounded, and with boundary C 0,1, m 1 an integer and 1 p < +. Then the next claims are true : a) If mp < n and 1 q < p(n 1) n mp, then there exists a unique mapping γ 0 : W m,p () L q ( ), linear and continuous, such that if u C ( ) then γ 0 u = u. If p > 1 then γ 0 is compact. b) If mp = n, then the affirmation (a) is valid for every q 1. Revista Colombiana de Matemáticas

4 130 RAFAEL A. CASTRO T. c) If mp > n, the trace γ 0 u of u W m,p () C ( ) is the classic restriction u. Proof. See [10, Theorem 6.2 p. 107], [8, Theorem ] or [3]. For the reader s convenience, we recall some notation and definitions of normality and covering. If x = (x 1, x 2,..., x n ) R n and α = (α 1,..., α n ) an n tuple of nonnegative integers α j, then x α denotes the monomial x α1 1 xα2 α = n j=1 α j. Similarly, if D j = x j is the differential operator of order α xαn n, which has degree for 1 j n, then D α = D α Dαn n Let A be a partial differential linear operator of order 2m, defined by Au = ( 1) k D k ( a kh D h u ), (2.3) k, h m where a kh C ( ), with real values. Let B 0,..., B m 1 be m boundary operators defined by B j u = b jh D h u, j = 0, 1,..., m 1, (2.4) h m j where 0 m j 2m 1 and the functions b jh are infinitely differential functions over. Definition 2.1. We say that the set {B j } m 1 j=0 is normal according to Aronszajn and Milgram, if for j i, m j m i, x, and normal vector η 0 to in x, then Q j (x, η) = b jh η h 0, j = 0, 1,..., m 1. h =m j Definition 2.2. The set {B j } m 1 j=0 covers the operator A defined in (2.3), if for each x and every pair of real non-null vectors ξ, ξ tangent and normal to in x respectively, then the polynomial in the variable τ, P (x, ξ + τξ ) = a kh (ξ + τξ ) k+h, k = h =m j has m roots: τ 1 (ξ, ξ ),, τ m (ξ, ξ ), with positive imaginary parts, and the m polynomials in τ Q j (τ) := Q j (x, ξ + τξ ) = b j,h (ξ + τξ ) h, h =m j are linearly independent module the polynomial m i=1 (τ τ i(ξ, ξ )). Definition 2.3. We say that the set of operators satisfies condition (N R) if: {A, B 0, B 1,, B m 1 }, Volumen 42, Número 2, Año 2008

5 REGULARITY OF THE SOLUTIONS FOR A ROBIN PROBLEM 131 a) The set of boundary operators B = {B j : j = 0, 1,, m 1} is normal according to Aronszajn and Milgram, and b) The set B covers operator A. Proposition 2.1. The operators n 2 =, and the boundary operator B 0 u = x 2 j=1 j n j=1 η j (x) u x j, where η(x) = (η 1,, η n ) is a unit normal vector field over and external to, satisfies Condition (N R). Proof. a) Let x and v(x) = (v 1,, v N ) 0 a normal vector field to, N Q 0 (x, η) = η j (x) v j (x) = η v 0, j=1 therefore, the operator {B 0 } is normal according to Aronszajn and Milgram. b) Now, the operator B 0 covers the operator. In fact, let ξ = (ξ 1,, ξ n ) and ξ = (ξ 1,, ξ n ) be tangent and normal vectors respectively to at the point x. By the ellipticity of the operator we have P (x, ξ + τξ ) = ξ + τξ 2 = ξ 2 τ 2 ξ 2, then P (x, ξ + τξ ) has 2 roots: τ 1 = ξ ξ i, and τ 2 = τ 1. On the other hand, the polynomial in τ N Q 0 (x, ξ + τξ ) = n j (x)(ξ j + τξ j ) = (η(x) ξ )τ = η ξ τ, j=1 is not multiple of the polynomial τ τ 1. Let 1 < p <, r 0 and η(x) = (η 1,, η n ) be a real vector normal field over and external to. We define : n N = u W 2,p () : u = 0, B 0 u = η j (x) u = 0 x j=1 j, { } N = u W 2,p () : ( ) u = 0V, B0u = 0, where p = p p 1 and (( ), B0) is the formal adjoint operator of (, B 0 ). It is easy to see that (, B 0 ) is self-adjoint. Spaces N and N are subspaces of C ( ), and thus they neither depend on p nor on p. If is simply connected then N = N is the space of the constants. In this work, the set is simply connected. We will consider the sets: W 2+r,p B 0 () = { u W 2+r,p () : B 0 u = 0 }, and {W r,p (); N } = { u W r,p () : u(x)dx = 0}. Revista Colombiana de Matemáticas

6 132 RAFAEL A. CASTRO T. Theorem 2.5. The operator is an isomorphism on {W r,p (); N } of ()/N for all real r 0 and 1 < p < +. W 2+r,p B 0 Proof. See [9, Theorem 3.1]. 3. Regularity results In Theorem 3.1 we establish the derivative of the functional Φ : H 1 () R, corresponding to our problem (P). In Theorem 3.2 we show that the critical points of Φ belong to the space H 2 (). Finally in Theorem 3.3 we obtain the variational formulation of the problem (P). Thus Theorem 3.2 is a regularity result for weak solutions of the problem (P). Theorem 3.1. The functional Φ : H 1 () R, defined by Φ(u) = 1 u 2 + α (γ 0 u) 2 ds F (x, u), 2 2 where α R, F (x, s) = s 0 f(x, t) dt belongs to class C1 (H 1 (), R) and its derivative is Φ (u), v = u v + α (γ 0 u)(γ 0 v)ds f(x, u)v, u, v H 1 (). (3.1) Proof. If Φ 1 (u) = 1 2 Φ 1 (u + tv) Φ 1 (u) = t (γ 0 u) 2 ds, {γ 0 u.γ 0 v + t2 } (γ 0v) 2 ds, then, Φ Φ 1 (u + tv) Φ 1 (u) 1 (u), v = lim = (γ 0 u)(γ 0 v)ds. t 0 t Using Theorem 2.1, the Gateaux derivative of Φ 1 is continuous; indeed, Φ 1(u), v γ 0 u L 2 ( ) γ 0 v L 2 ( ) C u v. Hence, Φ 1 C 1 ((), R). It is well known that the functional Φ 2 (u) = 1 u 2 F (x, u), 2 belongs to the class C 1 (H 1 (), R) whose derivative is Φ 2 (u), v = u. v f(x, u)v. Then we obtain (3.1). Volumen 42, Número 2, Año 2008

7 REGULARITY OF THE SOLUTIONS FOR A ROBIN PROBLEM 133 Theorem 3.2. Let R n, n 2 be a bounded and simply connected domain with smooth boundary, f : R R a function satisfying (f 0 ) and (f 1 ), and α a real number. If u H 1 () satisfies Φ (u), v = u v + α (γ 0 u)(γ 0 v)ds f(x, u)v = 0, (3.2) for all v H 1 (), then u H 2 (). Proof. Since γ 1 : H 2 () H 1/2 ( ) is onto (Trace Theorem), and αγ 0 u H 1/2 ( ), there exists w H 2 () such that w η = αγ 0u. (3.3) From (3.2), (3.3) and the Green formula we obtain h v = gv, v H 1 (), (3.4) where (i) First we prove that h = u w, and g(x) = f(x, u) + w(x). (3.5) g { W 0,p1 (); N }, where 1 < p 1 2. (3.6) Using Condition (f 1 ) in the cases: a) If n = 2, then b) If n > 2, taking into account that If p 1 = 2 σ then 1 < 2n Consequently f L 2 (), and so g L 2 (). (3.7) 1 < σ < n + 2 n 2, and 2 = 2n n 2. n + 2 < p 1 < 2n n 2. (3.8) ( f(x, u) p1 C 1 + u 2 ) < +, (3.9) thus, f L p1 (), then g L p1 () for 1 < p 1 2. From (3.4) with v = 1 over, we have Thus, g { W 0,p1 (); N }. (ii) Now, we will prove that g(x)dx = 0. u W 2,p1 (). (3.10) Revista Colombiana de Matemáticas

8 134 RAFAEL A. CASTRO T. By virtue of Theorem 2.5, there exists a unique h W 2,p1 B 0 ()/N such that { h = g, in, h (3.11) η = 0, on. In the case n 3, we have 1 p = p 1 > 1 n+2 2n = 1 2, then p 1 < 2, and hence H 1 () L p 1 (). If n = 2, H 1 () L p 1 (). From (3.11) we get v h = vg, v H 1 (). (3.12) Integrating by parts, the first integral of (3.12) gives h v = gv, v H 1 (). (3.13) Subtracting (3.13) from (3.4) we obtain (h h) v = 0, v H 1 (). (3.14) Let us see that h W 1,2 (). Since h W 2,p1(), if p 1 = 2 then h W 1,2 (); if p 1 < 2, then we have the cases: n = 2 and n > 2. If n = 2, given that p 1 > 1 then 2 < 2p1 2 p 1 ; if n > 2, as p 1 > 2n np1 n+2, then 2 < n p 1. Therefore, in both cases W 2,p1 () W 1,2 (), then h W 1,2 (). Making v = h h in (3.14) we obtain v 2 dx = 0. By the Poincaré inequality we have that v(x) = v = 1 v dx, x. Then h(x) = h(x)+v, thus D α h = D α h for any index α, with α 2. Since h W 2,p1 (), then h W 2,p1 (), therefore u W 2,p1 (). (iii) Finally we will prove that u W 2,2 (). (3.15) If n = 2, by (3.7) and case (i), g { W 0,2 (); N }, and by case (ii), u W 2,2 (). If n > 2, we have the following cases: a) 2p 1 > n, b) 2p 1 = n, c) 2p 1 < n. In case a), W 2,p1 () C ( ) L (), thus u L q (), q [1, + ), and on the other hand, we have f(x, u) 2 ( c 1 + u 2σ ) < +, (3.16) then f L 2 (), g L 2 (), thus by cases (i) and (ii), u W 2,2 (). Volumen 42, Número 2, Año 2008

9 REGULARITY OF THE SOLUTIONS FOR A ROBIN PROBLEM 135 In case b), W 2,p1 () L q (), q [p 1, ), 2σ > 2 > p 1 and by (3.16), then f L 2 (). Similarly to the last case u W 2,2 (). In case c), we use bootstrapping. As W 2,p1 () L p 1 (), where p 1 = thus u L p 1 (). Let p2 = p 1 σ be, and f(x, u) p2 c ( ) 1 + u p 1 < +. np1 n 2p 1, Then, f L p2 (), g L p2 (), thus u W 2,p2 (). If p 2 < 2 and 2p 2 < n then u L p 2 (), p 2 = np2 n 2p 2. Considering p 3 = p 2 σ then f L p3, g L p3 (), thus u W 2,p3 (). Therefore, we obtain the sequence {p j }, such that 1 < p 1 < p 2 < < p s, where p s 2. In fact, since p 1 > 2n n+2, there exists ε > 0 such that p 1 = (1 + ε) and we obtain 2n n+2 p 2 = p 1 /σ (n 2)(1 + ε) p 1 2 = > 1 + ε, /σ n 2 4ε then p 2 p 1 > ε. If 1 < p 1 < p 2 < < p i 1 < p i, where j = 2, 3,, i, then p i+1 p i = p i p i 1 = p i(n 2p i 1 ) p i 1 (n 2p i ) > p i > 1 + ε. p i 1 p j p j 1 > 1 + ε for Hence {p j } is an strictly increasing sequence, therefore, there exists p s 2 and u L p s 1 (), then f L p s (), g L ps () and by case (ii) u W 2,ps (). Therefore u W 2,2 (). Theorem 3.3. Based on the conditions in Theorem 3.2 on, f and α, the problems (P) and (Q) are equivalent: i) u H 1 (, ), (P) ii) u = f(x, u(x)), x, iii) γ 1 u + αγ 0 u = 0, on, and (Q) { i) u H 1 (), ii) u v = f(x, u)v α (γ 0u)(γ 0 v)ds, v H 1 (). Proof. Let u be a solution to problem (P), then ( u)v = f(x, u(x))v, v H 1 (). Using the Green Formula (2.2) we obtain u v γ 1 u, γ 0 v = f(x, u)vdx, v H 1 (). Now, taking the boundary condition, u is a solution to problem (Q). Revista Colombiana de Matemáticas

10 136 RAFAEL A. CASTRO T. thus If u is a solution to the problem (Q) then u. v = f(x, u)v α (γ 0 u)(γ 0 v)ds, v H 1 (), (3.17) u. v = f(x, u)v, v C c (). (3.18) By Theorem 3.2 u H 2 (), and using the Green formula, we obtain v u = f(x, u)v, v Cc (), then u = f(x, u(x)). Changing f for u in (3.17) and using the Green formula, we obtain γ 1 u + αγ 0 u = Applications The purpose of this section is to give the first step for the proof of the existence of solutions of our the problem (P) (see [4]), which consists in expressing the Sobolev space H 1 () as the orthogonal sum of two subspaces. To achieve this goal we consider the problem (P) with f(x, u(x)) = µu(x) where µ is a parameter. Making use of spectral analysis of compact symmetric operator, we obtain the existence of eigenvalues and by virtue of theorems 3.2 and 3.3, we establish some properties of the first eigenvalue and its associated eigenfunctions Existence of eigenvalues. Theorem 4.1. Let α be a real parameter, α 0. The eigenvalue problem { u = µu, in, (4.1.1) γ 1 u + αγ 0 u = 0, on, has an increasing sequence of eigenvalues. In the case α < 0 the first eigenvalue is negative, and in the case α > 0 the first eigenvalue is positive. Proof. 1) Case α < 0. In this case, to establish the existence of the eigenvalues of the problem (4.1.1) first we introduce an scalar product (.,.) k in H 1 (). To make this, first we have that given ε > 0, there exists a constant C(ε) > 0 such that α (γ 0 u) 2 ds αε u 2 + αc(ε) u 2, u H 1 (). (4.1.2) Choosing k and ε such that 0 < ε < 1 α and k > αc(ε), (4.1.3) Volumen 42, Número 2, Año 2008

11 REGULARITY OF THE SOLUTIONS FOR A ROBIN PROBLEM 137 we define the symmetric bilinear form in H 1 () in this way for u, v H 1 (): a k [u, v] = u v + α (γ 0 u)(γ 0 v)ds + k uv. (4.1.4) This defines an scalar product and we denote it by (.,.) k. In fact, the form a k [.,.] is coercive: using (4.1.2) we have a k [u, u] δ u 2, u H 1 (), with δ = min{1+αε, k+αc(ε)}. On the other hand, using the inequality γ 0 u L 2 ( ) c 1 u, u H 1 (), and the Schwarz inequality we have a k [u, v] c 3 u v, where c 3 = 1 + α c k, i.e. a k [.,.] is continuous. We define for each u H 1 () a norm u k = (u, u) k. Then δ u 2 u 2 k c 3 u 2, u H 1 (), (4.1.5) i.e., the norm k is equivalent to the usual norm of H 1 (). With our k chosen in (4.1.3), the problem (4.1.1) is equivalent to { u + ku = (µ + k)u, in, γ 1 u + αγ 0 u = 0, on. (4.1.6) We prove that the eigenvalues µ + k are positive. Let u be a weak solution of (4.1.6), then (u, v) k = (µ + k) uv, v H 1 (). (4.1.7) If moreover, u 0 and v = u in (4.1.7) we get that µ + k > 0. For fixed u H 1 (), the map v uv is a bounded linear functional in H 1 (), uv u v u v 1 δ u k v k, v H 1 (). (4.1.8) So, by the Riesz-Frechet Representation Theorem, there is an element in H 1 () that we denote by T u, such that uv = (T u, v) k, v H 1 (). (4.1.9) Thus (4.1.9) defines the operator T : H 1 () H 1 () that is linear and symmetric: (T u, v) k = (u, T v) k, u, v H 1 (). By virtue of (4.1.8) and (4.1.9) we have T u k 1 δ u k, u H 1 (), i.e. T is bounded. Now, we are going to prove that the operator T is compact. Let {u n } be a bounded sequence in H 1 (). By the theorem on compact imbedding of H 1 () in L 2 () there exists a subsequence {u nj } and a element u in H 1 () such that u nj u in H 1 (), and u nj u in L 2 (), when j. Revista Colombiana de Matemáticas

12 138 RAFAEL A. CASTRO T. By (4.1.9) and the definition of the operator T, we have T unj T u 2 k = (u nj u)t (u nj u) Then u nj u L2 () T (u nj u) L2 () u nj u L 2 () T (u nj u) 1 δ u nj u L 2 () T (u nj ) T (u) k. lim j T unj T u k = 0. Let us see that the eigenvalues of problem (4.1.6) are the inverse eigenvalues of the operator T. In fact, if u is a weak solution of (4.1.6), then by (4.1.7) and (4.1.9) we obtain ( ) 1 T u = u. (4.1.10) µ + k Reciprocally, if u satisfies (4.1.10) then u verifies (4.1.7) by (4.1.9). Our next step is to establish the existence of eigenvalues of operator T. Since, 1 µ 1 + k := sup {(T u, u) k : u k = 1} > 0, (4.1.11) then by [6, Lemma 1.1, p. 36], there exists ϕ 1 H 1 () with ϕ 1 k = 1 such that (T ϕ 1, ϕ 1 ) k = 1 ( ) 1 µ 1 + k, T ϕ 1 = ϕ 1. (4.1.12) µ 1 + k On the other hand, we have hence 1 µ 1 + k = sup {(T u, u) k : u k = 1} ( = sup T u ) u, u 0 u k u k k = sup u2 u 0 u 2, k (4.1.13) Then µ 1 + k = inf u 0 u H 1 () µ 1 = inf u 0 u H 1 () u 2 k. (4.1.14) u2 u 2 + α (γ 0u) 2 ds u2, (4.1.15) Volumen 42, Número 2, Año 2008

13 REGULARITY OF THE SOLUTIONS FOR A ROBIN PROBLEM 139 in particular, for the constant function ϕ(x) = b, x, b 0, we have µ 1 ϕ 2 + α (γ 0ϕ) 2 ds = αb2 n 1 ϕ2 b 2 < 0, where n 1 denotes the (n 1)-dimensional Lebesgue s measure and the Lebesgue s measure n-dimensional of. Let ϕ 1,, ϕ j 1 be the j 1 eigenfunctions that correspond to the j 1 eigenvalues of T. We define 1 µ j + k := sup{(t u, u) k : u k = 1, u ϕ 1,, ϕ j 1 } > 0, (4.1.16) then by [6, Proposition 1.3, p. 37], there exists ϕ j H 1 () with ϕ j k = 1 such that (T ϕ j, ϕ j ) k = 1 µ j + k, T ϕ j = 1 µ j + k ϕ j. (4.1.17) Thus, the eigenvalues of Problem (4.1.6) are positive and form the sequence 0 < µ 1 + k µ 2 + k, then the eigenvalue problem (4.1.1) has a sequence k < µ 1 µ 2 µ 3. (4.1.18) 2) Case α > 0. In this case, the bilinear form a k [u, v] = u v + k uv + α (γ 0 u)(γ 0 v)ds, u, v H 1 (), is coercive for every value of the constant k > 0. In fact a k [u, u] u 2 + k u 2 m u 2, where m = min{1, k}. On the other hand, a k [.,.] is continuous, hence a k [u, v] m 2 u v, where m 2 = 1 + k + αc 2 1. This implies m u 2 u 2 k m 2 u 2, u H 1 (), (4.1.19) i.e., the norm. k is equivalent to the usual norm of H 1 (). The procedure to determine the existence of the eigenvalues of Problem (4.1.1) with α > 0 continues in analogous form to the case α < 0. Considering the eigenvalue problem (4.1.6) with an arbitrary positive constant k, we find that it has a sequence of eigenvalues β 1 + k β 2 + k β 3 + k. Hence inf u 0 u H 1 () β 1 β 2 β 3 (4.1.20) are the eigenvalues of problem (4.1.1). The first eigenvalue β 1 is positive. In fact, by virtue of (4.1.15) we have β 1 = u 2 + α (γ 0u) 2 ds 0, (4.1.21) u2 Revista Colombiana de Matemáticas

14 140 RAFAEL A. CASTRO T. if β 1 = 0, let ϕ be an eigenfunction associated to β 1. Then 0 = ϕ 2 + α (γ 0ϕ) 2 ds, ϕ2 thus γ 0 ϕ = 0, ϕ H 1 0 () and ϕ = 0. Then ϕ 0 which is absurd because ϕ k = The first eigenvalue and its associated eigenfunctions. Theorem 4.2. If ϕ is an eigenfunction of problem (4.1.1) associated to the first eigenvalue λ 1, then ϕ C ( ). Proof. By Theorem 3.3 ϕ verifies ϕ v + α γ 0 ϕ γ 0 v = µ 1 ϕv, v H 1 (). (4.1.22) Using Theorem 3.2 with f(x, ϕ(x)) = λ 1 ϕ(x) we have ϕ H 2 (), γ 0 ϕ H 2 1/2 ( ). Then there exists w 1 H 3 () (Trace Theorem) such that γ 1 w 1 = αγ 0 ϕ. By the Green formula (2.2) we have ϕ. v = {λ 1 ϕ + w 1 }v + w 1 v. Making h = ϕ w 1 and G = λ 1 ϕ + w 1, we obtain { h v = Gv, v H 1 (), γ 1 (h) = 0, (4.1.23) thus, G { H 1 (); N }. By Theorem 2.5 there exists h H 3 () such that { h = G in, (4.1.24) h n = 0 on. Multiplying the first equality in (4.1.24) by v and integrating we obtain h. v = Gv, v H 1 (). (4.1.25) Subtracting (4.1.25) from the first equation in (4.1.23) we have (h h) v = 0, v H 1 (). Making v = h h we obtain (h h) 2 ( = 0, then h h ) = 0, h h = M for some constant M, then ϕ = w 1 + h + M H 3 (). Since ϕ H 3 () then γ 0 ϕ H 3 1/2 ( ). Using again Trace Theorem there exists w 2 H 4 () such that γ 1 w 2 = αγ 0 ϕ and proceeding as in the above case, we conclude that ϕ H 4 (). Continuing on this way we obtain that ϕ H m () for any integer m 0. Thus, for D β ϕ there exists an integer m β such that 2m β > n and Volumen 42, Número 2, Año 2008

15 REGULARITY OF THE SOLUTIONS FOR A ROBIN PROBLEM 141 D β ϕ H m β (). Then by the inclusion of Sobolev and smoothness of the boundary, D β ϕ C ( ). Theorem 4.3. The first eigenvalue of problem (4.1.1) is simple. Proof. i) We claim that if ϕ is an eigenfunction associated to the first eigenvalue λ 1, then ϕ does not change sign. Looking for by contradiction, suppose that ϕ changes sign. Then ϕ verifies the equality (ϕ, ϕ) k = (λ 1 + k) ϕ 2. (4.1.26) Writing the function ϕ in the form ϕ(x) = ϕ + (x) + ϕ (x), where ϕ + (x) = max x {ϕ(x), 0} and ϕ (x) = min x {ϕ(x), 0}, the left member of (4.1.26) can be written (ϕ, ϕ) k = ( ϕ +, ϕ +) k + ( ϕ, ϕ ) k, because ϕ + and ϕ are orthogonals, i.e. ( ϕ +, ϕ ) k = ϕ + ϕ + k = 0. ϕ + ϕ + α γ 0 ϕ + γ 0 ϕ ds Indeed, we have ϕ+ ϕ = 0 and ϕ+ ϕ = 0. Furthermore, by Theorem 4.2, ϕ C ( ). Then γ 0 ϕ + and γ 0 ϕ are restrictions of ϕ + and ϕ on, so γ 0 ϕ + γ 0 ϕ ds = ϕ + ϕ ds = 0. Let b 1 = (ϕ +, ϕ + ) k and b 2 = (ϕ, ϕ ) k, then b 1 > 0 and b 2 > 0. On the other hand, ϕ 2 ( = ϕ + ) 2 ( + ϕ ) 2 = a1 + a 2. Then using the above notation we can write (4.1.26) in the following way: b 1 + b 2 = (λ 1 + k)(a 1 + a 2 ). (4.1.27) From (4.1.27) and (4.1.14) we have 1 λ 1 + k = a 1 + a 2 = sup u2 b 1 + b 2 u 0 u 2. (4.1.28) k For the numbers a1 = a2 b 2. So a 1 b 1 b 1, a2 b 2, and a1+a2 b 1+b 2, we have the unique possibility a1+a2 b 1+b 2 = 1 λ 1 + k = (ϕ+ ) 2 ϕ + 2 = (ϕ ) 2 k ϕ 2. (4.1.29) k Using (4.1.13), the previous equalities imply that ϕ + and ϕ are eigenfunctions corresponding to λ 1. Then v H 1 () we have λ 1 ϕ + v = ϕ + v + α γ 0 ϕ + γ 0 v, (4.1.30) Revista Colombiana de Matemáticas

16 142 RAFAEL A. CASTRO T. By Theorem 4.2, we have that ϕ + C ( ). On the other side we observe that: ϕ + + kϕ + = (λ 1 + k)ϕ + 0, in. Since ϕ changes sign in, there exists x 0 inside of such that ϕ + (x 0 ) = 0 which is its minimum value in. Then by the Strong Maximum Principle ϕ + is constant in, thus ϕ + = 0 in, which is absurd. Therefore ϕ can not change sign in. ii) Next we prove that the geometric multiplicity of λ 1 is one. Let ϕ 1 and ϕ 2 be two eigenfunctions associate to the eigenvalue λ 1. By step i), for each t R the eigenfunction ϕ 1 + tϕ 2 has definite sign in, so the sets A = {t R : ϕ 1 + tϕ 2 0, in } and B = {t R : ϕ 1 + tϕ 2 0, in } are non empty, closed and A B = R. Since the set of real number R is connected, there exists t A B, t 0 such as ϕ 1 + tϕ 2 = 0, i.e. ϕ 1 and ϕ 2 are linearly dependent. Therefore, the eigenspace associate to the eigenvalue λ 1 is generated by a single eigenfunction, that we denote by ϕ 1. iii) The algebraic multiplicity of λ 1 is one, (λ 1 is simple). Let N(λ 1 I + ) = {u : λ 1 u + u = 0 and γ 1 u + αγ 0 u = 0}, N(λ 1 I + ) 2 = {u : (λ 1 I + )(λ 1 u + u) = 0 and γ 1 u + αγ 0 u = 0}. It is clear that N(λ 1 I + ) N(λ 1 I + ) 2. Next we show that N(λ 1 I + ) 2 N(λ 1 I + ). Indeed, let ϕ N(λ 1 I + ) 2 be, then λ 1 ϕ + ϕ N(λ 1 I + ), λ 1 ϕ + ϕ = tϕ 1 for some real t. By the other side, we have t(ϕ 1, ϕ 1 ) k = (tϕ 1, ϕ 1 ) k = (λ 1 ϕ + ϕ, ϕ 1 ) k = λ 1 (ϕ, ϕ 1 ) k + ( ϕ, ϕ 1 ) k = λ 1 (ϕ, ϕ 1 ) k + (λ 1 + k) ϕ 1 ϕ, (by (4.1.7)) = (λ 1 λ 1 )(ϕ, ϕ 1 ) k = 0. Hence t(ϕ 1, ϕ 1 ) k = 0, then t = 0. Thus λ 1 ϕ+ ϕ = 0 and ϕ N(λ 1 I + ) Orthogonal sum. Finally we express the Sobolev space H 1 () as the orthogonal sum of two subspaces, where an addend has finite dimension. In the case α < 0, from (4.1.18) and Theorem (4.3), the sequence of eigenvalues has the form k < µ 1 < µ 2 µ 3 (4.2.1) Let X 1 be the space associated to the first eigenvalue µ 1 and X 2 = X 1 the orthogonal complement of X 1 with respect to the inner product of (, ) k defined in (4.1.4). Then we have H 1 () = X 1 X 2, (4.2.2) Volumen 42, Número 2, Año 2008

17 REGULARITY OF THE SOLUTIONS FOR A ROBIN PROBLEM 143 and ϕ 2 + α (γ 0 ϕ) 2 ds = λ 1 ϕ 2, ϕ X 1. (4.2.3) In the case α > 0, from (4.1.20) and Theorem (4.3), the sequence of eigenvalues has the form β 1 < β 2 β 3 (4.2.4) Let Y 1 be the space associate to β 1 and Y 2 = Y 1 the orthogonal complement of Y 1 with respect to the inner product (, ) k, but here k > 0 is an arbitrary constant. Then H 1 () = Y 1 Y 2, (4.2.5) and ϕ 2 + α (γ 0 ϕ) 2 ds = β 1 ϕ 2, ϕ Y 1. (4.2.6) by virtue of Theorem 4.3, the subspaces X 1 and Y 1 have dimension one. Acknowledgement. The author is deeply thankful to Helena J. Nussenzveig Lopes, and to Djairo G. De Figueiredo for their valuable comments. References [1] Arcoya, D., and Villegas, S. Nontrivial solutions for a Neumann problem with a nonlinear term asymptotically linear at and superlinear at +. Math.-Z 219, 4 (1995), [2] Aubin, J. P. Approximation of elliptic boundary value problems. Wiley-Interscience, New York, [3] Baiocchi, C., and Capelo, A. Variational and quasivariational inequalities, aplications to free-boundary problems. John Wiley and Sons, New York, [4] Castro, R. A. Nontrivial solutions for a Robin problem with a nonlinear term asymptotically linear at and superlinear at. Revista Colombiana de Matemáticas (2008). Este volumen. [5] Dautray, R., and Lions, J. L. Analyse mathématique et calcul numérique; pour les sciences et les techniques. Masson, Paris, Vol. 3. [6] de Figueiredo, D. G. Positive solutions of semilinear elliptic problems, vol. 957 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, Heidelberg, New York, [7] Evans, L. C. Partial differential equations, vol. 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, [8] Kufner, A., John, O., and Fučik, S. Function spaces. Academia, Prague, [9] Lions, J. L., and Magenes, E. Problèmes aux limites non homogènes (vi). Journal D Analyse Mathématique XI (1963), [10] Nečas, J. Les méthodes directes en théorie des équations elliptiques. Masson, Prague, Paris/Academia. (Recibido en julio de Aceptado en agosto de 2008) Revista Colombiana de Matemáticas

18 144 RAFAEL A. CASTRO T. Escuela de Matemáticas Universidad Industrial de Santander Apartado Aéreo 678 Bucaramanga, Colombia Volumen 42, Número 2, Año 2008

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

More information

### Fourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks

Chapter 3 Fourier Series 3.1 Some Properties of Functions 3.1.1 Goal We review some results about functions which play an important role in the development of the theory of Fourier series. These results

More information

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

More information

### Course 221: Analysis Academic year , First Semester

Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................

More information

### 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

More information

### 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

More information

### 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,

More information

### 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

More information

### Recall that two vectors in are perpendicular or orthogonal provided that their dot

Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

More information

### DISTRIBUTIONS AND FOURIER TRANSFORM

DISTRIBUTIONS AND FOURIER TRANSFORM MIKKO SALO Introduction. The theory of distributions, or generalized functions, provides a unified framework for performing standard calculus operations on nonsmooth

More information

### FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

More information

### 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,

More information

### 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

More information

### 7 - Linear Transformations

7 - Linear Transformations Mathematics has as its objects of study sets with various structures. These sets include sets of numbers (such as the integers, rationals, reals, and complexes) whose structure

More information

### 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

More information

### The Heat Equation. Lectures INF2320 p. 1/88

The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)

More information

### 1 Sets and Set Notation.

LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

More information

### 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:

More information

### UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

More information

### 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

More information

### Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

More information

### Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1

Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing

More information

### 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

More information

### Inner Product Spaces and Orthogonality

Inner Product Spaces and Orthogonality week 3-4 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,

More information

### it is easy to see that α = a

21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

More information

### 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

More information

### 1 Completeness of a Set of Eigenfunctions. Lecturer: Naoki Saito Scribe: Alexander Sheynis/Allen Xue. May 3, 2007. 1.1 The Neumann Boundary Condition

MAT 280: Laplacian Eigenfunctions: Theory, Applications, and Computations Lecture 11: Laplacian Eigenvalue Problems for General Domains III. Completeness of a Set of Eigenfunctions and the Justification

More information

### 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

More information

### 1. Let P be the space of all polynomials (of one real variable and with real coefficients) with the norm

Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 005-06-15 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok

More information

### Classification of Cartan matrices

Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms

More information

### 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

More information

### 6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )

6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points

More information

### Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions

Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions D. R. Wilkins Copyright c David R. Wilkins 2016 Contents 3 Functions 43 3.1 Functions between Sets...................... 43 3.2 Injective

More information

### Linear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.

Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a sub-vector space of V[n,q]. If the subspace of V[n,q]

More information

### Math 5311 Gateaux differentials and Frechet derivatives

Math 5311 Gateaux differentials and Frechet derivatives Kevin Long January 26, 2009 1 Differentiation in vector spaces Thus far, we ve developed the theory of minimization without reference to derivatives.

More information

### by the matrix A results in a vector which is a reflection of the given

Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

More information

### 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

More information

### Chapter 20. Vector Spaces and Bases

Chapter 20. Vector Spaces and Bases In this course, we have proceeded step-by-step through low-dimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit

More information

### Section 6.1 - Inner Products and Norms

Section 6.1 - Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,

More information

### 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

More information

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

More information

### Sequences and Convergence in Metric Spaces

Sequences and Convergence in Metric Spaces Definition: A sequence in a set X (a sequence of elements of X) is a function s : N X. We usually denote s(n) by s n, called the n-th term of s, and write {s

More information

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

More information

### Chapter 5. Banach Spaces

9 Chapter 5 Banach Spaces Many linear equations may be formulated in terms of a suitable linear operator acting on a Banach space. In this chapter, we study Banach spaces and linear operators acting on

More information

### 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

More information

### Ideal Class Group and Units

Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

More information

### Introduction to Finite Fields (cont.)

Chapter 6 Introduction to Finite Fields (cont.) 6.1 Recall Theorem. Z m is a field m is a prime number. Theorem (Subfield Isomorphic to Z p ). Every finite field has the order of a power of a prime number

More information

### 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

More information

### 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

More information

### 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

More information

### PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

More information

### 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

More information

### 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

More information

### 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

More information

### Vector and Matrix Norms

Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty

More information

### 1 Norms and Vector Spaces

008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)

More information

### MATH 425, PRACTICE FINAL EXAM SOLUTIONS.

MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator

More information

### Riesz-Fredhölm Theory

Riesz-Fredhölm Theory T. Muthukumar tmk@iitk.ac.in Contents 1 Introduction 1 2 Integral Operators 1 3 Compact Operators 7 4 Fredhölm Alternative 14 Appendices 18 A Ascoli-Arzelá Result 18 B Normed Spaces

More information

### 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

More information

### MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION. Chapter 4: Fourier Series and L 2 ([ π, π], µ) ( 1 π

MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION Chapter 4: Fourier Series and L ([, π], µ) Square Integrable Functions Definition. Let f : [, π] R be measurable. We say that f

More information

### EXISTENCE AND NON-EXISTENCE RESULTS FOR A NONLINEAR HEAT EQUATION

Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 5 (7), pp. 5 65. ISSN: 7-669. UL: http://ejde.math.txstate.edu

More information

### CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

More information

### Adaptive Online Gradient Descent

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

More information

### ANALYTICAL MATHEMATICS FOR APPLICATIONS 2016 LECTURE NOTES Series

ANALYTICAL MATHEMATICS FOR APPLICATIONS 206 LECTURE NOTES 8 ISSUED 24 APRIL 206 A series is a formal sum. Series a + a 2 + a 3 + + + where { } is a sequence of real numbers. Here formal means that we don

More information

### 5. Convergence of sequences of random variables

5. Convergence of sequences of random variables Throughout this chapter we assume that {X, X 2,...} is a sequence of r.v. and X is a r.v., and all of them are defined on the same probability space (Ω,

More information

### ISOMETRIES OF R n KEITH CONRAD

ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x

More information

### Section 4.4 Inner Product Spaces

Section 4.4 Inner Product Spaces In our discussion of vector spaces the specific nature of F as a field, other than the fact that it is a field, has played virtually no role. In this section we no longer

More information

### In memory of Lars Hörmander

ON HÖRMANDER S SOLUTION OF THE -EQUATION. I HAAKAN HEDENMALM ABSTRAT. We explain how Hörmander s classical solution of the -equation in the plane with a weight which permits growth near infinity carries

More information

### Reference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.

5 Potential Theory Reference: Introduction to Partial Differential Equations by G. Folland, 995, Chap. 3. 5. Problems of Interest. In what follows, we consider Ω an open, bounded subset of R n with C 2

More information

### MATH 240 Fall, Chapter 1: Linear Equations and Matrices

MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS

More information

### LEARNING OBJECTIVES FOR THIS CHAPTER

CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional

More information

### Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.

More information

### arxiv:math/0010057v1 [math.dg] 5 Oct 2000

arxiv:math/0010057v1 [math.dg] 5 Oct 2000 HEAT KERNEL ASYMPTOTICS FOR LAPLACE TYPE OPERATORS AND MATRIX KDV HIERARCHY IOSIF POLTEROVICH Preliminary version Abstract. We study the heat kernel asymptotics

More information

### Systems of Linear Equations

Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

More information

### Weak topologies. David Lecomte. May 23, 2006

Weak topologies David Lecomte May 3, 006 1 Preliminaries from general topology In this section, we are given a set X, a collection of topological spaces (Y i ) i I and a collection of maps (f i ) i I such

More information

### 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.............................

More information

### 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

More information

### Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

More information

### 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

More information

### THE DIFFUSION EQUATION

THE DIFFUSION EQUATION R. E. SHOWALTER 1. Heat Conduction in an Interval We shall describe the diffusion of heat energy through a long thin rod G with uniform cross section S. As before, we identify G

More information

### 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

More information

### NOTES ON MEASURE THEORY. M. Papadimitrakis Department of Mathematics University of Crete. Autumn of 2004

NOTES ON MEASURE THEORY M. Papadimitrakis Department of Mathematics University of Crete Autumn of 2004 2 Contents 1 σ-algebras 7 1.1 σ-algebras............................... 7 1.2 Generated σ-algebras.........................

More information

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

More information

### Research Article Stability Analysis for Higher-Order Adjacent Derivative in Parametrized Vector Optimization

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 510838, 15 pages doi:10.1155/2010/510838 Research Article Stability Analysis for Higher-Order Adjacent Derivative

More information

### 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

More information

### Fourier series. Jan Philip Solovej. English summary of notes for Analysis 1. May 8, 2012

Fourier series Jan Philip Solovej English summary of notes for Analysis 1 May 8, 2012 1 JPS, Fourier series 2 Contents 1 Introduction 2 2 Fourier series 3 2.1 Periodic functions, trigonometric polynomials

More information

### Notes on Symmetric Matrices

CPSC 536N: Randomized Algorithms 2011-12 Term 2 Notes on Symmetric Matrices Prof. Nick Harvey University of British Columbia 1 Symmetric Matrices We review some basic results concerning symmetric matrices.

More information

### 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

More information

### On an isomorphic Banach-Mazur rotation problem and maximal norms in Banach spaces

On an isomorphic Banach-Mazur rotation problem and maximal norms in Banach spaces B. Randrianantoanina (joint work with Stephen Dilworth) Department of Mathematics Miami University Ohio, USA Conference

More information

### 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,

More information

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

More information

### Let H and J be as in the above lemma. The result of the lemma shows that the integral

Let and be as in the above lemma. The result of the lemma shows that the integral ( f(x, y)dy) dx is well defined; we denote it by f(x, y)dydx. By symmetry, also the integral ( f(x, y)dx) dy is well defined;

More information

### The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line

The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line D. Bongiorno, G. Corrao Dipartimento di Ingegneria lettrica, lettronica e delle Telecomunicazioni,

More information