Lectures on Periodic Homogenization of Elliptic Systems. Zhongwei Shen

Transcription

1 Lectures on Periodic Homogenization of Elliptic Systems arxiv: v1 [math.ap] 30 Oct 2017 Zhongwei Shen Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506, USA address:

2 2010 Mathematics Subject Classification. 35B27, 35J57, 74Q05 Supported in part by NSF Grant

3 Contents Preface 1 Chapter 1. Elliptic Systems of Second Order with Periodic Coefficients Weak solutions Two-scale asymptotic expansions and the homogenized operator Homogenization of elliptic systems Elliptic systems of linear elasticity Notes 23 Chapter 2. Convergence Rates, Part I Flux correctors and ε-smoothing Convergence rates in H 1 for Dirichlet problem Convergence rates in H 1 for Neumann problem Convergence rates in L p for Dirichlet problem Convergence rates in L p for Neumann problem Convergence rates for elliptic systems of elasticity Notes 50 Chapter 3. Interior Estimates Interior Lipschitz estimates A real-variable method Interior W 1,p estimates Asymptotic expansions of fundamental solutions Notes 78 Chapter 4. Regularity for Dirichlet Problem Boundary localization in the periodic setting Boundary Hölder estimates Boundary W 1,p estimates Green functions and Dirichlet correctors Boundary Lipschitz estimates Dirichlet problem in C 1 and C 1,η domains Notes 109 Chapter 5. Regularity for Neumann Problem Approximation of solutions at large scale Boundary Hölder estimates Boundary W 1,p estimates 118 v

4 vi CONTENTS 5.4. Boundary Lipschitz estimates Matrix of Neumann functions Elliptic systems of linear elasticity Notes 139 Chapter 6. Convergence Rates, Part II Convergence rates in H 1 and L Convergence rates of eigenvalues Asymptotic expansions of Green functions Asymptotic expansions of Neumann functions Convergence rates in L p and W 1,p Notes 170 Chapter 7. L 2 Estimates in Lipschitz Domains Lipschitz domains and nontangential convergence Estimates of fundamental solutions Estimates of singular integrals Method of layer potentials Laplace s equation Rellich property Well-posedness for small scales Rellich estimates for large scales L 2 boundary value problems L 2 estimates in arbitrary Lipschitz domains Square function and H 1/2 estimates Notes 238 Bibliography 239

5 Preface In recent years considerable advances have been made in quantitative homogenization of partial differential equations in the periodic and non-periodic settings. This monograph surveys the theory of quantitative homogenization for second-order linear elliptic systems in divergence form with rapidly oscillating periodic coefficients, L ε = div ( A(x/ε) ), in a bounded domain in R d. It begins with a review of the classical qualitative homogenization theory, and addresses the problem of convergence rates of solutions. The main body of the monograph investigates various interior and boundary regularity estimates (Hölder, Lipschitz, W 1,p, nontangnetial-maximal-function) that are uniform in the small parameter ε > 0. Additional topics include convergence rates for Dirichlet eigenvalues and asymptotic expansions of fundamental solutions, Green functions, and Neumann functions. In Chapter 1 we present the quantitative homogenization theory for L ε, which has been well understood since 1970 s. We start out with a review of basic facts on weak solutions for elliptic systems with bounded measurable coefficients, and use the method of (formal) asymptotic expansions to derive the formula for the homogenized (or effective) operator L 0. We then prove the classical results on the homogenization of Dirichlet and Neumann boundary value problems for L ε. In Chapter 2 we address the issue of convergence rates for solutions and two-scale expansions. Various estimates in L p and in H 1 are obtained without smoothness assumptions on the coefficient matrix A. Chapters 3, 4 and 5 are devoted to the study of sharp regularity estimates, which are uniform in ε > 0, for solutions of L ε (u ε ) = F. The case of interior estimates is treated in Chapter 3. We use a compactness method to establish the Lipschitz estimate down to the microscopic scale ε under the ellipticity and periodicity assumptions. With additional smoothness assumptions on A, this, together with a simple blow-up argument, leads the full-scale Hölder and Lipschitz estimates. The compactness method, which originated from the study of the regularity theory in the calculus of variations and minimal surfaces, was introduced to the study of homogenization problems by M. Avellaneda and F. Lin [9]. In this chapter we also introduce a real-variablemethodfor establishing L p andw 1,p estimates. The method, originated in a paper by L. Caffarelli and I. Peral [19], may be regarded a refined and dual version of the celebrated Calderón-Zygmund Lemma. As corollaries of interior estimates, we obtain asymptotic expansions for the fundamental solution Γ ε (x,y) and its derivatives x Γ ε (x,y), y Γ ε (x,y) and x y Γ ε (x,y), as ε 0. In Chapter 4 we study the boundary regularity estimates for solutions of L ε (u ε ) = F in with the Dirichlet condition u ε = f on. The boundary Lipschitz estimate is proved by 1

6 2 PREFACE the compactness method mentioned above. A key step is to prove the Lipschitz estimate for the so-called Dirichlet correctors. The real-variable method introduced in Chapter 3 is used to establish the boundary W 1,p estimates. It effectively reduces to the problem to certain (weak) reverse Hölder inequalities. In Chapter 5 we prove the boundary Hölder, Lipschitz, and W 1,p estimates for solutions of L ε (u ε ) = F in with the Neumann condition uε ν ε = g in. Here we introduce a general scheme, recently developed by S. N. Armstrong and C. Smart [5] in the study of stochastic homogenization, for establishing regularity estimates at large scale. Roughly speaking, the scheme states that if a function u is well approximated by C 1,α functions at every scale greater than ε, then u is Lipschitz continuous at every scale greater than ε. In Chapter 6 we revisit the problem of convergence rates. We establish an O(ε) error estimate in H 1 for a two-scale expansion involving the Dirichlet correctors anduse it to prove a convergence result for the Dirichlet eigenvalue λ ε,k. We derive the asymptotic expansions for the Green function G ε (x,y) and its derivatives, as ε 0. Analogous results are also obtained for the Neumann function N ε (x,y). Chapter 7 is devoted to the study of L 2 boundary value problems for L ε (u ε ) = 0 in a Lipschitz domain. We establish optimal estimates in terms of nontangential maximal functions for Dirichlet problems with boundary data in L 2 ( ) and in H 1 ( ) as well as the Neumann problem with boundary data in L 2 ( ). This is achieved by the method of layer potentials - the classical method of integral equations. The asymptotic results for the fundamental solution Γ ε (x,y) in Chapter 3 are used to obtain the L p boundedness of singular integrals on, associated with the single and double potentials. The proof of Rellich estimates, (0.0.1) u ε L tan u ε L ν 2 ( ), ε 2 ( ) which are crucial in the use of the method of layer potentials in Lipschitz domains, is divided into two cases. In the small-scale case, where diam() ε, the estimates are obtained by using Rellich identities and a three-step approximation argument. The proof of (0.0.1) for the large-scale case, where diam() > ε, uses an error estimate in H 1 () for a two-scale expansion obtained in Chapter 2. For reader s convenience we also include a section in which we prove (0.0.1) and solve the L 2 Dirichlet, Neumann and regularity problems in Lipschitz domains for the case L =. Part of this monograph is based on lecture notes for courses I taught at several summer schoolsandattheuniversityofkentucky. MuchofmaterialinChapters6and7istakenfrom my joint papers [51, 53] with Carlos Kenig and Fang-Hua Lin, and from [55] with Carlos Kenig. I would like to express my deep gratitude to Carlos Kenig and Fang-Hua Lin for introducing me to the research area of homogenization and for their important contribution to our joint work. Zhongwei Shen Lexington, Kentucky Fall 2017

7 CHAPTER 1 Elliptic Systems of Second Order with Periodic Coefficients In this monograph we shall be concerned with a family of second-order linear elliptic systems in divergence form with rapidly oscillating periodic coefficients, (1.0.1) L ε = div ( A(x/ε) ) = [ ( x ) ] a αβ ij, ε > 0, x i ε x j in R d (the summation convention that the repeated indices are summed is used throughout). We will always assume that the coefficient matrix (tensor) A(y) = ( a αβ ij (y)), with 1 i,j d and 1 α,β m, is real, bounded measurable, and satisfies certain ellipticity condition, to be specified later. We also assume that A is 1-periodic; i.e., (1.0.2) A(y +z) = A(y) for a.e. y R d and z Z d. Observe that by a linear transformation one may replace Z d in (1.0.2) by any lattice in R d. In this chapter we present the qualitative homogenization theory for L ε. We start out in Section 1.1 with basic facts on weak solutions of second-order elliptic systems in divergence form. In Section 1.2 we use the method of (formal) asymptotic expansions to derive the formula for the homogenized (or effective) operator L 0 with constant coefficients. In Section 1.3 we prove some classical theorems on the homogenization of boundary value problems for second-order elliptic systems. In particular, we will show that if u ε H0 1(;Rm ) and L ε (u ε ) = F inaboundedlipschitz domain, wheref H 1 (;R m ), thenu ε u 0 strongly in L 2 (;R m ) and weakly in H0(;R 1 m ), as ε 0. Moreover, the function u 0 H0(;R 1 m ) is a solution of L 0 (u 0 ) = F in. Section 1.4 is devoted to the qualitative homogenization of elliptic systems of linear elasticity. Throughout the monograph we will use C and c to denote positive constants that are independent of the parameter ε > 0. They may change from line to line and depend on A and/or. We will use u to denote the E L1 average of a function u over a set E; i.e. E u = 1 E E u Weak solutions In this section we review basic facts on weak solutions of second-order elliptic systems with bounded measurable coefficients. For convenience of reference it will be done in the context of operator L ε. However, the periodicity condition (1.0.2) is not used in the section. 3

8 4 1. ELLIPTIC SYSTEMS WITH PERIODIC COEFFICIENTS For a domain in R d and 1 p, let W 1,p (;R m ) = u L p (;R m ) : u L p (;R m d ). Equipped with the norm u W 1,p () := 1/p u p L p () + u p L p () for 1 p <, and u W 1, () := u L () + u L (), W 1,p (;R m ) is a Banach space. For 1 < p <, let W 1,p 0 (;R m ) denote the closure of C0 (;R m ) in W 1,p (;R m ) and W 1,p (;R m ) the dual of W 1,p 0 (;R m ), where p = p. If p = 2, we often use the p 1 usual notation: H 1 (;R m ) = W 1,2 (;R m ), H0(;R 1 m ) = W 1,2 0 (;R m ), and H 1 (;R m ) = W 1,2 (;R m ). Definition LetL ε = div(a(x/ε) )witha(y) = ( a αβ ij (y)). ForF H 1 (;R m ), we call u ε H 1 (;R m ) a weak solution of L ε (u ε ) = F in, if (1.1.1) A(x/ε) u ε ϕdx = F,ϕ H 1 () H0 1() for any test function ϕ C 0 (;R m ). To establish the existence of weak solutions for the Dirichlet problem, we introduce the following ellipticity condition: there exists a constant µ > 0 such that (1.1.2) A µ 1, (1.1.3) µ u 2 dx A u udx for any u C0 (R d ;R m ). R d R d Observe that the condition (1.1.2)-(1.1.3), which is referred as the V-ellipticity, is invariant under translation and dilation. In particular, if A = A(x) satisfies (1.1.2)-(1.1.3), so does A ε = A(x/ε) with the same constant µ. Lemma The integral condition (1.1.3) implies the following algebraic condition, (1.1.4) µ ξ 2 η 2 a αβ ij (y)ξ iξ j η α η β for a.e. y R d, where ξ = (ξ 1,...,ξ d ) R d and η = (η 1,...,η m ) R m. Proof. Since A is real, it follows from (1.1.3) that (1.1.5) µ u 2 dx Re A u udx for any u C0 (R d ;C m ). R d R d Fix y R d, ξ R d and η R m. Let u(x) = ϕ ε (x)t 1 e itξ x η in (1.1.5), where t > 0, ϕ ε (x) = ε d/2 ϕ(ε 1 (x y)) and ϕ is a function in C0 (Rd ) with ϕ 2 dx = 1. Using R d u β = iϕ ε e itξ x ξ j η β + ϕ ε t 1 e itξ x η β, x j x j

9 we see that as t, 1.1. WEAK SOLUTIONS 5 u 2 dx = ξ 2 η 2 ϕ ε 2 dx+o(t 1 ), R d R d Re A u udx = R d In view of (1.1.5) this implies that R d a αβ ij (x)ξ iξ j η α η β ϕ ε 2 dx+o(t 1 ). (1.1.6) µ ξ 2 η R 2 ϕ ε 2 dx a αβ ij (x)ξ iξ j η α η β ϕ ε 2 dx. d R d Since ϕ 2 ε (x) = ε d ϕ 2 (ε 1 (x y)) is a mollifier, the inequality (1.1.4) follows by letting ε 0 in (1.1.6). The ellipticity condition (1.1.4) is called the Legendre-Hadamard condition. It follows from Lemma that in the scale case (m = 1), the conditions (1.1.3) and (1.1.4) are equivalent. By using the Plancherel Theorem one may also show the equivalency when m 2 and A is constant. Theorem Suppose that A satisfies (1.1.2)-(1.1.3). Let u ε H 1 (;R m ) be a weak solution of L ε (u ε ) = F +div(g) in, where F L 2 (;R m ) and G = (G α i ) L2 (;R m d ). Then, for any ψ C 1 0(), (1.1.7) u ε 2 ψ 2 dx C where C depends only on µ. 1 u ε 2 ψ 2 dx+ G 2 ψ 2 dx+ F u ε ψ 2 dx, Proof. Note that by (1.1.1), (1.1.8) A(x/ε) u ε ϕdx = F α ϕ α dx G α i ϕ α x i dx for any ϕ = (ϕ α ) C 0 (;Rm ). Since u ε H 1 (;R m ), by a density argument, (1.1.8) continues to hold for any ϕ H 1 0(;R m ). Observe that (1.1.9) A(x/ε) (ψu ε ) (ψu ε ) =A(x/ε) u ε (ψ 2 u ε )+A(x/ε)( ψ)u ε (ψu ε ) A(x/ε) (ψu ε ) ( ψ)u ε +A(x/ε)( ψ)u ε ( ψ)u ε, 1 The constants C and c in this monograph may also depend on d and m. However, this fact is irrelevant to our investigation and will be ignored.

10 6 1. ELLIPTIC SYSTEMS WITH PERIODIC COEFFICIENTS where ψ C0(). 1 It follows that A(x/ε) (u ε ψ) (u ε ψ)dx = A(x/ε) u ε (u ε ψ 2 )dx+ A(x/ε)( ψ)u ε (ψu ε )dx A(x/ε) (ψu ε ) ( ψ)u ε dx+ A(x/ε)u ε ψ u ε ψdx = F α u α εψ 2 dx G α ( i u α x ε ψ 2) dx+ A(x/ε)( ψ)u ε (ψu ε )dx i A(x/ε) (ψu ε ) ( ψ)u ε dx+ A(x/ε)u ε ψ u ε ψdx, where we have used (1.1.8) with ϕ = u ε ψ 2 for the last step. Hence, by (1.1.2)-(1.1.3), µ (u ε ψ) 2 dx A(x/ε) (u ε ψ) (u ε ψ)dx F u ε ψ 2 dx+c Gψ (u ε ψ) dx+c Gψ u ε ψ dx +C (ψu ε ) u ε ψ dx+c u ε 2 ψ 2 dx. This yields (1.1.7) by applying the Cauchy inequality (1.1.10) ab δa 2 + b2 4δ, where a,b 0 and δ > 0. For a ball B = B(x 0,r) = x R d : x x 0 < r in R d, we will use tb to denote B(x 0,tr), the ball with the same center and t times the radius as B. Let u ε H 1 (2B;R m ) be a weak solution of L ε (u ε ) = F +div(g) in 2B, where F L 2 (2B;R m ) and G = (G α i ) L2 (2B;R m d ). Then (1.1.11) u ε 2 1 dx C u (t s) 2 r 2 ε E 2 dx+r 2 F 2 dx+ G 2 dx sb tb for any 1 < s < t < 2 and E R m, where C depends only on µ. The inequality (1.1.11) is called (interior) Caccioppoli s inequality. To see (1.1.11), one applies Theorem to u ε E and choose ψ C 1 0 (tb) so that 0 ψ 1, ψ = 1 on sb, and ψ C(t s) 1 r 1. Theorem (Reverse Hölder inequality). Suppose that A satisfies conditions (1.1.2)- (1.1.3). Let u ε H 1 (2B;R m ) be a weak solution of L ε (u ε ) = 0 in 2B, where B = B(x 0,r) for some x 0 R d and r > 0. Then there exists some p > 2, depending only on µ (and d,m), such that ( 1/p ( 1/2 (1.1.12) u ε dx) p C u ε dx) 2, B 2B tb tb

11 1.1. WEAK SOLUTIONS 7 where C depends only on µ. Proof. Suppose L ε (u ε ) = 0 in 2B. It follows from (1.1.11) by Sobolev-Poncaré inequality that ( ) 1/2 (1.1.13) u ε 2 dx C ( 1/q u ε dx) q, t s sb where 1 < s < t < 2 and 1 = This gives a reverse Hölder inequality, which has the q 2 d so-called self-improving property. We refer the reader to [37, Chapter V] for a proof of the property. We are interested in the Dirichlet boundary value problem, Lε (u ε ) = F +div(g) in, (1.1.14) u ε = f on, and the Neumann boundary value problem, L ε (u ε ) = F +div(g) in, (1.1.15) u ε = g n G ν ε on, with non-homogeneous boundary conditions, where the conormal derivative uε ν ε on is defined by ( ) α uε (1.1.16) = n i (x)a αβ ij ν (x/ε) uβ ε, ε x j and n = (n 1,...,n d ) denotes the outward unit normal to. Let be a bounded Lipschitz domain in R d. The space H 1/2 ( ) may be defined as the subspace of L 2 ( ) of functions f for which f H 1/2 ( ) := f 2 f(x) f(y) 2 1/2 dσ+ dσ(x)dσ(y) <. x y d Theorem Assume that A satisfies (1.1.2)-(1.1.3). Let be a bounded Lipschitz domain in R d. Then, for any f H 1/2 ( ;R m ), F L 2 (;R m ) and G L 2 (;R m d ), there exists a unique u ε H 1 (;R m ) such that L ε (u ε ) = F +div(g) in and u ε = f on in the sense of trace. Moreover, the solution satisfies the energy estimate (1.1.17) u ε H 1 () C where C depends only on µ and. tb F L 2 () + G L 2 () + f H 1/2 ( ) Proof. In the case where f = 0 on, this follows by applying the Lax-Milgram Theorem to the bilinear form (1.1.18) B[u, v] = A(x/ε) u vdx,

12 8 1. ELLIPTIC SYSTEMS WITH PERIODIC COEFFICIENTS on the Hilbert space H 1 0(;R m ). In general, if f H 1/2 ( ;R m ), then f is the trace of a function w in H 1 (;R m ) with w H 1 () C f H 1/2 ( ). By considering u ε w, one may reduce the general case to the case where f = 0. We now consider the Neumann boundary value problem. Let H 1/2 ( ;R m ) denote the dual of H 1/2 ( ;R m ). Definition We call u ε H 1 (;R m ) a weak solution of the Neumann problem (1.1.15) with data F L 2 (;R m ), G L 2 (;R m d ) and g H 1/2 ( ;R m ), if (1.1.19) A(x/ε) u ε ϕdx = F ϕdx G ϕdx+ g,ϕ H 1/2 ( ) H 1/2 ( ) for any ϕ C (R d ;R m ). If m 2, the ellipticity condition in (1.1.2)-(1.1.3) is not sufficient for solving the Neumann problem. As such, we introduce the very strong ellipticity condition, also called the Legendre condition: there exists a constant µ > 0 such that (1.1.20) µ ξ 2 a αβ ij (y)ξα i ξβ j 1 µ ξ 2 for a.e. y R d, where ξ = (ξ α i ) Rm d. It is easy to see that (1.1.20) = (1.1.2)-(1.1.3). Theorem Let be a bounded Lipschitz domain in R d and A satisfy (1.1.20). Assume that F L 2 (;R m ), G L 2 (;R m d ) and g H 1/2 ( ;R m ) satisfy the compatibility condition (1.1.21) F bdx+ g,b H 1/2 ( ) H 1/2 ( ) = 0 for any b R m. Then the Neumann problem (1.1.15) has a weak solution u ε, unique up to a constant in R m, in H 1 (;R m ). Moreover, the solution satisfies the energy estimate (1.1.22) u ε L 2 () C F L 2 () + G L 2 () + g H 1/2 ( ), where C depends only on µ and. Proof. Using (1.1.20), one obtains µ u 2 dx A(x/ε) u udx for any u H 1 (;R m ). The results follow from the Lax-Milgram Theorem by considering the bilinear form (1.1.18) on the Hilbert space H 1 (;R m )/R m.

13 1.2. ASYMPTOTIC EXPANSIONS Two-scale asymptotic expansions and the homogenized operator Let L ε = div(a(x/ε) ) with matrix A = A(y) satisfying (1.1.2)-(1.1.3). Also assume that A is 1-periodic. In this section we use the method of formal two-scale asymptotic expansions to derive the formula for the homogenized (effective) operator for L ε. Suppose that L(u ε ) = F in. Let (1.2.1) Y = [0,1) d = R d /Z d be the elementary cell for the lattice Z d. In view of the coefficients of L ε, one seeks a solution u ε in the form (1.2.2) u ε (x) = u 0 (x,x/ε)+εu 1 (x,x/ε)+ε 2 u 2 (x,x/ε)+, where the functions u j (x,y) are defined on R d and 1-periodic in y, for any x. Note that if φ ε (x) = φ(x,y) with y = x/ε, then It follows that (1.2.3) φ ε = 1 φ + φ. x j ε y j x j L ε ( uj (x,x/ε) ) = ε 2 L 0( u j (x,y) ) (x,x/ε)+ε 1 L 1( u j (x,y) ) (x,x/ε) where the operators L 0,L 1,L 2 are defined by (1.2.4) L 0 (φ(x,y)) = y i L 1 (φ(x,y)) = x i L 2 (φ(x,y)) = x i a αβ ij (y) φβ y j a αβ ij (y) φβ y j a αβ ij (y) φβ x j,. +L 2( u j (x,y) ) (x,x/ε), a αβ ij y (y) φβ, i x j In view of (1.2.2) and (1.2.3) we obtain, at least formally, L ε (u ε ) = ε 2 L 0 (u 0 )+ε 1 L 1 (u 0 )+L 0 (u 1 ) (1.2.5) + L 2 (u 0 )+L 1 (u 1 )+L 0 (u 2 ) +. Since L ε (u ε ) = F, by identifying the powers of ε, it follows from (1.2.5) that (1.2.6) L 0 (u 0 ) = 0, (1.2.7) L 1 (u 0 )+L 0 (u 1 ) = 0, (1.2.8) L 2 (u 0 )+L 1 (u 1 )+L 0 (u 2 ) = F. Using the fact that u 0 (x,y) is 1-periodic in y, we may deduce from (1.2.6) that A(y) y u 0 y u 0 dy = 0. Y

14 10 1. ELLIPTIC SYSTEMS WITH PERIODIC COEFFICIENTS Under the ellipticity condition (1.1.2)-(1.1.3) and periodicity condition (1.0.2), we will show that (1.2.9) µ y φ 2 dy A(y) y φ y φdy for any φ Hper 1 (Y;Rm ). Y Y See Lemma It follows that y u 0 = 0. Thus u 0 (x,y) is independent of y; i.e., (1.2.10) u 0 (x,y) = u 0 (x). Here and henceforth, H k per (Y;Rm ) denotes the closure in H k (Y;R m ) of C per (Y;Rm ), the set of C and 1-periodic functions in R d. To derive the equation for u 0, we first use (1.2.10) and (1.2.7) to obtain (1.2.11) ( L 0 (u 1 ) )α = ( L 1 (u 0 ) )α = u β a αβ ij y (y) 0. i x j By the Lax-Milgram Theorem and (1.2.9) one can show that if h L 2 loc (Rd ;R m d ) is 1- periodic, the cell problem (1.2.12) L 0 (φ) = div(h) in Y, φ H 1 per(y;r m ), has a unique (up to constants) solution. In view of (1.2.11) we may write (1.2.13) u α 1 (x,y) = χαβ j (y) uβ 0 (x)+ũ α 1 x (x), j where, for each 1 j d and 1 β m, the function χ β j = (χ1β j,...,χmβ j ) Hper(Y;R 1 m ) is the unique solution of the following cell problem: L 0 (χ β j ) = L0 (P β j ) in Y, (1.2.14) χ β j (y) is 1-periodic, dy = 0. χ β j Y In (1.2.14) and henceforth, P β j = P β j (y) = y je β, where e β = (0,...,1,...,0) with 1 in the β th position. Note that the α th component of L 0 (P β j ) is ( y i a αβ ij (y)). We now use the equations (1.2.8) and (1.2.13) to obtain ( L 0 (u 2 ) )α = F α ( L 2 (u 0 ) )α ( L 1 (u 1 ) ) α = F α (x)+a αβ ij (y) 2 u β 0 +a αβ ij x i x (y) 2 u β 1 + a αβ ij j x i y j y (y) uβ 1 i x j = F α (x)+a αβ ij (y) 2 u β 0 +a αβ ij x i x (y) χβγ k j y j 2 u γ 0 + x i x k y i a αβ ij (y) uβ 1 x j.

15 Y 1.2. ASYMPTOTIC EXPANSIONS 11 It follows by an integration in y over Y that [ ] (1.2.15) a αβ ij (y)+aαγ ik (y) χγβ j 2 u β 0 dy (x) = F α (x) y k x i x j in. Definition Let  = (âαβ ij ), where 1 i,j d, 1 α,β m, and [ (1.2.16) â αβ ij = a αβ ij +a αγ ( ) ] ik χ γβ j dy, y k and define (1.2.17) L 0 = div(â ). Y In summary we have formally deduced that the leading term u 0 in the expansion (1.2.2) depends only on x and that u 0 is a solution of L 0 (u 0 ) = F in. As we shall prove in the next section, the constant coefficient operator L 0 is indeed the homogenized operator for L ε. Correctors and effective coefficients. The constant matrix  is called the matrix of effective or homogenized coefficients. Because of (1.2.13) we call the 1-periodic matrix χ(y) = ( χ β j (y)) = ( χ αβ j (y) ), with 1 j d and 1 α,β m, the matrix of (first-order) correctors for L ε. Define ( ) (1.2.18) a per φ,ψ = a αβ ij (y) φβ ψα dy y j y i for φ = (φ α ) and ψ = (ψ α ). In view of (1.2.14) the corrector χ β j Hper 1 (Y;Rm ) such that Y χβ j = 0 and (1.2.19) a per ( χ β j,ψ) = a per ( P β j,ψ) for any ψ H 1 per(y;r m ). It follows from (1.2.9) and (1.2.19) with ψ = χ β j that (1.2.20) χ β j H 1 (Y) C, Y is the unique function in where C depends only on µ. With the summation convention the first equation in (1.2.14) may be written as [ (1.2.21) a αβ ij +a αγ ( ) ] ik χ γβ j = 0 in R d ; y i y k i.e., L 1 (χ β j +Pβ j ) = 0 in Rd. It follows from the reverse Hölder estimate (1.1.12) and (1.2.20) that (1.2.22) χ β j L p (Y) C 0 for some p > 2,

16 12 1. ELLIPTIC SYSTEMS WITH PERIODIC COEFFICIENTS where p and C 0 depend only on µ. This implies that χ β j are Hölder continuous if d = 2. By the classical De Giorgi - Nash theorem, χ β j is also Hölder continuous if m = 1 and d 3. If m 2 and d 3, we may use Sobolev imbedding and (1.2.22) to obtain (1.2.23) χ β j L q (Y) C for some q > 2d d 2. We further note that by rescaling, (1.2.24) L ε P β j (x)+εχβ j (x/ε) = 0 in R d for any ε > 0. We now proceed to prove the inequality (1.2.9), on which the existence of correctors (χ β j ) depends. The proof uses the property of weak convergence for periodic functions. Proposition Leth l be asequence of1-periodicfunctions. Assume that h l L 2 (Y) C and h l (y)dy c 0 as l. Y Let ε l 0. Then h l (x/ε l ) c 0 weakly in L 2 () as l, where is a bounded domain in R d. In particular, if h is 1-periodic and h L 2 (Y), then h(x/ε) h weakly in L 2 (), as ε 0. Y Proof. By considering the periodic function h l Y h l, we may assume that Y h l = 0 and hence c 0 = 0. Let u l H 2 per (Y) be a 1-periodic function such that u l = h l in Y. Let g l = u l. Then h l = div(g l ) and g l L 2 (Y) C h l L 2 (Y) C. Note that h l (x/ε l ) = ε l div g l (x/ε l ). It follows that, if ϕ C0(), 1 (1.2.25) h l (x/ε l )ϕ(x)dx = ε l as ε l 0. This is because, if B(0,R), g l (x/ε l ) 2 dx ε d l g l (x/ε l ) ϕ(x)dx 0, B(0,R/ε l ) C g l 2 L 2 (Y) C, g l (y) 2 dy where we have used the periodicity of g l for the second inequality and C depends on R. Similarly, (1.2.26) h l (x/ε l ) L 2 () C h l L 2 (Y) C. In view of (1.2.25) and (1.2.26) we may conclude that h l (x/ε l ) 0 weakly in L 2 (). Lemma Suppose that A = A(y) is 1-periodic and satisfies the ellipticity condition (1.1.2)-(1.1.3). Then the inequality (1.2.9) holds for any φ H 1 per (Y;Rm ).

17 1.2. ASYMPTOTIC EXPANSIONS 13 Proof. Let u ε (x) = εη(x)φ(x/ε), where φ is a 1-periodic function in C (R d ;R m ) and η C0 (Rd ) with η 2 dx = 1. Since A(x/ε) satisfies the condition (1.1.3), It follows that R d (1.2.27) µ u ε 2 dx A(x/ε) u ε u ε dx. R d R d We now take limits by letting ε 0 on both sides of (1.2.27). Using u ε (x) = η(x) φ(x/ε)+ε η(x) φ(x/ε) and Proposition 1.2.2, we see that as ε 0, A(x/ε) u ε u ε dx A φ φdy η 2 dx, R d Y R d u ε 2 dx φ 2 dy η 2 dx. R d Y R d This, together with (1.2.27), yields (1.2.9). The following lemma gives the ellipticity for L 0. Lemma Suppose that A = A(y) is 1-periodic and satisfies (1.1.2)-(1.1.3). Then (1.2.28) µ ξ 2 η 2 â αβ ij ξ iξ j η α η β µ 1 ξ 2 η 2 for any ξ = (ξ 1,...,ξ d ) R d and η = (η 1,...,η m ) R m, where µ 1 depends only on µ (and d, m). Proof. Thesecondinequalityin(1.2.28)followsreadilyfromtheenergyestimate χ β j H 1 (Y) C, where C depends only on µ. To prove the first inequality, we will show that (1.2.29) µ φ 2 dx  φ φdx for any φ C0 (R d ;R m ). R d R d As we pointed out earlier, since  is constant, this is equivalent to the first inequality in (1.2.28). To establish (1.2.29), we fix φ = (φ α ) C0 (Rd ;R m ) and let u ε = φ+εχ β j (x/ε) φβ x j in (1.2.27) and then take the limits as ε 0. Using u ε = φ+ χ β j (x/ε) φβ x j +εχ β j (x/ε) x j φ β = ( P β j +χβ j ) (x/ε) φ β x j +εχ β j (x/ε) x j φ β and Proposition 1.2.2, we see that as ε 0, A(x/ε) u ε u ε dx A ( ( P β j +χβ j ) Rd P α i +χ α φ β i )dy φα dx R d Y x j x i =  φ φdx. R d

18 14 1. ELLIPTIC SYSTEMS WITH PERIODIC COEFFICIENTS Observe that u ε φ weakly in L 2 (R d ;R m d ). It follows that µ φ 2 dx liminf µ u ε 2 dx R d ε 0 R d This completes the proof. lim A(x/ε) u ε u ε dx R d =  φ φdx. R d ε 0 We end this section with a useful observation on the homogenized matrix for the adjoint operator L ε = div ( A (x/ε) ). Lemma Let A = ( a αβ ) ij denote the adjoint of A, where a αβ ij  = (  ). In particular, if A(y) is symmetric, i.e. a αβ ij 1 α,β m, so is Â. (y) = aβα ji = a βα ji. Then (y) for 1 i,j d and Proof. Let χ (y) = ( χ β j (y)) = ( χ αβ j (y) ) denote the matrix of correctors for L ε; i.e. χ β j is the unique function in Hper 1 (Y;Rm ) such that Y χ β j = 0 and (1.2.30) a per (χ β j,ψ) = a per (Pβ j,ψ) for any ψ H1 per (Y;Rm ), where a per (φ,ψ) = a per(ψ,φ). Observe that by (1.2.19) and (1.2.30), ( ) ( ) â αβ ij = a per P β j +χβ j,pα i = aper P β j +χβ j,pα i +χ α i ( = a per P α i +χ α i,p β ) ( j (1.2.31) +χβ j = a per P α i +χ α i,p β ) j ( = a per P α i +χ α i,p β ) j +χ β j = â βα ji, for 1 α,β m and 1 i,j d. This shows that (  ) = Â. Then We start with a Div-Curl Lemma Homogenization of elliptic systems Theorem Let u l and v l be two bounded sequences in L 2 (;R d ). Suppose that (1) u l u and v l v weakly in L 2 (;R d ); (2) curl(u l ) = 0 in and div(v l ) f strongly in H 1 (). (u l v l )ϕdx as l, for any scalar function ϕ C 1 0(). (u v)ϕdx

19 Proof. By considering 1.3. HOMOGENIZATION OF ELLIPTIC SYSTEMS 15 u l v l = (u l u) (v l v) u v +u l v +u v l, we may assume that u l 0, v l 0 weakly in L 2 (;R d ) and that div(v l ) 0 strongly in H 1 (). By a partition of unity we may also assume that ϕ C0 1 (B) for some ball B. Since curl(u l ) = 0 in, there exists U l H 1 (B) such that u l = U l in B and U B ldx = 0. It follows that (u l v l )ϕdx = ( U l v l )ϕdx B B = div(v l ),U l ϕ H 1 (B) H0 1(B) U l (v l ϕ)dx. Hence, (1.3.1) B (u l v l )ϕdx div(v l ) H 1 (B) U l ϕ H 1 0 (B) + U l L 2 (B) v l ϕ L 2 (B). We will show that both terms in the RHS of (1.3.1) converge to zero. By Poincaré inequality, Thus, U l L 2 (B) C u l L 2 (B) C. div(v l ) H 1 (B) U l ϕ H 1 0 (B) 0. Using U l L 2 (B) C, U l = u l 0 weakly in L 2 (B;R d ), and B U l = 0, we may deduce that if U lk is a subsequence of U l and converges weakly in L 2 (B), then it must converge weakly to zero. This implies that the full sequence U l 0 weakly in L 2 (B). It follows that U l 0 weakly in H 1 (B) and therefore U l 0 strongly in L 2 (B). Consequently, as l. This completes the proof. U l L 2 (B) v l ϕ L 2 (B) C U l L 2 (B) 0 The next theorem shows that the sequence of operators L l ε l is G-compact in the sense of G-convergence. Theorem Let A l (y) be a sequence of 1-periodic matrices satisfying (1.1.2)- (1.1.3) with the same constant µ. Let F l H 1 (;R m ). Suppose that (1.3.2) L l ε l (u l ) = F l in, where ε l 0, u l H 1 (;R m ), and We further assume that (1.3.3) L l ε l = div ( A l (x/ε l ) ). F l F in H 1 (;R m ), u l u weakly in H 1 (;R m ), Â l A 0, B

20 16 1. ELLIPTIC SYSTEMS WITH PERIODIC COEFFICIENTS where Âl denotes the matrix of effective coefficients for A l. Then (1.3.4) A l (x/ε l ) u l A 0 u weakly in L 2 (;R m d ), A 0 is a constant matrix satisfying the ellipticity condition (1.2.28), and u is a weak solution of (1.3.5) div(a 0 u) = F in. Proof. We first note that since Âl A 0 and Âl satisfies (1.2.28), so does A 0. Also, (1.3.5) follows directly from (1.3.2) and (1.3.4). To see (1.3.4), we let u l be a subsequence such that A l (x/ε l ) u l H weakly in L 2 (;R m d ) for some H L 2 (;R m d ) and show that H = A 0 u. This would imply that the whole sequence A l (x/ε l ) u l converges weakly to A 0 u in L 2 (;R m d ). With loss of generality we assume that (1.3.6) A l (x/ε l ) u l H weakly in L 2 (;R m d ) for some H = (Hi α) L2 (;R m d ). Let χ l (y) = ( χ β k,l (y)) denote the correctors associated with the matrix A l, the adjoint of A l. Fix 1 k d, 1 γ m and consider the identity ( ) A l (x/ε l ) u l P γ k +ε lχ γ k,l (x/ε l) ψdx (1.3.7) ( ) = u l A l (x/ε l) P γ k +ε lχ γ k,l (x/ε l) ψdx, where ψ C0 1 (). By Proposition 1.2.2, ( ) P γ k (1.3.8) +ε lχ γ k,l (x/ε l) = P γ k + χ γ k,l (x/ε l) P γ k weakly in L 2 (), where we have used the fact Y χ γ k,l dy = 0. Since Ll ε l (u εl ) = F l in, in view of (1.3.6) and (1.3.8), it follows by Theorem that the LHS of (1.3.7) converges to (H P γ k )ψdx = H γ k ψdx. Similarly, note that u l u and ( ) A l(x/ε l ) P γ k +ε lχ γ k,l (x/ε l) lim A l l Y ( = lim l  l Pγ k =(A 0 ) P γ k ) P γ k + χ γ k,l dy weakly in L 2 (), where we have used Proposition as well as Lemma Since L l ε l P γ k +ε lχ γ k (x/ε l) = 0 in R d,

21 1.3. HOMOGENIZATION OF ELLIPTIC SYSTEMS 17 we may use Theorem again to claim that the RHS of (1.3.7) converges to ( u (A 0 ) P γ ) k ψdx. As a result, since ψ C0 1 () is arbitrary, it follows that (1.3.9) H γ k = u (A0 ) P γ k = A0 u P γ k in. This shows that H = A 0 u and completes the proof. We now use Theorem to establish the qualitative homogenization of the Dirichlet and Neumann problems for L ε. The proof only uses a special case of Theorem 1.3.2, where A l = A is fixed. The general case is essential in a compactness argument we will use in Chapters 3 and 4 for regularity estimates that are uniform in ε > 0. Homogenization of Dirichlet Problem (1.1.14). Assume that A satisfies the elliptic condition (1.1.2)-(1.1.3) and is 1-periodic. Let F L 2 (;R m ), G L 2 (;R m d ) and f H 1/2 ( ;R m ). By Theorem there exists a unique u ε H 1 (;R m ) such that L ε (u ε ) = F +div(g) in and u ε = f on (the boundary data is taken in the sense of trace). Furthermore, the solution u ε satisfies u ε H 1 () C F L 2 () + G L 2 () + f H 1/2 ( ), where C depends only on µ and. Let u ε be a subsequence of u ε such that as ε 0, u ε u weakly in H 1 (;R m ) for some u H 1 (;R m ). It follows readily from Theorem that A(x/ε ) u ε  u and L 0 (u) = F + div(g) in. Since f H 1/2 ( ;R m ), there exists Φ H 1 (;R m ) such that Φ = f on. Using the facts that u ε Φ u Φ weakly in H 1 (;R m ) and u ε Φ H 1 0(;R m ), we see that u Φ H 1 0(;R m ). Hence, u = f on. Consequently, u is the unique weak solution to the Dirichlet problem, L 0 (u 0 ) = F +div(g) in and u 0 = f on. Since u ε is bounded in H 1 (;R m ) and thus any sequence u εl with ε l 0 contains a subsequence that converges weakly in H 1 (;R m ), one may conclude that as ε 0, A(x/ε) uε  u 0 weakly in L 2 (;R m d ), (1.3.10) u ε u 0 weakly in H 1 (;R m ). By the compactness of the embedding H 1 (;R m ) L 2 (;R m ), we also obtain (1.3.11) u ε u 0 strongly in L 2 (;R m ). Homogenization of Neumann Problem (1.1.15). Assume that A satisfies the Legendre ellipticity condition (1.1.20) and is 1-periodic. To establish the homogenization theorem for the Neumann problem, we first show that the homogenized matrix  also satisfies the Legendre condition. This ensures that the corresponding Neumann problem for L 0 is well posed.

22 18 1. ELLIPTIC SYSTEMS WITH PERIODIC COEFFICIENTS Lemma Suppose that A = A(y) is 1-periodic and satisfies the Legendre condition (1.1.20). Then  also satisfies the Legendre condition. In fact, (1.3.12) µ ξ 2 â αβ ij ξα i ξβ j µ 1 ξ 2 for any ξ = (ξ α i ) R m d, where µ 1 > 0 depends only on µ (and d,m). Proof. The proof for the second inequality in (1.3.12) is the same as in the proof of Lemma To see the first, we fix ξ = (ξi α) Rm d and let φ = ξi αpα i, ψ = ξα i χα i. Observe that by (1.1.20), â αβ ij ξα i ξβ j = a per(φ+ψ,φ+ψ) µ φ+ ψ 2 dy Y = µ φ 2 dy +µ Y Y ψ 2 dy, where we have also used the fact Y χα i dy = 0. It follows that â αβ ij ξα i ξβ j µ φ 2 dy This finishes the proof. Y = µ ξ 2. LetF L 2 (;R m ), G L 2 (;R m d ), andg H 1/2 ( ;R m ), thedualofh 1/2 ( ;R m ). Assume that F, G and g satisfy the compatibility condition (1.1.21). By Theorem the Neumann problem (1.1.15) has a unique (up to a constant in R m ) solution. Furthermore, if u εdx = 0, by (1.1.22) and Poincaré inequality, u ε H 1 () C F L 2 () + G L 2 () + g H 1/2 ( ) where C depends only on µ and. Let u ε be a subsequence of u ε such that u ε u 0 weakly in H 1 (;R m ) for some u 0 H 1 (;R m ). It follows from Theorem that A(x/ε ) u ε  u 0 weakly in L 2 (;R m d ). By taking limits in (1.1.19) we see that u 0 is a weak solution to the Neumann problem:, (1.3.13) L 0 (u 0 ) = F +div(g) in and and that u 0dx = 0, where u 0 ν 0 = g n G on, (1.3.14) ( u0 ν 0 ) α = n i â αβ u β 0 ij x j is the conormal derivative associated with the operator L 0. Since such u 0 is unique, we may conclude that as ε 0, u ε u 0 weakly in H 1 (;R m ) and thus strongly in L 2 (;R m ). We also obtain A(x/ε) u ε  u 0 weakly in L 2 (;R m d ).

23 1.4. ELLIPTIC SYSTEMS OF LINEAR ELASTICITY Elliptic systems of linear elasticity In this section we consider the elliptic system of linear elasticity L ε = div ( A(x/ε) ). We assume that the coefficient matrix A(y) = ( a αβ ij (y)), with 1 i,j,α,β d, is 1-periodic and satisfies the elasticity condition, denoted by A E(κ 1,κ 2 ), (1.4.1) a αβ ij (y) = aβα ji (y) = aiβ αj (y), κ 1 ξ 2 a αβ ij (y)ξα i ξ β j κ 2 ξ 2 for a.e. y R d and for any symmetric matrix ξ = (ξ α i ) R d d, where κ 1,κ 2 are positive constants. Lemma Let be a bounded domain in R d. Then (1.4.2) 2 u L 2 () u+( u) T L 2 () for any u H 1 0 (;Rd ), where ( u) T denotes the transpose of u. Proof. By a density argument, to prove (1.4.2), which is called the first Korn inequality, we may assume that u C0 (;Rd ). This allows us to use integration by parts to obtain ( )( ) u u+( u) T 2 α dx = + ui u α + ui dx x i x α x i x α = 2 u 2 u α u i dx+2 dx x i x α = 2 = 2 2 u 2 dx 2 u 2 dx+2 u 2 dx, u α x α ( div(u) ) dx div(u) 2 dx from which the inequality (1.4.2) follows. Lemma Suppose A = ( a αβ ) ij E(κ1,κ 2 ). Then κ 1 (1.4.3) 4 ξ +ξt 2 a αβ ij ξα i ξ β j κ 2 4 ξ +ξt 2 for any ξ = (ξi α ) R d d. Proof. Note that by the symmetry conditions in (1.4.1), (1.4.4) a αβ ij = a βα ji = a iβ αj = aβi jα = aji βα = aij αβ = aαj iβ. It follows that for any ξ = (ξ α i ) R d d, a αβ ij ξα i ξ β j = 1 4 aαβ ij (ξα i +ξ i α)(ξ β j +ξj β ), from which (1.4.3) follows readily from (1.4.1).

24 20 1. ELLIPTIC SYSTEMS WITH PERIODIC COEFFICIENTS It follows from Lemmas and that A u udx κ 1 u+( u) T 2 dx R 4 d R d κ 1 u 2 dx, 2 R d for any u C 1 0(R d ;R d ). This shows that the elasticity condition (1.4.1) implies the ellipticity condition (1.1.2)-(1.1.3) for some µ > 0 depending only on κ 1 and κ 2. Consequently, all results proved in previous sections under the condition (1.1.2)-(1.1.3) hold for the elasticity system. In particular, the matrix of homogenized coefficients may be defined and satisfies the ellipticity condition (1.2.28). However, a stronger result can be proved. Theorem Suppose that A = ( a αβ ) ij E(κ1,κ 2 ) and is 1-periodic. Let  = ( â αβ ) ij be its matrix of effective coefficients. Then  E(κ 1,κ 2 ). (1.4.5) Proof. Let the bilinear form a per (, ) be defined by (1.2.18). Observe that ( ) â αβ ij = a per P β j +χβ j,pα i = a per ( P β j +χβ j,pα i ±χ α i where we have used (1.2.19) and χ α i Hper(Y;R 1 d ) for the second equality. Since a αβ ij = a βα ji, we have a per (φ,ψ) = a per (ψ,φ). It follows that â αβ ij = â βα ji ),. Also, using aαβ ij = a iβ αj and (1.2.16), we obtain â αβ ij = â iβ αj. Let ξ = (ξi α ) R d d be a symmetric matrix. Let φ = ξ β j Pβ j and ψ = ξ β j χβ j from (1.4.5) and (1.4.3) that ij ξα i ξ β j = a ( ) per φ+ψ,φ+ψ κ 1 4 φ+ ψ +( φ) T +( ψ) T 2 dy Y = κ 1 4 φ+( φ) T 2 dy + κ 1 4 ψ +( ψ) T 2 dy, â αβ Y Y. It follows where we have used the observation Y χβ j dy = 0 for the last step. Since φ = ξ = ξt, this implies that (1.4.6) â αβ ij ξα i ξ β j κ 1 4 Y = κ 1 4 ξ +ξt 2 = κ 1 ξ 2. φ+( φ) T 2 dy

25 1.4. ELLIPTIC SYSTEMS OF LINEAR ELASTICITY 21 Also, note that by (1.4.5), â αβ ij ξα i ξβ j = a ( ) per φ+ψ,φ ψ = a per (φ,φ) a(ψ,ψ) a per (φ,φ) κ 2 ξ 2, where we have used the fact a per (φ,ψ) = a per (ψ,φ) and a per (ψ,ψ) 0. As we pointed out earlier, the results for the Dirichlet problem in Section 1.3 hold for the elasticity operator. Additional work is needed for the Neumann problem (1.1.15), as the elasticity condition (1.4.1) does not imply the Legendre ellipticity condition. Let (1.4.7) R = φ = Bx+b : B R d d is skew-symmetric and b R d denote the space of rigid displacements, with dim(r) = d(d+1). 2 Using the symmetric condition a αβ ij = a iβ αj, we see that A(x/ε) u φ = 0 for any φ R. Consequently, the existence of solutions of (1.1.15) implies that (1.4.8) F φdx G φdx+ g,φ H 1/2 ( ) H 1/2 ( ) = 0 for any φ R. Theorem Let be a bounded Lipschitz domain in R d and A E(κ 1,κ 2 ). Assume that F L 2 (;R d ), G L 2 (;R d d ) and g H 1/2 ( ;R d ) satisfy the compatibility condition (1.4.8). Then the Neumann problem (1.1.15) has a weak solution u ε, unique up to an element of R, in H 1 (;R d ). Moreover, the solution satisfies the energy estimate, (1.4.9) u ε L 2 () C F L 2 () + G L 2 () + g H 1/2 ( ), where C depends only on κ 1,κ 2 and. Proof. This again follows from the Lax-Milgram Theorem by considering the bilinear form (1.1.18) on the Hilbert space H 1 (;R d )/R. To prove B[u,v] is coercive, one applies the second Korn inequality (1.4.10) u 2 dx C u+( u) T 2 dx for any u H 1 (;R d ) with the property that u R in H 1 (;R d ) or in L 2 (;R d ). We refer the reader to [63] for a proof of (1.4.10). Theorem Assume that A is 1-periodic and satisfies the elasticity condition(1.4.1). Let u ε H 1 (;R d ) be the weak solution to the Neumann problem (1.1.15) with u ε φ = 0

26 22 1. ELLIPTIC SYSTEMS WITH PERIODIC COEFFICIENTS for any φ R, given by Theorem Then u ε u 0 weakly in H 1 (;R d ), (1.4.11) A(x/ε) u ε  u 0 weakly in L 2 (;R d d ), where u 0 is the unique weak solution to the Neumann problem, (1.4.12) L 0 (u 0 ) = F +div(g) in and with u 0 φ = 0 for any φ R. u 0 ν 0 = g n G on, Proof. The proof is similar to that for the Neumann problem under the Legendre ellipticity condition. We point out that Theorem is needed for the existence and uniqueness of the Neumann problem (1.4.12) for L 0. We end this section with some observations on systems of linear elasticity. Let A(y) = ( a αβ ij (y)) E(κ 1,κ 2 ). Define (1.4.13) ã αβ ij (y) = aαβ ij (y)+µδ iαδ jβ µδ iβ δ jα, where 0 < µ κ 1 /2. The following proposition shows that à = (ãαβ ij ) is symmetric and satisfies the Legendre ellipticity condition. Proposition Let A E(κ 1,κ 2 ) and ã αβ ij be defined by (1.4.13). Then ã αβ ij = ã βα ji, and (1.4.14) ã αβ ij ξα i ξβ j µ ξ 2 for any ξ = (ξ α i ) R d d. Proof. The symmetry property is obvious. To see (1.4.14), we let ξ = (ξi α) Rd d and recall that by (1.4.3), a αβ ij ξα i ξβ j κ 1 4 ξ +ξt 2. It follows that ã αβ ij ξα i ξβ j κ 1 4 ξ +ξt 2 +µ ξ 2 µξ j i ξi j = κ 1 2 ( ) ξ 2 +ξ j i ξi j +µ ξ 2 µξ j i ξi j ( κ1 ) 2 µ ξ +ξ T 2 = µ ξ µ ξ 2, where we have used the assumption µ κ 1 /2 for the last step. Proposition Let Ã(y) = (ãαβ ij ) be defined by (1.4.13). Then (1.4.15) A(x/ε) u ϕdx = Ã(x/ε) u ϕdx for any u H 1 loc (;Rd ) and ϕ C 0 (;R d ).

27 (1.4.16) 1.5. NOTES 23 Proof. Let u C (;R d ) and ϕ C0 (;R d ). Note that ) (Ã(x/ε) A(x/ε) u ϕdx = µ (δ iα δ jβ δ iβ δ jα ) uβ ϕα dx x j x i = µ = 0, div(u) div(ϕ)dx µ u β x α ϕα x β dx where we have used integration by parts for the last step. By a density argument one may deduce that (1.4.16) continues to hold for any u Hloc 1 (;Rd ) and ϕ C0 (;Rd ). Let L ε = div(ã(x/ε) ). It follows from Proposition that if u ε H 1 loc (;Rd ), then (1.4.17) L ε (u ε ) = F in if and only if Lε (u ε ) = F in, where F ( C0 (;R d ) ) is a distribution. In view of Proposition this allows us to treat the system of linear elasticity as a special case of elliptic systems satisfying the Legendre condition and the symmetry condition. Indeed, the approach works well for interior regularity estimates as well as for boundary estimates with the Dirichlet condition. However, we should point out that since the Neumann boundary condition depends on the coefficient matrix, the re-writing of the system of elasticity changes the Neumann problem. More precisely, let u ε ν ε denote the conormal derivative associated with L ε, then ( ) α ( ) α uε uε u β ε = +µn α div(u ε ) µn β. ν ε ν ε x α 1.5. Notes Material in Section 1.1 is standard for second-order linear elliptic systems in divergence form with bounded measurable coefficients. The formal asymptotic expansions as well as other results in Section 1.2 may be found in the classical book [15]. Much of the material in Sections 1.3 and 1.4 is more or less well known and may be found in books [89, 63].

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29 CHAPTER 2 Convergence Rates, Part I Let L ε = div(a(x/ε) ) for ε > 0, where A(y) = ( a αβ ij (y)) is 1-periodic and satisfies certain ellipticity condition. Let L 0 = div(â ), where  = ( â αβ ) ij denotes the matrix of effective coefficients, given by (1.2.16). For F L 2 (;R m ) and ε 0, consider the Dirichlet problem Lε (u ε ) = F in, (2.0.1) u ε = f on, and the Neumann problem (2.0.2) L ε (u ε ) = F in, u ε = g on, ν ε u ε R m in L 2 (;R m ), where f H 1 ( ;R m ), and g L 2 ( ;R m ). It is shown in Section 1.3 that as ε 0, u ε converges to u 0 weakly in H 1 (;R m ) and strongly in L 2 (;R m ). In this chapter we investigate the problem of convergence rates in H 1 and L 2. In Section 2.1 we introduce the flux correctors and study the properties of an ε-smoothing operator S ε. Sections 2.2 and 2.3 are devoted to error estimates of two-scale expansions in H 1 for Dirichlet and Neumann problems in a Lipschitz domain, respectively. We will show that (2.0.3) u ε u 0 εχ(x/ε)η ε S 2 ε ( u 0) H 1 () C ε u 0 H 2 (), where η ε is a cut-off function satisfying (2.2.1). Moreover, if A satisfies the symmetry condition A = A, we obtain C ε F L q (2.0.4) u ε u 0 εχ(x/ε)η ε Sε 2 ( u () + f H 1 ( ) for (2.0.1), 0) H 1 () C ε F L q () + g L 2 ( ) for (2.0.2), for 0 < ε < 1, where q = 2d. The constant C in (2.0.3)-(2.0.4) depends only on the d+1 ellipticity constant µ and. The O( ε) rate in H 1 () given by (2.0.3) and (2.0.4) is more or less sharp. Note that the error estimates (2.0.3)-(2.0.4) also imply the O( ε) rate for u ε u 0 in L 2 (). However, this is not sharp. In fact, it will be proved in Sections 2.4 and 2.5 that if is a bounded Lipschitz domain and A = A, the scaling-invariant estimate (2.0.5) u ε u 0 L p () Cε u 0 W 2,q (), 25

30 26 2. CONVERGENCE RATES, PART I holds for p = 2d and q = d 1 p = 2d, where C depends only on µ and. Without the d+1 symmetry condition, it is shown that (2.0.6) u ε u 0 L 2 () Cε u 0 H 2 (), under the assumption that is a bounded C 1,1, domain. In Section 2.6 we address the problem of convergence rates for elliptic systems of linear elasticity. No smoothness conditiononawill beimposedonthecoefficient matrixainthischapter. Further results on convergence rates may be found in Chapter 6 under additional smoothness conditions on A Flux correctors and ε-smoothing Throughout this section we assume that A = A(y) is 1-periodic and satisfies the V- ellipticity condition (1.1.2)-(1.1.3). For 1 i,j d and 1 α,β m, let (2.1.1) b αβ ij (y) = aαβ ij (y)+aαγ ik (y) ( ) χ γβ j (y) â αβ ij y, k where the repeated index k is summed from 1 to d and γ from 1 to m. Observe that the matrix B(y) = ( b αβ ij (y)) is 1-periodic and that B L p (Y) C 0 for some p > 2 and C 0 > 0 depending on µ. Moreover, it follows from the definitions of χ β j and âαβ ij in Section 1.2 that (2.1.2) ( y i b αβ ij ) = 0 and Y b αβ ij (y)dy = 0. Proposition There exist φ αβ kij H1 per (Y), where 1 i,j,k d and 1 α,β m, such that (2.1.3) b αβ ij = y k ( φ αβ kij ) and φ αβ kij = φαβ ikj. Moreover, if χ = (χ β j ) is Hölder continuous, then φαβ kij L (Y). Proof. Since Y bαβ ij dy = 0, there exists fαβ ij H 2 per(y) such that Y fαβ ij dy = 0 and (2.1.4) f αβ ij = b αβ ij in Y. Moreover, Define f αβ ij H 2 (Y) C b αβ ij L 2 (Y) C. φ αβ kij (y) = y k ( f αβ ij ) y i ( f αβ kj ).

31 2.1. FLUX CORRECTORS AND ε-smoothing 27 Clearly, φ αβ kij H1 per (Y) and φαβ kij = φαβ ikj. Using y i b αβ ij = 0, we may deduce from (2.1.4) that y i f αβ ij is a 1-periodic harmonic function and thus is constant. Hence, φ αβ kij = f αβ ij 2 f αβ kj y k y k y i = f αβ ij = b αβ ij. ( ) Suppose that the corrector χ is Hölder continuous. Recall that L 1 χ β j +Pβ j = 0 in R d. By Caccioppoli s inequality, χ 2 dx C χ(x) χ(y) 2 dx+cr d. r 2 B(y,r) B(y,2r) This implies that χ is in the Morrey space L 2,ρ (Y) for some ρ > d 2; i.e., χ 2 dx Cr ρ for y Y and 0 < r < 1. B(y,r) Consequently, b αβ ij L 2,ρ (Y) for some ρ > d 2 and sup x Y Y b αβ ij (y) dy C. x y d 1 In view of (2.1.4), using a potential representation for Laplace s equation, one may deduce that f αβ ij L (Y) C f αβ ij b αβ ij L 2 (Y) +Csup (y) dy x Y Y x y d 1 C. It follows that φ αβ kij L (Y). Remark Recall that if d = 2 or m = 1, the function χ is Hölder continuous. As a result, we obtain φ αβ kij C. In the case where d 3 and m 2, we have b αβ ij L p (Y) for some p > 2. It follows that 2 f αβ ij L p (Y) for some p > 2. By Sobolev imbedding this implies that φ αβ kij Lq (Y) for some q > 2d. We mention that if the coefficient matrix A is in d 2 VMO(R d ) (see (3.0.1) for the definition), then χ(y) is Hölder continuous and consequently, φ = ( φkij) αβ is bounded. Remark A key property of φ = ( φkij) αβ, which follows form (2.1.3), is the identity: (2.1.5) b αβ ψα ij (x/ε) = ε φ αβ ψα kij (x/ε) x i x k x i for any ψ = (ψ α ) H 2 (;R m ).

32 28 2. CONVERGENCE RATES, PART I Let u ε H 1 (;R m ), u 0 H 2 (;R m ), and A direct computation shows that w ε = u ε u 0 εχ(x/ε) u 0. (2.1.6) A(x/ε) u ε Â u 0 B(x/ε) u 0 = A(x/ε) w ε +εa(x/ε)χ(x/ε) 2 u 0, where B(y) = ( b αβ ij (y)) is defined by (2.1.1). It follows that (2.1.7) A(x/ε) u ε Â u 0 B(x/ε) u 0 L 2 () C w ε L 2 () +Cε χ(x/ε) 2 u 0 L 2 () C u ε u 0 χ(x/ε) u 0 L 2 () +Cε χ(x/ε) 2 u 0 L 2 (), where C depends only on µ. This indicates that the 1-periodic matrix-valued function B(y) plays the same role for the flux A(x/ε) u ε as χ(y) does for u ε. Since b αβ ( ij = y k φ αβ kij), the 1-periodic function φ = ( φkij) αβ is called the flux corrector. To deal with the fact that the correctors χ and φ may be unbounded (if d 3 and m 2), we introduce an ε-smoothing operator S ε. Definition Fix ρ C0 (B(0,1/2)) such that ρ 0 and ρdx = 1. For ε > 0, R d define (2.1.8) S ε (f)(x) = ρ ε f(x) = f(x y)ρ ε (y)dy, R d where ρ ε (y) = ε d ρ(y/ε). The following two propositions contain the most useful properties of S ε for us. Proposition Let f L p loc (Rd ) for some 1 p <. Then for any g L p loc (Rd ), ( ) 1/p (2.1.9) g(x/ε)s ε (f) L p (O) C sup g p f L p (O ε/2 ), x R d where O R d is open, and C depends only on p. B(x,1/2) O t = x O : dist(x,o) < t, Proof. By Hölder s inequality, S ε (f)(x) p f(y) p ρ ε (x y)dy. R d This, together with Fubini s Theorem, gives (2.1.9) for the case O = R d. The general case follows from the observation that S ε (f)(x) = S ε (fχ Oε/2 )(x) for any x O. It follows from (2.1.9) that if g is 1-periodic and belongs to L p (Y), then (2.1.10) g(x/ε)s ε (f) L p (O) C g L p (Y) f L p (O ε/2 ), where C depends only on p. A similar argument gives (2.1.11) g(x/ε) S ε (f) L p (O) Cε 1 g L p (Y) f L p (O ε/2 ).