Hội tụ hai-kích thước mạnh cho một trường hợp hai chiều

Nội dung text: Hội tụ hai-kích thước mạnh cho một trường hợp hai chiều

1. Tina Mai / Tạp chớ Khoa học và Cụng nghệ Đại học Duy Tõn 5(48) (2021) 121-125 121 5(48) (2021) 121-125 Strong two-scale convergence for a two-dimensional case Hội tụ hai-kớch thước mạnh cho một trường hợp hai chiều a,b, Tina Mai ∗ a,b, Mai Ti Na ∗ aInstitute of Research and Development, Duy Tan University, Da Nang, 550000, Vietnam bFaculty of Natural Sciences, Duy Tan University, Da Nang, 550000, Vietnam aViện Nghiờn cứu và Phỏt triển Cụng nghệ Cao, Trường Đại học Duy Tõn, Đà Nẵng, Việt Nam bKhoa Khoa học Tự nhiờn, Trường Đại học Duy Tõn, Đà Nẵng, Việt Nam (Ngày nhận bài: 16/06/2021, ngày phản biện xong: 19/06/2021, ngày chấp nhận đăng: 20/10/2021) Abstract In this paper, we present definitions and some properties of the classical strong two-scale convergence for component-wise vector or matrix functions in a two-dimensional case. Keywords: two-scale homogenization; strong two-scale convergence; two-dimensional Túm tắt Trong bài bỏo này, chỳng tụi trỡnh bày cỏc định nghĩa và một số tớnh chất của hội tụ hai-kớch thước mạnh cổ điển cho cỏc hàm vectơ hoặc ma trận trong một trường hợp hai chiều. Từ khúa: đồng nhất húa hai-kớch thước; hội tụ hai-kớch thước mạnh; hai chiều 1. Introduction functions [1, 2], in a two-dimensional case. We are given in dimension two, a bounded reference domain Ω Ω1 Ω2 R R and a vari- 2. Preliminaries = ì ∈ ì able x (x1,x2) Ω. In two-scale homogeniza- = ∈ tion theory, strong two-scale convergence can be Latin indices are in the set {1,2}. The space viewed as an intermediate property between the of functions, vector fields in R2, and 2 2 ma- ì usual (one-scale) weak and strong convergence. trix fields, defined over Ω are represented respec- In light of this spirit, we first give a necce- tively by italic capitals (e.g. L2(Ω)), boldface Ro- sary review of the usual weak convergence in man capitals (e.g. V ), and special Roman capitals the Hilbert space L2(Ω) then the definitions and (e.g. S). properties of the classical strong two-scale con- In the rest of this paper, we use the following vergence for component-wise vector or matrix notations [1]: ∗Corresponding Author: Tina Mai; Institute of Research and Development, Duy Tan University, Da Nang, 550000, Viet- nam; Faculty of Natural Sciences, Duy Tan University, Da Nang, 550000, Vietnam; Email: maitina@duytan.edu.vn
2. 122 Tina Mai / Tạp chớ Khoa học và Cụng nghệ Đại học Duy Tõn 5(48) (2021) 121-125 2 à 1 ả • Y : [0,1] is the reference periodic cell. x 1 = period ², equivalently, u u(y ) is a peri- ² = • C 0(Ω) is the space of functions that vanish odic function in y1 with period 1. That is, for any at infinity. integer k, 1 1 1 • C ∞ (Y ) denotes the Y -periodic C ∞ vector- u (x ) u (x ²) u (x k²), per ² = ² + = ² + valued functions in R2. Here, Y -periodic equivalently, means 1-periodic in each variable yi ,i = 1,2. à x1 ả à x1 ả à x1 ả u u 1 u k1 u(y1 k). 1 ² = ² + = ² + = + • The notation Hper(Y ), as the closure for 1 the H -norm of C per∞ (Y ), is the space of 3. Weak convergence vector-valued functions v L2(Y ) such ∈ 2 that v(y) is Y -periodic in R2. In the Hilbert space L (Ω), we describe the basic notions of the usual weak convergence, • which is defined below [8]. 1 Consider a sequence of functions u L2(Ω). v y v(y)dy . ² 〈 〉 = Y ˆ 2 ∈ | | Y Then, (u²) is said to be bounded in L (Ω) if • 2 limsup u² dx c , 1 ² 0 ˆΩ | | ≤ < ∞ Hper(Y ): {v Hper(Y ) v y 0}. → = ∈ |〈 〉 = for some positive constant c. 2 By definition, a sequence (u²(x)) L (Ω) is • We use for the canonical inner products 2 ∈ 2 ã 2 2 weakly convergent to u(x) L (Ω) as ² 0, de- in R and R ì , respectively. ∈ → noted by u² * u, if • The notation stands for up to a multi- . ≤ plicative constant that only depends on Ω lim u²(x) φdx u φdx , (1) ² 0 ˆ ã = ˆ ã when applicable. → Ω Ω for any test function φ L2(Ω). ∈ 2 The Sobolev norm W 1,2(Ω) has the form Furthermore, a sequence (u ) in L (Ω) is k ã k 0 ² called strongly convergent to u L2(Ω) as ² 0, 2 2 1 ∈ → 2 denoted by u u, if v W 1,2(Ω) ( v 2 v L2 ) ; ² k k 0 = k kL (Ω) + k∇ k (Ω) → here, v 2 : v 2 , where v represents lim u² v ² dx u v dx , (2) k kL (Ω) = k| |kL (Ω) | | ² 0 ˆ ã = ˆ ã the Euclidean norm of the 2-component vector- → Ω Ω 2 valued function v, and v 2 : v 2 , for every sequence (v ²) L (Ω) which is weakly k∇ kL (Ω) = k|∇ |kL (Ω) ∈ where v denotes the Frobenius norm of the convergent to v L2(Ω). |∇ | ∈ 2 2 matrix v. Recall that the Frobenius norm The following are well-known weak conver- ì ∇ on L2(Ω) is specified by X 2 : X X tr(X TX ). gence properties in L2(Ω). | | = ã = Let ² be some natural small scale. For po- (a) Any weakly convergent sequence is bounded tential applications in homogenization, based on in L2(Ω). [3, 4, 5, 6], we consider u (x) W 1,2(Ω) de- ² ∈ 0 pending on x1 only, that is, u (x) u (x1), with (b) Compactness principle: any bounded se- ² = ² boundary conditions of Neumann type. As re- quence in L2(Ω) has a weakly convergent marked in [7], we do not distinguish between subsequence. R R2 a function on and its extension to as a 2 function of the first variable. It is assumed that (c) If a sequence (u²) is bounded in L (Ω) and à x1 ả (1) is satisfied for all φ C ∞(Ω), then u² * 1 1 ∈ 0 u²(x ) u is a periodic function in x with u L2(Ω). = ² ∈
3. Tina Mai / Tạp chớ Khoa học và Cụng nghệ Đại học Duy Tõn 5(48) (2021) 121-125 123 (d) If u u L2(Ω) and v * v L2(Ω), then Definition 4.1. Let (u ) be a bounded sequence ² → ∈ ² ∈ ² in L2(Ω). If there exist a subsequence, still de- 1 2 1 lim u² v ² dx u v dx . noted by u², and a function u(x, y ) L (Ω Y ), ² 0 ˆ ã = ˆ ã 1 ∈ ì → Ω Ω where Y [0,1] such that = à à x1 ảả 2 lim u²(x) φ(x)h dx (e) Weak convergence of (u²) to u in L (Ω) to- ² 0 ˆ ² → Ω (4) gether with u(x, y1)(φ(x)h(y1))dx dy1 = ˆΩ Y 1 lim u 2 dx u 2 dx ì ² 1 ² 0 ˆΩ | | = ˆΩ | | for any φ C ∞(Ω) and any h C ∞ (Y ), then → ∈ 0 ∈ per such a sequence u² is said to weakly two-scale is equivalent to strong convergence of (u²) to 1 2 converge to u(x, y ). This convergence is denoted u in L (Ω). 1 by u²(x) u(x, y ) . Hereafter, we denote by Y [0,1]2 the cell of = For vectors u², equation (4) implies periodicity. (In our paper, a periodic cell has the à x1 ả form Y [0,1] [0,1].) The mean value of a 1- lim u (x) Φ x, dx = ì 1 ² periodic function ψ(y ) is denoted by ψ , that ² 0 ˆΩ ã ² 〈 〉 → (5) is, u(x, y1) Φ(x, y1)dx dy1 , 1 1 = ˆΩ Y 1 ã ψ ψ(y )dy . ì 〈 〉 ≡ ˆY 1 for every Φ L2(Ω;C (Y 1)), whose choice is 1 1 1 per Recall that y ²− x , and we do not distinguish ∈ = explained in [9] (p. 8). between a function on Y 1 and its extension to Y as a function of the first variable only. 5. Strong two-scale convergence Also, in our paper, the symbol L2(Y ) is used not only for functions defined on Y but also The further extension of the class of test func- for the space of functions in L2(Y ) extended by tions in Definition 4.1 leads to the basis of the 2 1-periodicity to all R . Similarly, C per∞ (Y ) rep- following notion of the classical strong two-scale resents the space of infinitely differentiable 1- convergence [8, 10]. periodic functions on the entire R2. Definition 5.1. A bounded sequence u L2(Ω) For later use, we need the following classical ² is called strongly two-scale convergent∈ if there result. exists u u(x, y1) L2(Ω Y 1) such that = ∈ ì Lemma 3.1 (The mean value property). Let 1 h(y ) be a 1-periodic function on R and h lim u²(x)v²(x)dx ∈ ² 0 ˆ L2(Y 1). Then, for any bounded domain Ω, there → Ω (6) holds the weak convergence u(x, y1)v(x, y1)dx d y1 = ˆΩ Y 1 à 1 ả ì x 2 2 h * h in L (Ω) as ² 0. (3) for any bounded sequence v²(x) L (Ω) such ² 〈 〉 → 1 2 ∈ that v²(x) v(x, y ) L (Ω). This convergence ∈ 1 Proof. The proof is based on property (c) and is denoted by u²(x) u(x, y ). →→ can be found in [8]. For vector (or matrix) u², equation (6) im- plies 4. Weak two-scale convergence lim u²(x) v ²(x)dx We have the following definition of weak ² 0 ˆ ã → Ω two-scale convergence in L2( ) (introduced by (7) Ω 1 1 1 in 1989 by Nguetseng) [1, 2]. u(x, y ) v(x, y )dx d y . = ˆΩ Y 1 ã ì
4. 124 Tina Mai / Tạp chớ Khoa học và Cụng nghệ Đại học Duy Tõn 5(48) (2021) 121-125 In the next well-known results, weak and The converse of (ii) is also true as follows [8]. strong two-scale convergence can be viewed as Lemma 5.3. Weak two-scale convergence intermediate properties between the usual (one- u (x) u(x, y1) together with the relation (8) scale) weak and strong convergence. ² implies strong two-scale convergence u²(x) 2 1 →→ Proposition 5.2. Let (u²) be a sequence in L (Ω) u(x, y ). and u L2(Ω Y 1). Then, ∈ ì Proof. The proof is based on [8]. Consider an (i) u u in L2(Ω) u u in L2(Ω Y 1), arbitrary subsequence (still denoted by ²) ² 0 ² ² → → =⇒ →→ ì such that there exist limits whenever u is independent of y1, the con- 2 lim u²(x)v²(x)dx α, lim v²(x) dx β, verse also holds, ² 0 ˆ = ² 0 ˆ | | = → Ω → Ω 2 1 2 1 (ii) u² in L (Ω Y ) u² u in L (Ω where v²(x) v(x, y ). Then, using the lower 1→→ ì =⇒ ì Y ), semicontinuity property [8] for tv u , we ob- ² + ² tain (iii) u u in L2(Ω Y 1) u * ² ì =⇒ ² u( , y1)d y1 in L2(Ω). 2 Y 1 lim tv²(x) u²(x) dx ã ² 0 ˆ | + | ´ → Ω Proof. For (i), the proof is readily followed from 1 1 2 1 Definition 4.1, the mean value property (3), and tv(x, y ) u(x, y ) dxd y . ≥ ˆΩ Y 1 | + | the property (d) of convergence in L2. ì For (ii), it is obvious. Indeed, it suffices to Applying (8), we get take, in Definition 5.1, t 2β 2tα t 2 v 2dx d y1 1 1 + ≥ ˆΩ Y 1 | | v²(x) φ(x)h(²− x ), ì = 1 2 1 2t uv dx d y . φ C ∞(Ω),h L (Y ), and recall (3), to derive + ˆ 1 ∈ 0 ∈ Ω Y (4) as desired. Moreover, from u u in L2(Ω ì ²→→ ì Hence, Y 1) (6), taking v u , one obtains the relation ² = ² à ả 1 2 1 2 1 2t α uv dx d y lim u²(x) dx u(x, y ) dx d y . − ˆΩ Y 1 ² 0 ˆ | | = ˆ 1 | | ì → Ω Ω Y à ả ì (8) t 2 β v 2dx d y1 . ≥ − + ˆΩ Y 1 | | For (iii), by the definition of weak two-scale ì convergence 4.1, it follows that On the right hand side of this inequality, we ap- ply the lower semicontinuity property [8] again à x1 ả lim u²(x)Φ x, dx for v ². Then, with the arbitrariness of t, we must ² 0 ˆ ² → Ω (9) have 1 1 1 u(x, y )Φ(x, y )dx d y , α uv dx d y1 , = ˆΩ Y 1 = ˆΩ Y 1 ì ì 2 1 which is our desired result. for every Φ L (Ω;Cper(Y )). Choosing Φ 1 in ∈ = (9) and applying the property (c) of weak conver- The following theorem is stated and proved gence, one obtains in [8]. 1 1 Theorem 5.4. Let u (x) L2(Ω), u (x) u²(x) * u(x, y )d y u(x, ) , (10) ² ∈ ² →→ ˆY 1 = 〈 ã 〉 u(x, y1). Suppose in addition that u(x, y1) is a Carathộodory function, u(x, y1) Φ (y1), Φ which implies that one can reach the usual weak ≤ 0 0 ∈ limit from the two-scale limit by taking the aver- L2(Y 1). Then, age over the cell of periodicity. 1 1 2 lim u²(x) u(x,²− x ) dx 0. (11) ² 0 ˆ | − | = → Ω
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