Periodic Homogenization of Elliptic Problems (Draft)

Transcription

1 Periodic Homogenization of Elliptic Problems (Draft) Zhongwei Shen 1 Department of Mathematics University of Kentucky 1 Supported in part by NSF grant DMS

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3 Contents 1 Elliptic Operators with Periodic Coefficients Weak solutions Asymptotic expansions and the homogenized operator Homogenization of elliptic systems Convergence results Notes Interior Estimates Interior Hölder estimates Interior Lipschitz estimates A real-variable method Interior W 1,p estimates Asymptotic expansions of fundamental solutions Notes Regularity for the Dirichlet Problem Boundary Hölder estimates Correctors for the Dirichlet problem Boundary Lipschitz Estimates The Dirichlet problem in C 1,η domains Notes Regularity for the Neumann Problems Boundary Hölder estimates W 1,p estimates Matrix of Neumann functions Correctors for the Neumann problems Boundary Lipschitz Estimates Lipschitz estimates in C 1,η domains Notes

4 4 CONTENTS

5 Chapter 1 Second Order Elliptic Operators with Periodic Coefficients In this monograph we shall be concerned with a family of second-order elliptic operators in divergence form with rapidly oscillating periodic coefficients, L ε = div ( A (x/ε) ) = [ ( x ) ] a αβ ij, ε > 0. 1 (1.0.1) x i ε x j We will always assume that the coefficient matrix A(y) = (a αβ ij (y)) with 1 i, j d and 1 α, β m is real, bounded measurable, and satisfies the ellipticity condition µ ξ 2 a αβ ij (y)ξα i ξ β j 1 µ ξ 2 for y R d and ξ = (ξ α i ) R dm, (1.0.2) where µ > 0, and the periodicity condition A(y + z) = A(y) for y R d and z Z d. (1.0.3) Some smoothness condition as well as the symmetry condition will be imposed on A, as needed. The goal of this chapter is to introduce the 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 (effective) operator L 0 of constant coefficients. In Section 1.3 we prove some classical theorems on homogenization of elliptic systems. In particular, we will show that if u ε H0(Ω; 1 R m ) and L ε (u ε ) = F in Ω, where F H 1 (Ω; R m ), then u ε u 0 strongly in L 2 (Ω; R m ) and weakly in H0(Ω; 1 R m ), as ε 0. Moreover, the function u 0 H0(Ω; 1 R m ) is a solution of L 0 (u 0 ) = F in Ω. Finally in Section 1.4 we study the problem of convergence rates of u ε in L 2 and H 1, an important topic to which we will devote Chapter 7. 1 The summation convention is used throughout the monograph. 5

6 6 CHAPTER 1. ELLIPTIC OPERATORS WITH PERIODIC COEFFICIENTS 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 or/and Ω. 1.1 Weak solutions In this section we review some basic facts on weak solutions of second-order elliptic systems in divergence form with bounded measurable coefficients. This will be done in the context of L ε. We note that the periodicity condition (1.0.3) is not used in the section. For a bounded domain Ω in R d and 1 p, let Equipped with the norm W 1,p (Ω; R m ) = u L p (Ω; R m ) : u L p (Ω). u W 1,p (Ω) = u p L p (Ω) + u p L p (Ω) 1/p, W 1,p (Ω; R m ) is a Banach space. Let W 1,p 0 (Ω; R m ) denote the closure of C0 (Ω; R m ) in W 1,p (Ω; R m ) and W 1,p (Ω; R m ) the dual of W 1,p 0 (Ω; R m ), where p = p. If p = 2, we p 1 will use the usual notation: H 1 (Ω; R m ) = W 1,2 (Ω; R m ), H0(Ω; 1 R m ) = W 1,2 0 (Ω; R m ), and H 1 (Ω; R m ) = W 1,2 (Ω; R m ). Let L ε = div(a(x/ε) ) with A(y) = ( a αβ ij (y)). Let F W 1,p (Ω; R m ). We say u ε = (u α ε ) W 1,p (Ω; R m ) is a weak solution of L ε (u ε ) = F in Ω, if for any ϕ = (ϕ α ) C 0 (Ω; R m ). Ω a αβ ij (x/ε) uβ ε x j ϕ α x i dx =< F, ϕ > W 1,p (Ω) W 1,p 0 (Ω) (1.1.1) Theorem Let Ω be a bounded domain in R d. Suppose that A satisfies the ellipticity condition (1.0.2). Then, for any F H 1 (Ω; R m ), the system L ε (u ε ) = F in Ω has a unique weak solution in H 1 0(Ω; R m ). Moreover, the solution satisfies the estimate where C depends only on µ and Ω. 2 u ε H 1 0 (Ω) C F H 1 (Ω), (1.1.2) Proof. This follows from the Lax-Milgram Theorem (see e.g. [5, pp ]). 2 The constant C, which may change from line to line in this monograph, may also depend on d and m. However, this fact is not relevant to our investigation and will be ignored.

7 1.1. WEAK SOLUTIONS 7 We will be interested in the Dirichlet problem Lε (u ε ) = F in Ω, u ε = f on Ω, (1.1.3) and the Neumann problem L ε (u ε ) = F in Ω, u ε = g on Ω, ν ε (1.1.4) with non-homogeneous boundary conditions, where the conormal derivative uε ν ε on Ω is defined by ( ) α uε = n i (x)a αβ ij ν (x/ε) uβ ε, (1.1.5) ε x j and n = (n 1,..., n d ) denotes the outward unit normal to Ω. Let Ω be a bounded Lipschitz domain. The space H 1/2 ( Ω) may be defined as the subspace of L 2 ( Ω) of functions f for which f 2 H 1/2 ( Ω) = f 2 L 2 ( Ω) + f(x) f(y) 2 dσ(x)dσ(y) <. x y d+1 Ω Ω Theorem Let Ω be a bounded Lipschitz domain in R d. Suppose that A satisfies (1.0.2). Then, for any f H 1/2 ( Ω; R m ) and F H 1 (Ω; R m ), there exists a unique u ε H 1 (Ω; R m ) such that L ε (u ε ) = F in Ω and u ε = f on Ω in the sense of trace. Moreover, the solution satisfies where C depends only on µ and Ω. u ε H 1 (Ω) C F H 1 (Ω) + f H 1/2 ( Ω), (1.1.6) Proof. Since f H 1/2 ( Ω; R m ), it is the trace of a function w in H 1 (Ω; R m ) with w H 1 (Ω) C f H 1/2 ( Ω). By considering u ε w, one may reduce Theorem to Theorem Let H0 1 (Ω; R m ) denote the dual of H 1 (Ω; R m ) and H 1/2 ( Ω; R m ) the dual of H 1/2 ( Ω; R m ). We say that u ε = (u α ε ) H 1 (Ω; R m ) is a weak solution of the Neumann problem (1.1.4) with data F H0 1 (Ω; R m ) and g H 1/2 ( Ω; R m ), if Ω a αβ ij (x/ε) uβ ε x j ϕ α x i dx =< F, ϕ > H 1 0 (Ω) H1 (Ω) + < g, ϕ > H 1/2 ( Ω) H 1/2 ( Ω) (1.1.7) for any ϕ = (ϕ α ) C 0 (R d ; R m ). Clearly, the existence of solutions implies that for any e R m. < F, e > H 1 0 (Ω) H1 (Ω) + < g, e > H 1/2 ( Ω) H 1/2 ( Ω)= 0 (1.1.8)

8 8 CHAPTER 1. ELLIPTIC OPERATORS WITH PERIODIC COEFFICIENTS Theorem Let Ω be a bounded Lipschitz domain. Suppose that A satisfies (1.0.2). Assume that F H0 1 (Ω; R m ) and g H 1/2 ( Ω; R m ) satisfy the compatibility condition (1.1.8). Then the Neumann problem (1.1.4) has a unique (up to constants) solution u ε in H 1 (Ω; R m ), and the solution satisfies u ε L 2 (Ω) C where C depends only on µ and Ω. Proof. This again follows from the Lax-Milgram Theorem. F H 1 0 (Ω) + g H 1/2 ( Ω), (1.1.9) 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 ) = 0 in 2B. Then B u ε 2 dx C r 2 2B u ε 2 dx, (1.1.10) where C depends only on µ. Using the Cacciopoli s inequality (1.1.10) and a reverse Hölder argument, one may show that there exists p > 2, depending on µ, such that ( 1/p ( 1/2 1 1 u ε dx) p C u ε dx) 2 (1.1.11) B B 2B 2B (see e.g. [6, Chapter V]). By Sobolev imbedding, it follows that if d = 2, local weak solutions of L ε (u ε ) = 0 are Hölder continuous, uniformly in ε. More precisely, one has ( ) ρ ( 1/2 x y 1 u ε (x) u ε (y) C u ε dx) 2 (1.1.12) r 2B 2B for any x, y B, where ρ and C are positive constants depending only on µ (not on ε). The same is true for scalar equations (m = 1) if d 3. This is the classical De Giorgi -Nash Hölder estimate for second-order elliptic equations in divergence form with bounded measurable coefficients. We will show in Chapter 2 that under the periodicity condition (1.0.3) and some smooth condition on A, weak solutions of L ε (u ε ) = 0 in 2B are Lipschitz continuous: u ε L (B) C r uniformly in ε, for d 2 and m 1. ( 1/2 1 u ε dx) 2, 2B 2B

9 1.2. ASYMPTOTIC EXPANSIONS AND THE HOMOGENIZED OPERATOR Asymptotic expansions and the homogenized operator In this section we use the method of formal asymptotic expansions to derive the formula for the homogenized (effective) operator for L ε. Let Y = [0, 1) d = R d /Z d. (1.2.1) A function h in R d is said to be Y -periodic if h(y + z) = h(y) for y R d and z Z d. Let L ε = div(a(x/ε) ) with matrix A(y) satisfying (1.0.2)-(1.0.3). Suppose that L(u ε ) = F in Ω. In view of the coefficients of L ε, we seek a solution u ε in the form u ε (x) = u 0 (x, x/ε) + εu 1 (x, x/ε) + ε 2 u 2 (x, x/ε) +, (1.2.2) where the functions u j (x, y) are defined on Ω Y and are Y-periodic in y, for any x Ω. Note that if φ ε (x) = φ(x, y) with y = x/ε, then It follows that 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 L 0 (φ(x, y)) = a αβ ij y (y) φβ, i y j L 1 (φ(x, y)) = a αβ ij x (y) φβ i y j L 2 (φ(x, y)) = x i a αβ ij (y) φβ x j + L 2( u j (x, y) ) (x, x/ε),. y i 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 ) a αβ ij (y) φβ, x j + L 2 (u 0 ) + L 1 (u 1 ) + L 0 (u 2 ) +. (1.2.3) (1.2.4) (1.2.5) Since L ε (u ε ) = F, by identifying the powers of ε, it follows from (1.2.5) that L 0 (u 0 ) = 0, (1.2.6) L 1 (u 0 ) + L 0 (u 1 ) = 0, (1.2.7)

10 10 CHAPTER 1. ELLIPTIC OPERATORS WITH PERIODIC COEFFICIENTS L 2 (u 0 ) + L 1 (u 1 ) + L 0 (u 2 ) = F. (1.2.8) Using the fact that u 0 (x, y) is Y-periodic in y, we may deduce from (1.2.6) that u 0 (x, y) is independent of y; i.e., u 0 (x, y) = u 0 (x). (1.2.9) This, together with (1.2.7), gives L 0 (u 1 ) = L 1 (u 0 ) = u β a αβ ij y (y) 0. (1.2.10) i x j Let Hper(Y 1 ; R m ) denote the closure in H 1 (Y ; R m ) of Cper(Y ; R m ), the set of C and Y-periodic functions in R d. Using the Lax-Milgram Theorem and the ellipticity condition (1.0.2), one can show that if f L 2 loc (Rd ; R dm ) is Y-periodic, the cell problem L 0 (φ) = div(f) in Y, φ Hper(Y 1 ; R m (1.2.11) ) has a unique (up to constants) solution. In view of (1.2.10) this implies that u α 1 (x, y) = χ αβ j (y) uβ 0 x j (x) + ũ α 1 (x), (1.2.12) where, for each 1 j d and 1 β m, the function χ β j = (χ1β j,..., χmβ j ) Hper(Y 1 ; R m ) is the unique solution of the following cell problem: L 1 (χ β j ) = L 1(P β j ) in Y, χ β j (y) is Y-periodic, (1.2.13) χ β j dy = 0 Y (note that L 1 = L 0 ). Here P β j = P β j (y) = y j(0,..., 1,..., 0) with 1 in the β th position. We now use the equations (1.2.8) and (1.2.12) to obtain L 0 (u 2 ) = F L 2 (u 0 ) L 1 (u 1 ) = F (x) a αβ ij (y) 2 u β 0 x i x j a αβ ij (y) uβ 1 x i y j y i = F (x) a αβ ij (y) 2 u β 0 a αβ ij x i x (y) χβγ k j Y y j 2 u β 0 x i x k y i a αβ ij (y) uβ 1 x j a αβ ij (y) uβ 1 x j It follows by an integration in y over Y that [ ] a αβ ij (y) + aαγ ik (y) χγβ j 2 u β 0 dy (x) = F (x) (1.2.14) y k x i x j.

11 1.2. ASYMPTOTIC EXPANSIONS AND THE HOMOGENIZED OPERATOR 11 in Ω. Let  = (âαβ ij ), where 1 i, j d, 1 α, β m, and and [ â αβ ij = a αβ ij Y + a αγ ik ( y k χ γβ j ) ] dy, (1.2.15) L 0 = div(â ). (1.2.16) We have formally deduced that the leading term u 0 in the expansion (1.2.2) depends only x and is a solution of L 0 (u 0 ) = F in Ω. As we shall prove in the next section, the constant coefficient operator L 0 is indeed the homogenized operator for L ε. The constant matrix  will be called the matrix of effective coefficients. Because of (1.2.12) we will call the matrix χ(y) = ( χ β j (y)) = ( χ αβ j (y) ) H 1 per(y ) with 1 j d and 1 α, β m the matrix of (first-order) correctors for L ε. With the summation convention the first equation in (1.2.13) may be written as y i [ a αβ ij + a αγ ik ( y k χ γβ j ) ] = 0 in R d. (1.2.17) Since L 1 (χ β j + P β j ) = 0 in Rd, there exist p > 2 and C 0 > 0, depending only on µ, such that χ β j L p (Y ) C 0. It follows that the correctors χ β j are Hölder continuous if d = 2. By the De Giorgi -Nash estimates, χ j are also Hölder continuous for d 3 and m = 1. Observe that L ε P β j (x) + εχβ j (x/ε) = 0 in R d. The ellipticity of L 0. It follows from (1.2.15) that â αβ ij c ξ 2, we write â αβ ij = Y C, where C depends only on µ. To see aαβ ij ξα i ξ β j y k δ sα y i a sγ kl y l δ γβ y j + χ γβ dy, where we have used the Kronecker delta: δ αβ = 1 if α = β, and δ αβ = 0 if α β. Using L 1 P β j + χβ j = 0, we then obtain â αβ ij = Y y k δ sα y i + χ sα i a sγ kl y l j δ γβ y j + χ γβ j dy. (1.2.18)

12 12 CHAPTER 1. ELLIPTIC OPERATORS WITH PERIODIC COEFFICIENTS Thus, for ξ = (ξi α ) R dm, we have a αβ ij ξα i ξ β j = δ sα y i ξi α Y y k m µ s=1 Y + χ sα i ξi α a sγ kl y l y i ξi s + χ sα i ξi α 2 dy 0. δ γβ y j ξ β j + χγβ j ξ β j dy (1.2.19) Now, suppose that a αβ ij ξα i ξ β j = 0 for some ξ = (ξi α ) R dm. Then, by (1.2.19), y i ξi s + χ sα i (y)ξi α must be constant. It follows that the linear function y i ξi s is Y-periodic. Hence ξ = 0. This shows that there exists c > 0 such that a αβ ij ξα i ξ β j c ξ 2 for any ξ = (ξi α ) R dm. Let A = (a αβ ij ) and B = (bαβ ij ). We say B A if bαβ ij ξα i ξ β j aαβ ij ξα i ξ β j for any ξ = (ξi α ) R dm. It can be shown that if A and B satisfy the conditions (1.0.2) and (1.0.3) and B A, then B Â, provided B is symmetric, i.e. bαβ ij = b βα ji. By taking B = µ(δ ijδ αβ ), this implies that if µ ξ 2 a αβ ij ξα i ξ β j, then  satisfies the condition with the same µ. Furthermore, if A is symmetric, periodic and µ 1 ξ 2 a αβ ij ξα i ξj α µ 2 ξ 2, then µ 1 ξ 2 â αβ ij ξα i ξ β j µ 2 ξ 2 with the same µ 1 and µ 2. See e.g. [2, Section 3.4]. Theorem Suppose that A = ( ) a αβ ij satisfies the conditions (1.0.2)-(1.0.3) with the ellipticity constant µ. Let  = ( ) â αβ ij be its matrix of effective coefficients. Then µ ξ 2 â αβ ij ξα i ξ β j µ 1 ξ 2 (1.2.20) for any ξ = (ξ α i ) R dm, where µ 1 depends only on µ. Moreover, if A is symmetric, one may take µ 1 = µ 1. The homogenized operator of the adjoint operator. Let χ (y) = (χ αβ j (y)) denote the matrix of correctors for L ε = div(a (x/ε) ), the adjoint of L ε ; i.e. χ Hper(Y 1 ), y i [ a βα ji + a γα li ( y l χ γβ j ) ] = 0 in R d, (1.2.21) and Y χ (y) dy = 0. It follows from (1.2.21) that a βα ji + a γα ( ) li χ γβ ( ) j χ αt k dy = 0. (1.2.22) y l y i Similarly, we may deduce from (1.2.17) that a αβ ij + a αγ ( il y l Y Y χ γβ j By comparing (1.2.22) with (1.2.23) we obtain a βα ( ) ji χ αt k dy = y i Y ) ( ) χ αt k dy = 0. (1.2.23) y i Y a αt ik ( y i χ αβ j ) dy. (1.2.24)

13 1.3. HOMOGENIZATION OF ELLIPTIC SYSTEMS 13 As a result we may rewrite the homogenized coefficients â αβ ij â αβ ij = a αβ ij Y + a γβ lj (χ γα y l i ) as dy. (1.2.25) This shows that the homogenized operator for L ε is given by L 0, the adjoint of L 0. In particular, if A(y) is symmetric, i.e. a αβ ij (y) = aβα ji (y) for 1 i, j d and 1 α, β m, so is Â. 1.3 Homogenization of elliptic systems Let F H 1 (Ω; R m ) and u ε H 1 0(Ω; R m ) be the solution of L ε (u ε ) = F in Ω. Let u 0 H 1 0(Ω; R m ) be the unique solution to L 0 (u 0 ) = F in Ω, where L 0 is given by (1.2.16). In this section we will show that u ε converges to u 0 weakly in H 1 0(Ω; R m ) and strongly in L 2 (Ω; R m ). We start with the Div-Curl Lemma. Theorem Let u j, v j L 2 (Ω; R d ). Suppose that Then 1. u j u and v j v weakly in L 2 (Ω; R d ); 2. curl(u j ) = 0 in Ω and div(v j ) f strongly in H 1 (Ω). as j, for any ϕ C 1 0(Ω). Proof. By considering Ω (u j v j )ϕ dx Ω (u v)ϕ dx u j v j = (u j u) (v j v) u v + u j v + u v j, we may assume that u j 0, v j 0 weakly in L 2 (Ω; R d ), and div(v j ) 0 strongly in H 1 (Ω). By partition we may also assume that ϕ C0(B) 1 for some ball B Ω. Since curl(u j ) = 0 in Ω, there exists U j H 1 (B) such that u j = U j and U B j dx = 0. It follows that (u j v j )ϕ dx = ( U j v j )ϕ dx B B = < div(v j ), U j ϕ > H 1 (B) H0 1(B) U j (v j ϕ) dx. Hence, (u j v j )ϕ dx div(v j ) H 1 (B) U j ϕ H 1 0 (B) + U j L 2 (B) v j ϕ L 2 (B). (1.3.1) B B

14 14 CHAPTER 1. ELLIPTIC OPERATORS WITH PERIODIC COEFFICIENTS We will show that both terms in the right hand side of (1.3.1) converge to zero. By Poincaré inequality, U j L 2 (B) C u j L 2 (B) C. Thus, div(v j ) H 1 (B) U j ϕ H 1 0 (B) 0. Since U j L 2 (B) C, U j = u j 0 weakly in L 2 (B; R d ), and B U j = 0, we may deduce that U j 0 weakly in L 2 (B). It follows that U j 0 weakly in H 1 (B) and therefore U j 0 strongly in L 2 (B). Consequently, as j. This completes the proof. U j L 2 (B) v j ϕ L 2 (B) C U j L 2 (B) 0 For k > 0, we denote by H k per(y ) the closure of C per(y ) in W k,2 (Y ). Proposition Suppose that h j is Y-periodic and h j L 2 (Y ) C. Also assume that Y h j (y)dy c 0 as j. Let ε j 0. Then h j (x/ε j ) c 0 weakly in L 2 (Ω) as j. In particular, if h is Y-periodic and h L 2 (Y ), then h(x/ε) Y h weakly in L2 (Ω) as ε 0. Proof. By considering the periodic function h j Y h j, we may assume that Y h j = 0 and hence c 0 = 0. Let u j H 2 per(y ) be an Y -periodic function such that u j = h j in Y. Let g j = u j. Then h j = div(g j ) and g j L 2 (Y ) C h j L 2 (Y ) C. Note that It follows that, if ϕ C 1 0(Ω), as ε j 0. This is because Ω Ω h j (x/ε j ) = ε j div g j (x/ε j ). h j (x/ε j )ϕ(x) dx = ε j g j (x/ε j ) 2 dx = ε d j Ω g j (x/ε j ) ϕ(x) dx 0, (1.3.2) Ω j g j (y) 2 dy C g j 2 L 2 (Y ) C, where Ω j = x : ε j x Ω B(0, Cε 1 j ) and we have used the periodicity of g j for the first inequality. Similarly, h j (x/ε j ) L 2 (Ω) C h j L 2 (Y ) C. (1.3.3) In view of (1.3.2) and (1.3.3) we may conclude that h j (x/ε j ) 0 weakly in L 2 (Ω).

15 1.3. HOMOGENIZATION OF ELLIPTIC SYSTEMS 15 Theorem Suppose that A(y) satisfies (1.0.2) and (1.0.3). Let Ω be a bounded Lipschitz domain and F H 1 (Ω; R m ). Let u ε H 1 0(Ω; R m ) be the weak solution of L ε (u ε ) = F in Ω and u ε = 0 on Ω. Then u ε u 0 strongly in L 2 (Ω; R m ) and weakly in H 1 0(Ω; R m ), as ε 0. Moreover, A(x/ε) u ε Â u 0 weakly in L 2 (Ω), and u 0 is the weak solution of the homogenized problem: L 0 (u 0 ) = F in Ω and u 0 = 0 on Ω, where L 0 is given by (1.2.16). Proof. Let u ε be a subsequence of u ε such that as ε 0, u ε u weakly in H 1 0(Ω; R m ), u ε u strongly in L 2 (Ω; R m ) (1.3.4) and a αβ ij (x/ε ) uβ ε x j f α i weakly in L 2 (Ω) (1.3.5) for some u H0(Ω; 1 R m ) and fi α L 2 (Ω). We will show that u = u 0 is the weak solution of L 0 (u) = 0 in Ω and u = 0 on Ω. Since u ε H 1 0 (Ω) is uniformly bounded and thus any sequence u εl with ε l 0 contains a subsequence satisfying (1.3.4)-(1.3.5), we may conclude that u ε converges to u 0 weakly in H0(Ω; 1 R m ) and strongly in L 2 (Ω; R m ), as ε 0. Fix 1 k d, 1 γ m and consider the identity a αβ ij (x/ε ) uβ ε x k δ αγ + ε χ αγ k (x/ε ) ψ dx Ω x j x i u β ε = a αβ ij x (x/ε ) (1.3.6) x k δ αγ + ε χ αγ k (x/ε ) ψ dx, j x i where ψ C0(Ω). 1 By Proposition 1.3.2, x k δ αγ + ε χ αγ k (x/ε ) x i Ω Y y i y k δ αγ + χ αγ k (y) dy = δ ik δ αγ (1.3.7) weakly in L 2 (Ω). Since L ε (u ε ) = F in Ω, in view of (1.3.5) and (1.3.7), it follows from Theorem (the Div-Curl Lemma) that the left hand side of (1.3.6) converges to fi α δ ik δ αγ ψ dx = f γ k ψ dx. Similarly, note that uβ ε x j a αβ ij (x/ε ) x i uβ x j Ω and x k δ αγ + ε χ αγ k (x/ε ) Ω a αβ ij (y) y k δ αγ + χ αγ k (y) dy Ω y i = â γβ kj weakly in L 2 (Ω), where we have used Proposition and (1.2.25). Since L ε P γ k + εχ γ k (x/ε) = 0 in R d,

16 16 CHAPTER 1. ELLIPTIC OPERATORS WITH PERIODIC COEFFICIENTS we may use Theorem again to claim that the right hand side of (1.3.6) converges to u β â γβ kj x ψ dx. j Since ψ C 1 0(Ω) is arbitrary, we obtain Ω f γ k = u β âγβ kj in Ω. (1.3.8) x j Finally, note that since L ε (u ε ) = F in H 1 (Ω; R m ), (1.3.5) imply that f γ k x k H 1 (Ω). In view of (1.3.8) we obtain x k â γβ kj u β = F γ in Ω, x j = F γ in i.e. L 0 (u) = F in Ω. This completes the proof. Remark In general u ε does not converge strongly to u 0 in H 1 (Ω; R m ). However, Theorem gives Since < F, u ε > < F, u 0 >, it follows that A(x/ε) u ε Â u 0 weakly in L 2 (Ω; R dm ). Ω a αβ ij In fact, it may be shown that Ω a αβ ij ( x ε ( x ε ) u β ε ) u β ε x j x j u α ε dx x i Ω u α ε ψ dx x i Ω for any ψ C 0 (Ω) (convergence of energy) (see [4, p.143]). â αβ u β 0 u α 0 ij dx. x j x i â αβ u β 0 u α 0 ij ψ dx x j x i Theorem Assume that A and Ω satisfy the same assumptions as in Theorem Let ε l 0. Suppose that w l H 1 (Ω; R m ) and div A(x/ε l )( w l + g l ) = F l in Ω. Further assume that w l w weakly in H 1 (Ω; R m ), g l g strongly in L 2 (Ω; R dm ) and F l F strongly in H 1 (Ω; R m ). Then div Â( w + g) = F in Ω, and A(x/ε l )( w l + g l ) Â( w + g) weakly in L2 (Ω; R dm ).

17 1.3. HOMOGENIZATION OF ELLIPTIC SYSTEMS 17 Proof. Let f l = A(x/ε l )( w l + g l ). We will show that if a subsequence of f l converges weakly to f in L 2 (Ω). Then f = Â( w + g). This would imply that f l converges weakly to Â( w + g) and div Â( w + g) = F in Ω. The proof is similar to that of Theorem Without loss of generality we may assume that f l converges weakly to f in L 2 (Ω). Consider the identity A(x/ε l ) w l + g l P k + ε l χ k(x/ε l ) ψ dx Ω = w l A (x/ε l ) P k + ε l χ k(x/ε l ) ψ dx (1.3.9) Ω + g l A (x/ε l ) P k + ε l χ k(x/ε l ) ψ dx, Ω where ψ C0(Ω). 1 By Theorem the left hand side of (1.3.9) converges to fψ dx Ω and the first term in the right hand side of (1.3.9) converges to  w ψ dx. Since g Ω l converges strongly to g in L 2 (Ω; R dm ) and A (x/ε l ) P k + ε l χ k (x/ε l) converges weakly to  in L 2 (Ω; R dm ), the second term in the right hand side of (1.3.9) converges to Âg ψ dx. Ω It follows that fψ dx =  wψ dx + Âgψ dx for any ψ C0(Ω). 1 This yields f = Â( w + g). Ω Ω We remark that in the proof of Theorems and 1.3.5, Proposition is used only in the special case h j = h. The general case may be used to prove the following result needed in the next chapter. We leave the details to the reader. Theorem Let A k (y) be a sequence of matrices satisfying (1.0.2) and (1.0.3). Suppose that L k ε k (u k ) = F in Ω, where ε k 0 and L k ε k = div ( A k (x/ε k ) ). We further assume that u k u weakly in H 1 (Ω; R m ) and Âk A 0, where Âk is the coefficient matrix of the homogenized operator associated with div(a k (x/ε) ). Then A k (x/ε k ) u k A 0 u weakly in L 2 (Ω; R dm ) Ω and div(a 0 u) = F in Ω. Homogenization of the Dirichlet Problem (1.1.3). Let F H 1 (Ω; R m ) and f H 1/2 ( Ω; R m ). By Theorem there exists a unique u ε H 1 (Ω; R m ) such that L ε (u ε ) = F in Ω and u ε = f on Ω

18 18 CHAPTER 1. ELLIPTIC OPERATORS WITH PERIODIC COEFFICIENTS (the boundary data is taken in the sense of trace). Furthermore, the solution u ε satisfies u ε H 1 (Ω) C F H 1 (Ω) + f H 1/2 ( Ω), where C depends only on µ and Ω. Choose G H 1 (Ω; R m ) such that G = f on Ω and G H 1 (Ω) 2 f H 1/2 ( Ω). Let w ε = u ε G. Then w ε H 1 0(Ω; R m ), w ε H 1 0 (Ω) C and div A(x/ε)( w ε + G) = F in Ω. Suppose that w ε w weakly in H 1 0(Ω; R m ). Then, by Theorem 1.3.5, div Â( w + G) = F in Ω. Since such w is unique in H 1 0(Ω; R m ), we may conclude that w ε w weakly in H 1 0(Ω; R m ) as ε 0. It follows that u ε u 0 = w + G weakly in H 1 (Ω; R m ) and strongly in L 2 (Ω; R m ). Note that u 0 is the weak solution of the Dirichlet problem: L 0 (u 0 ) = F in Ω and u 0 = f on Ω. Homogenization of the Neumann Problem (1.1.4). Let F H 1 0 (Ω; R m ), the dual of H 1 (Ω; R m ), and g H 1/2 ( Ω; R m ), the dual of H 1/2 ( Ω; R m ). Assume that F and g satisfy the compatibility condition (1.1.8). By Theorem 1.1.3, the Neumann problem (1.1.4) has a unique (up to constants) solution. Furthermore, if Ω u ε dx = 0, then u ε H 1 (Ω) C F H 1 0 (Ω) + g H 1/2 ( Ω), where C depends only on µ and Ω. Let u ε be a subsequence of u ε such that u ε u 0 weakly in H 1 (Ω; R m ). It follows from Theorem that A(x/ε ) u ε Â u 0 weakly in L 2 (Ω; R dm ). By taking limit in (1.1.7) we see that u 0 is the unique weak solution of the Neumann problem L 0 (u 0 ) = F in Ω and u 0 ν 0 = g on Ω, which satisfies Ω u 0 dx = 0. As a result we may conclude that u ε u 0 weakly in H 1 (Ω; R m ) and strongly in L 2 (Ω; R m ).

19 1.4. CONVERGENCE RESULTS Convergence results For 1 i, j d and 1 α, β m, let Note that b αβ ij b αβ ij L 2 (Y ), b αβ ij (y) = âαβ ij a αβ ij (y) aαγ ik (y) ( y k χ γβ j is Y-periodic. It follows from the definitions of χ and â αβ ij ( y i b αβ ij ) = 0 and Lemma There exist F αβ kij H1 per(y ) such that b αβ ij = y k ( F αβ kij ) and Y ). (1.4.1) in Section 1.2 that b αβ ij (y) dy = 0. (1.4.2) F αβ kij Moreover, if χ is Hölder continuous, then F αβ kij L (Y ). Proof. Since Y bαβ ij Define = F αβ ikj. (1.4.3) αβ dy = 0, there exists fij Hper(Y 2 ) such that f αβ Y ij dy = 0 and F αβ kij (y) = f αβ ij ( y k = b αβ ij in Y. (1.4.4) f αβ ij ) y i ( Clearly, F αβ kij H1 per(y ) and F αβ αβ kij = Fikj. Using y i b αβ ij = 0, we may deduce from (1.4.4) that y i f αβ ij is an Y-periodic harmonic function and thus is constant. Hence, y k F αβ kij = f αβ ij 2 y k y i f αβ kj f αβ kj ). = f αβ ij = b αβ ij. ) Finally, suppose that the function χ is Hölder continuous. Recall that L 1 (χ β j + P β j = 0 in R d. By Cacciopoli s inequality, this implies that χ is in the Morrey space L 2,ρ (Y ) for some ρ > d 2; i.e., χ 2 dy C r ρ for x Y and 0 < r < 1. Hence, b αβ ij B(x,r) L 2,ρ (Y ) for some ρ > d 2. In view of (1.4.4) we may deduce that f αβ ij It follows that F αβ kij L (Y ). L (Y ) C f αβ ij L 2 (Y ) + C sup x Y Y b αβ ij (y) x y d 1 dy C.

20 20 CHAPTER 1. ELLIPTIC OPERATORS WITH PERIODIC COEFFICIENTS Remark In the scalar case (m = 1), the function χ is Hölder continuous, by the classical De Giorgi -Nash estimate. As a result, we have F αβ kij C(µ). The same is true if m 2 and d = 2. Beginning with Chapter 2, we will assume that the coefficient matrix A(y) is Hölder continuous. Under this additional smoothness condition we see that χ and hence b αβ ij are Hölder continuous. It follows that F αβ kij is Hölder continuous. Recall that P β j (x) = x j(δ 1β,..., δ mβ ). Lemma Suppose that u ε H 1 (Ω; R m ), u 0 H 2 (Ω; R m ), and L ε (u ε ) = L 0 (u 0 ) in Ω. Let w ε (x) = u ε (x) u 0 (x) V β ε,j (x) P β j (x) uβ 0, (1.4.5) x j where V β 1β ε,j = (V 1 β m. Then ε,j,..., V mβ ε,j (L ε (w ε )) α =ε x i + x i + a αβ ij ) H1 (Ω; R m ) and L ε ( V β ε,j) = 0 in Ω for each 1 j d and [F αγ jik (x/ε)] 2 u γ 0 x j x k a αβ ij (x/ε) [ V βγ ε,k (x) x kδ βγ ] 2 u γ 0 x j x k (x/ε) x j [ V βγ ε,k (x) x kδ βγ εχ βγ k (x/ε) ] 2 u γ 0 x i x k, where F αβ kij is given by Lemma and δβγ = 1 if β = γ, and zero otherwise. Proof. Note that a αβ ij (x/ε) wβ ε x j = a αβ ij ( x ε Lε (w ε ) α = x i ) u β ( ε x ) u β a αβ 0 ij x j ε a αβ ij This, together with L ε (u ε ) = L 0 (u 0 ), gives [ â αβ ij a αβ ij + a αβ ij + x i ( x a αβ ij x j ε ( x ) V βγ ε,k ε x kδ βγ ) x j V βγ ε,k x kδ βγ uγ 0 x k 2 u γ 0 x k x j. ] u β (x/ε) 0 + L ε (V γ ε,k x P γ k ) α u γ 0 j x k ε,k x kδ βγ 2 u γ 0 (x/ε) V βγ x j [ ] a αβ ij (x/ε) V βγ ε,k x kδ βγ x i x k 2 u γ 0 x k x j. (1.4.6) Since L ε ( V γ ε,k P γ k ) ( ) = Lε P γ k = Lε εχ γ k (x/ε),

21 1.4. CONVERGENCE RESULTS 21 we obtain Lε (w ε ) α = x i + a αβ ij b αβ ij (x/ε) uβ 0 x j (x/ε) V βγ x j + [ ] a αβ ij x (x/ε) V βγ ε,k x kδ βγ i ε,k (x) x kδ βγ χβγ k 2 u γ 0 (x/ε) x j x i x k 2 u γ 0, x k x j (1.4.7) where b αβ ij (y) is defined by (1.4.1). In view of (1.4.3), we may re-write the first term in the right hand side of (1.4.7) as x i The formula (1.4.6) now follows. [ ] εf αβ kij x (x/ε) uβ 0 k x j = ε F αβ kij x (x/ε) 2 u β 0. i x k x j Theorem Suppose that A(y) satisfies conditions (1.0.2) and (1.0.3). Also assume that χ = (χ αβ l ) is Hölder continuous (if m 2 and d 3). Let Ω be a bounded Lipschitz domain. For ε 0, let u ε H 1 (Ω; R m ) be the weak solution of the Dirichlet problem: L ε (u ε ) = F in Ω and u ε = g on Ω, where F L 2 (Ω; R m ) and g H 1/2 ( Ω; R m ). Then, if u 0 H 2 (Ω; R m ), uε u 0 εχ β j (x/ε) uβ 0 x j εv ε H 1 0 (Ω) C ε 2 u 0 L 2 (Ω), (1.4.8) where v ε H 1 (Ω; R m ) satisfies L ε (v ε ) = 0 in Ω, and C depends only on A and Ω. v ε = χ β j (x/ε) uβ 0 x j on Ω, (1.4.9) Proof. We first point out that under the assumption that χ L (Y ) and u 0 H 2 (Ω; R m ), χ γ k (x/ε) uγ 0 x k H 1 (Ω; R m ). (1.4.10) To see this we only need to show that χ(x/ε) u 0 L 2 (Ω). Let F = χ γ k + P γ k 1 k d and 1 γ m. Since L 1 (F ) = 0 in R d, it follows that µ F 2 ψ 2 dx a αβ α ij (x) F F β ψ 2 dx R d R x d i x j = 2 F α a αβ β ij (x) F ψ ψ dx R x d j x i for some

22 22 CHAPTER 1. ELLIPTIC OPERATORS WITH PERIODIC COEFFICIENTS for any ψ C0(R 1 d ). By Hölder s inequality this gives R d F 2 ψ 2 dx C F 2 R d ψ 2 dx. (1.4.11) Note that since C0(R 1 d ) is dense in H 1 (R d ), the inequality (1.4.11) holds for any ψ H 1 (R d ). Now, given any ψ H 1 (Ω), there exists ψ H 1 (R d ) such that ψ = ψ in Ω and ψ H 1 (R d ) C ψ H 1 (Ω). This allows us to deduce from (1.4.11) that F 2 ψ 2 dx C F 2 ψ 2 H 1 (Ω) (1.4.12) Ω for any ψ H 1 (Ω). The observation (1.4.10) follows readily from (1.4.12). Finally, let h ε (x) = u ε (x) u 0 (x) εχ β j (x/ε) uβ 0 x j εv ε (x) H 1 0(Ω; R m ), where v ε is defined by (1.4.9). By taking in Lemma 1.4.3, we obtain Since χ αβ j ( Lε (h ε ) ) α = ε x i V αβ ε,j (x) = x jδ αβ + εχ αβ j (x/ε) [ ] F αγ 2 jik (x/ε) + aαβ ij (x/ε)χβγ k (x/ε) u γ 0. (1.4.13) x j x k is Hölder continuous, by Lemma 1.4.1, F αβ jik L (Y ). It follows from (1.4.13) that This finishes the proof. A few remarks are in order. h ε H 1 0 (Ω) C ε 2 u 0 L 2 (Ω). Remark Let m = 1 and Ω be a bounded Lipschitz domain. satisfies (1.0.2) and (1.0.3). Then Suppose that A(y) u ε u 0 L 2 (Ω) C ε 2 u 0 L 2 (Ω) + u 0 L (Ω). (1.4.14) To see this, we note that by Theorem uε u 0 εχ j (x/ε) u 0 x j εv ε L 2 (Ω) C ε 2 u 0 L 2 (Ω). This, together with the estimate v ε L (Ω) C u 0 L ( Ω) obtained by the maximum principle, yields (1.4.14). Also, observe that if Ω 1 Ω, then v ε H 1 (Ω 1 ) C v ε L 2 (Ω) C u 0 L (Ω), where C depends on Ω 1. It follows that uε u 0 εχ j (x/ε) u 0 H x C ε 1 (Ω 1 2 u ) 0 L 2 (Ω) + u 0 L (Ω). (1.4.15) j

23 1.5. NOTES 23 Remark Let m 1 and Ω be a bounded Lipschitz domain. Assume that χ is Hölder continuous. Observe that v ε H 1 (Ω) C χ(x/ε) u 0 H 1/2 ( Ω) C ψ ε χ(x/ε) u 0 H 1 (Ω), where ψ ε is a scalar function in C 1 0(R d ) such that ψ ε (x) = 1 if dist(x, Ω) < ε, ψ ε (x) = 1 if dist(x, Ω) 2ε, and ψ ε Cε 1. It follows that v ε H 1 (Ω) Cε 1/2 u 0 L (Ω) + C 2 u 0 L 2 (Ω). In view of (1.4.8) this leads to an O(ε 1/2 ) estimate in H 1 (Ω): uε u 0 εχ β j (x/ε) uβ 0 H x 1 (Ω) Cε1/2 u 0 L (Ω) + ε 1/2 2 u 0 L 2 (Ω). (1.4.16) j Remark Various convergence theorems will be proved in Chapter??, using uniform regularity results obtained in Chapters 2-6. In particular, we will show that if A satisfies (1.0.2)-(1.0.3), A = A, and is Hölder continuous, then 1/2 w ε H 1/2 (Ω) + w ε 2 δ(x) dx C ε 2 u 0 L 2 (Ω) + u 0 L 2 ( Ω) Ω (1.4.17) where w ε = u ε u 0 εχ β j (x/ε) uβ 0 x j and δ(x) = dist(x, Ω). This should be compared with the interior O(ε) estimates (1.4.15) and the O(ε 1/2 ) estimate (1.4.16) in H 1 (Ω). Observe that estimate (1.4.17) yields u ε u 0 L 2 (Ω) C ε u 0 H 2 (Ω). (1.4.18) 1.5 Notes Materials in Sections are standard and may be found in books [2] [9] [3].

24 24 CHAPTER 1. ELLIPTIC OPERATORS WITH PERIODIC COEFFICIENTS

25 Chapter 2 Interior Estimates In this chapter we establish uniform interior Hölder estimates, W 1,p estimates, and Lipschitz (C 0,1 ) estimates for solutions of L ε (u ε ) = F by a compactness method. As a result we obtain uniform size estimates of Γ ε (x, y), x Γ ε (x, y), y Γ ε (x, y), and x y Γ ε (x, y), where Γ ε (x, y) denotes the matrix of fundamental solutions for L ε in R d. This in turn allows us to establish asymptotic expansions, as ε 0, of Γ ε (x, y), x Γ(x, y), y Γ ε (x, y), and x y Γ ε (x, y). The compactness method, which originated from the regularity theory in the calculus of variations and minimal surface, was introduced to the study of homogenization problems by M. Avellaneda and F. Lin [1]. It also will be used in Chapters 3 and 4 to establish uniform boundary Hölder and Lipschitz estimates for solutions of L ε (u ε ) = F with Dirichlet or Neumann boundary conditions. We point out that since u ε in general does not converge strongly in L 2, no uniform C 1,α estimates of u ε should be expected. Throughout this chapter, unless otherwise indicated, we will assume that the coefficient matrix A(y) satisfies the ellipticity condition (1.0.2), the periodicity condition (1.0.3), and the Hölder continuity condition, A(x) A(y) τ x y λ for any x, y R d, (2.0.1) where τ 0 and λ (0, 1]. We shall call A Λ(µ, λ, τ) if it satisfies (1.0.2), (1.0.3) and (2.0.1). We will use u to denote the average of u over E; i.e. E u = 1 u. E E E 2.1 Interior Hölder estimates As an intermediate step we establish uniform interior Hölder estimates in this section. Recall that for a ball B = B(x, r) = y R d : y x < r and t > 0, we use tb to denote the ball B(x, tr). 25

26 26 CHAPTER 2. INTERIOR ESTIMATES Theorem Suppose that A Λ(µ, λ, τ). Let u ε H 1 (2B; R m ) be a weak solution to L ε (u ε ) = 0 in 2B for some ball B = B(x 0, r). Then, for any 0 < ρ < 1, u ε (x) u ε (y) C ρ ( x y r ) ρ for any x, y B, where C ρ depends only on µ, λ, τ, and ρ. In particular, u ε L (B) C 2B 2B u ε 2 1/2 (2.1.1) u ε 2 1/2. (2.1.2) Remark We will show in the next section that estimate (2.1.1) in fact holds for ρ = 1. In the scalar case (m = 1), the Hölder estimate (2.1.1) (and hence (2.1.2)) holds for some ρ > 0 depending on µ, under the ellipticity condition only. This is the De Giorgi-Nash estimate. Theorem is proved by a three-step compactness argument (one-step improvement, iteration, and blow-up), which relies on Theorem and takes advantage of the fact that solutions of elliptic systems with constant coefficients are smooth. The same line of argument will also be used to establish interior Lipschitz estimates as well as boundary Hölder and Lipschitz estimates. Lemma (One-step improvement). Let 0 < ρ < 1. Then there exist constants ε 0 > 0 and θ (0, 1/4), depending only on µ, λ, τ, and ρ, such that u ε u ε 2 θ 2ρ (2.1.3) B(0,θ) B(0,θ) for any 0 < ε < ε 0, whenever u ε H 1 (B(0, 1); R m ), L ε (u ε ) = 0 in B(0, 1) and u ε 2 1. B(0,1) Proof. We prove the estimate (2.1.3) by contradiction, using Theorem1.3.6 and the following observation: for any θ (0, 1/4), u u 2 C 0 θ 2 u 2, (2.1.4) B(0,θ) B(0,θ) B(0,1/2) where L 0 (u) = 0 in B(0, 1/2), L 0 is a second-order elliptic operator with constant coefficients, and C 0 depends only on the ellipticity constant of L 0. Estimate (2.1.4) follows from the interior Lipschitz estimate u L (B(0,1/4)) C u 2 B(0,1/2)

27 2.1. INTERIOR HÖLDER ESTIMATES 27 for solutions of second-order elliptic systems with constant coefficients. Recall that if A Λ(µ, λ, τ), then the matrix  = ( ) â αβ ij of effective coefficients satisfies the ellipticity condition (1.2.20) for some µ 1 > 0 depending only on µ. Let C 0 be the constant in (2.1.4) for elliptic operators with constant coefficients satisfying (1.2.20). Choose θ (0, 1/4) so small that 2 d C 0 θ 2 < θ 2ρ. (2.1.5) We claim that estimate (2.1.3) holds for this θ and some ε 0 > 0, which depends only on µ, λ, τ and ρ. Suppose this is not the case. Then there exist sequences ε k, A k Λ(µ, λ, τ) and u k H 1 (B(0, 1); R m ) such that ε k 0, div (A k (x/ε k ) u k ) = 0 in B(0, 1), u k 2 1, and B(0,1) B(0,θ) u k u k 2 > θ 2ρ. (2.1.6) B(0,θ) Note that u k is bounded in L 2 (B(0, 1); R m ) and hence in H 1 (B(0, 1/2); R m ) by the Cacciopoli s inequality (1.1.10). By passing to a subsequence, we may assume that u k u weakly in L 2 (B(0, 1); R m ) and in H 1 (B(0, 1/2); R m ). Since Âk is bounded, we may further assume that Âk A. Clearly, the constant matrix A satisfies the ellipticity condition (1.2.20). It follows from Theorem that div(a u) = 0 in B(0, 1/2). Since u k 2 dx B(0, 1), B(0,1) and u k u 0 weakly in L 2 (B(0, 1); R m ), we obtain u 2 dx B(0, 1). (2.1.7) B(0,1) Also, note that u k u strongly in L 2 (B(0, 1/2); R m ). In view of (2.1.6), this gives u u 2 θ 2ρ. (2.1.8) B(0,θ) B(0,θ) However, by (2.1.4), (2.1.7), and (2.1.5), u u 2 2 d C 0 θ 2 B(0,θ) which contradicts (2.1.8). B(0,θ) B(0,1) u 2 2 d C 0 θ 2 < θ 2ρ,

28 28 CHAPTER 2. INTERIOR ESTIMATES A very important feature of the family of operators L ε, ε > 0 is the following rescaling property: ) if L ε (u ε = F and v(x) = uε (rx), then L ε (v) = G, where G(x) = (2.1.9) r2 F (rx). r It will be used in the proof of the next lemma as well as in numerous other rescaling arguments in this monograph. The property of translation will also be important to us: if div ( A(x/ε) u ε ) = F and v(x) = u ε (x x 0 ), then div ( B(x/ε) v ) = G, where B(x) = A(x + ε 1 x 0 ) and G(x) = F (x x 0 ). (2.1.10) Note that the matrix B(x) satisfies the ellipticity condition (1.0.2) with the same µ and the periodicity condition (1.0.3). Lemma (Iteration). Fix 0 < ρ < 1. Let ε 0 and θ be given by Lemma Suppose that u ε H 1 (B(0, 1); R m ) and L ε (u ε ) = 0 in B(0, 1). Then, if 0 < ε < ε 0 θ k 1 for some k 1, B(0,θ k ) u ε B(0,θ k ) u ε 2 θ 2kρ u ε 2. (2.1.11) B(0,1) Proof. We prove the estimate (2.1.11) by an induction argument on k, using Lemma and the rescaling property (2.1.9). The case k = 1 follows directly from Lemma by applying estimate (2.1.3) to ( v ε = u ε / B(0,1) u ε 2 ) 1/2. Now suppose that the estimate (2.1.11) is true for some k 1. Let L ε (u ε ) = 0 in B(0, 1) and 0 < ε < ε 0 θ k. Define 1/2 w(x) = θ u ρk ε (θ k x) u ε / u ε 2. B(0,θ k ) B(0,1) Then and by the induction assumption, Since ε θ k L ε (w) = 0 in B(0, 1), θk w 2 1. B(0,1) < ε 0, we may apply Lemma to obtain w w 2 θ 2ρ. B(0,θ) B(0,θ)

29 2.1. INTERIOR HÖLDER ESTIMATES 29 This gives the desired estimate u ε and completes the proof. B(0,θ k+1 ) B(0,θ k+1 ) u ε 2 θ 2(k+1)ρ u ε 2, B(0,1) With Lemma at our disposal, we complete the proof of Theorem by a blow-up argument. We also use the Campanato s characterization of Hölder spaces. For E R d, let u(x) u(y) u C 0,ρ (E) = sup : x, y R d and x y. (2.1.12) x y ρ If Ω is a bounded Lipschitz domain, then u 2 C 0,ρ (Ω) r sup 2ρ Ω(x,r) where Ω(x, r) = Ω B(x, r) (see e.g. [6, pp.70-72]). u u 2, x Ω and 0 < r < diam(ω) Ω(x,r), (2.1.13) Proof of Theorem By translation and dilation we may assume that x 0 = 0 and r = 1. Let ε 0 and θ be given by Lemma We will show that u ε u ε 2 Cr 2ρ u ε 2 (2.1.14) B(0,r) B(0,r) B(0,1) for any 0 < r < θ, whenever u ε H 1 (B(0, 1); R m ) and L ε (u ε ) = 0 in B(0, 1). By translation this implies that u ε u ε 2 Cr 2ρ u ε 2 (2.1.15) B(x,r) B(x,r) B(0,2) for any x B(0, 1) and 0 < r < 1/2, if u ε H 1 (B(0, 2); R m ) and L ε (u ε ) = 0 in B(0, 2). The Hölder estimate (2.1.1) follows from (2.1.15) by the Campanato s characterization (2.1.13) of Hölder spaces. To see (2.1.14), we first note that if ε θε 0, A(x/ε) A(y/ε) τε 1 x y λ τ(θε 0 ) λ x y λ. Estimate (2.1.14) thus follows directly from the standard Lipschitz estimates for second order elliptic systems in divergence form with Hölder continuous coefficients (see e.g. [6, p.88]). Assume now that 0 < ε < θε 0. Consider the case ε/ε 0 r < θ. There exists k 1 such that θ k+1 r < θ k. Since θ k > ε/ε 0, by Lemma 2.1.4, B(0,θ k ) u ε B(0,θ k ) u ε 2 θ 2kρ B(0,1) u ε 2 Cr 2ρ u ε 2. (2.1.16) B(0,1)

30 30 CHAPTER 2. INTERIOR ESTIMATES Using we see that B(0,t) 2 E u ε E u ε 2 = E u ε u ε 2 B(0,t) E u ε (x) u ε (y) 2 dxdy, ( ) 2d t1 t B(0,t 1 ) u ε u ε 2 (2.1.17) B(0,t 1 ) for any 0 < t < t 1. In view of (2.1.16) and (2.1.17), we have proved the estimate (2.1.14) for any ε/ε 0 r < θ. Finally, consider the case 0 < r < ε/ε 0. Let w(x) = u ε (εx) u ε. B(0,2ε/ε 0 ) Since L 1 (w) = 0 in B(0, 2/ε 0 ), it follows from the standard regularity theory for L 1 that Hence, if 0 < r < ε/ε 0, B(0,r) B(0,r) w B(0,r) ( r 2ρ u ε u ε 2 C B(0,r) ε) w 2 Cr 2ρ w 2. B(0,2/ε 0 ) B(0,2ε/ε 0 ) Cr 2ρ u ε 2, B(0,1) u ε u ε 2 B(0,2ε/ε 0 ) where we haves used (2.1.14) for r = 2ε/ε 0 in the last inequality. This completes the proof of Theorem Remark Observe that the smoothness condition (2.0.1) is used in the blow-up argument as well as for the case ε θε 0, not in the proofs of Lemmas and As a result, the Hölder estimate (2.1.1) in fact holds under the conditions (1.0.2)-(1.0.3) and A C(R d ). Furthermore, since solutions of second-order elliptic systems in divergence form with VMO coefficients are Hölder continuous, one may replace the continuity condition by A V MO(R d ); i.e., lim sup t 0 x R d,0<r<t B(x,r) 2.2 Interior Lipschitz estimates In this section we establish interior Lipschitz estimates. A A = 0. (2.1.18) B(x,r)

31 2.2. INTERIOR LIPSCHITZ ESTIMATES 31 Theorem (Interior Lipschitz estimate). Suppose that A Λ(µ, λ, τ). Let u ε H 1 (B; R m ) be a weak solution to L ε (u ε ) = 0 in B for some ball B = B(x 0, r). Then where C depends only on µ, λ, and τ. u ε (x 0 ) C 1/2 u ε 2, (2.2.1) r The proof of Theorem is parallel to that of Theorem However, correctors χ β l = (χαβ l ) are needed to handle the Lipschitz estimates. Recall that P β l (x) = x l(δ αβ ). Lemma (One-step improvement). Let 0 < ρ < 1. There exist constants ε 0 > 0 and θ (0, 1/4), depending only on µ, λ, τ, and ρ, such that for 0 < ε < ε 0, sup uε (x) u ε (0) P βl (x) + εχβl (x/ε) u β ε θ 1+ρ u ε L x θ x (B(0,1)), (2.2.2) l B B(0,θ) whenever u ε H 1 (B(0, 1); R m ) and L ε (u ε ) = 0 in B(0, 1). Proof. Estimate (2.2.2) is proved by contradiction, using Theorem and the following observation: for any θ (0, 1/4), sup u(x) u(0) xl x θ B(0,θ) u x l C0 θ 2 ( B(0,1/2) u 2 ) 1/2, (2.2.3) where L 0 (u) = 0 in B(0, 1/2), L 0 is a second-order elliptic operator with constant coefficients, and C 0 depends only on the ellipticity constant of L 0. Estimate (2.2.3) is a direct consequence of the standard regularity estimate 2 u L (B(0,1/4)) C u L 2 (B(0,1/2)) for solutions of second-order elliptic systems with constant coefficients. Choose θ (0, 1/4) such that 2 d C 0 θ 2 < θ 1+ρ. We claim that the estimate (2.2.2) holds for this θ and some ε 0 > 0, which depends only on µ, λ, τ, and ρ. Suppose this is not the case. Then there exist sequences ε k, A k Λ(µ, λ, τ) and u k H 1 (B(0, 1); R m ) such that ε k 0, div (A k (x/ε k ) u k ) = 0 in B(0, 1), and sup x θ u k L (B(0,1)) 1, uk (x) u k (0) P β l (x) + ε kχ k,β l (x/ε k ) u β k > θ 1+ρ, (2.2.4) B(0,θ) x l where χ k,β l are the correctors associated with the matrix A k. By passing to a subsequence, as in the proof of Lemma 2.1.3, we may assume that u k u weakly in L 2 (B(0, 1); R m ) and

32 32 CHAPTER 2. INTERIOR ESTIMATES H 1 (B(0, 1/2); R m ), where L 0 (u) = 0 in B(0, 1/2) for some second order elliptic operator L 0 with constant coefficients satisfying (1.2.20). Note that by Theorem 2.1.1, u k C 0,ρ (B(0,1/4)) C. Consequently, by passing to a subsequence, we may also assume that u k u uniformly in B(0, 1/4). Now, let k in (2.2.4). This gives sup u(x) u(0) xl x θ B(0,θ) u x l θ 1+ρ. (2.2.5) Also, note that B(0,1) u 2 B(0, 1). It then follows from (2.2.5) and (2.2.3) that θ 1+ρ C 0 θ ( 2 B(0,1/2) ) 1/2 ) 1/2 u 2 2 d C 0 θ ( 2 u 2 2 d C 0 θ 2, B(0,1) which is in contradiction with the choice of θ. This completes the proof. Lemma (Iteration). Let 0 < ρ < 1 and ε 0 > 0, θ (0, 1/4) be given by Lemma Suppose that L ε (u ε ) = 0 in B(0, 1) and 0 < ε < θ k 1 ε 0 for some k 1. Then there exist constants b(ε, k) R and E(ε, k) = (E β l (ε, k)) Rdm, such that b(ε, k) C E(ε, k 1), E(ε, k) C 1 + θ + + θ (k 1)ρ u ε L (B(0,1)) and sup uε (x) u ε (0) εb(ε, k) P β l (x) + εχβ l (x/ε) E β l (ε, k) x θ k where C depends only on µ, λ, τ, and ρ. θ k(1+ρ) u ε L (B(0,1)), (2.2.6) Proof. We prove (2.2.6) by an induction argument on k. If k = 1, let b(ε, k) = 0 and E β l (ε, k) = u β ε. x l B(0,θ) The desired estimate follows from Lemma (set E(ε, 0) = 0). Suppose now that (2.2.6) holds for some k 1. Let 0 < ε < θ k ε 0 and L ε (u ε ) = 0 in B(0, 1). Consider the function w(x) = u ε (θ k x) u ε (0) εb(ε, k) θ P k β l (x) + εθ k χ β l (θk x/ε) E β l (ε, k). Then L ε (w) = 0 in B(0, 1). θk

33 2.2. INTERIOR LIPSCHITZ ESTIMATES 33 Since εθ k < ε 0, it follows from Lemma that sup w(x) w(0) P β l (x)+εθ k χ β l (θk x/ε) x θ θ 1+ρ w L (B(0,1)). B(0,θ) w β x l (2.2.7) Note that by the induction assumption, w L (B(0,1)) = sup x θ k uε (x) u ε (0) θ k(1+ρ) u ε L (B(0,1). εb(ε, k) P β l (x) + εχβ l (x/ε) E β l (ε, k) This, together with (2.2.7), gives where sup u ε (x) u ε (0) εb(ε, k + 1) P β l (x) + εχβ l (x/ε) E β l (ε, k + 1) x θ k+1 θ (k+1)(1+ρ) u ε L (B(0,1)), b(ε, k + 1) = χ β l (0) Eβ l (ε, k), E β l (ε, k + 1) = Eβ l (ε, k) + θ k Since χ C, we see that b(ε, k + 1) C E(ε, k). Finally, we observe that by the divergence theorem, Hence, B(0,θ) B(0,θ) w β x l. w x l C w L (B(0,θ)) Cθ k(1+ρ) u ε L (B(0,1)). E(ε, k + 1) E(ε, k) + Cθ kρ u ε L (B(0,1)) C 1 + θ ρ + θ kρ u ε L (B(0,1)). This completes the induction argument and thus the proof. We now give the proof of Theorem Proof of Theorem We will show that if L ε (u ε ) = 0 in B(x 0, r), then u ε (x 0 ) Cr 1 u ε L (B(x 0,r)). (2.2.8) Estimate (2.2.2) follows from (2.2.8) and the L estimate (2.1.2). By translation and rescaling we may assume that x 0 = 0 and r = 1. We may further assume that 0 < ε < θε 0, as the case ε θε 0 follows directly from the standard Lipschitz estimates for elliptic systems in divergence form with Hölder continuous coefficients.

34 34 CHAPTER 2. INTERIOR ESTIMATES Suppose now that L ε (u ε ) = 0 in B(0, 1) and 0 < ε < θε 0. To apply Lemma we choose k 1 such that θ k+1 ε 0 ε < θ k ε 0. Note that It follows from the estimate (2.2.6) that Finally, we let b(ε, k + 1) + E(ε, k + 1) Cε u ε L (B(0,1)). sup u ε (x) u ε (0) sup u ε (x) u ε (0) Cε u ε L (B(0,1)). (2.2.9) x θε x θ k+1 ε 0 w(x) = u ε(εx) u ε (0). ε Then L 1 (w) = 0 in B(0, 2ε 1 0 ). Again, by the standard regularity theory, w(0) C w L (B(0,θε 1 0 )). (2.2.10) Since w(0) = u ε (0), in view of (2.2.9) and (2.2.10), we obtain u ε (0) Cε 1 This completes the proof of (2.2.8). sup x < θε 0 ε u ε (x) u ε (0) C u ε L (B(0,1)). 2.3 A real-variable method In this section we introduce a real-variable method for L p estimates. It may be regarded as a refined and dual version of the celebrated Calderón-Zygmund Lemma. For f L 1 loc (Rd ), the Hardy-Littlewood maximal function M(f) is defined by M(f)(x) = sup f : x B. (2.3.1) It is known that the operator M is bounded on L p (R d ) for 1 < p, and is of weak type (1, 1): x R d : M(f)(x) > t C f dx for any t > 0 (2.3.2) t R d (see e. g. [12, Chapter 1] for a proof). For a fixed ball B in R d, the localized Hardy- Littlewood maximal function M B (f) is defined by M B (f)(x) = sup f : x B and B B. (2.3.3) B B

35 2.3. A REAL-VARIABLE METHOD 35 Since M B (f)(x) M(fχ B )(x) for any x B, it follows that M B is bounded on L p (B) for 1 < p, and is of weak type (1, 1). In the proof of Theorem we will perform a Calderón-Zygmund decomposition. It will be convenient to work with (open) cubes Q in R d with sides parallel to the coordinate planes. By tq we denote the cube that has the same center and t times the side length as Q. We say Q is a dyadic subcube of Q if Q may be obtained from Q by repeatedly bisecting the sides of Q. If Q is obtained from Q by bisecting each side of Q once, we will call Q the dyadic parent of Q. Observe that if E Q is open and E < 2 d Q, then there exists a sequence of disjoint dyadic subcubes Q k of Q such that Q k E, the dyadic parent of Q k in Q is not contained in E, and E \ k Q k = 0. To see this, one simply collects all dyadic subcubes Q of Q with the property that Q E and its dyadic parent is not contained in E; i.e. Q is maximal among all dyadic subcubes of Q that are contained in E. Note that since E is open in Q, the set E \ k Q k is contained in the union Z of boundaries of all dyadic subcubes of Q, and Z has measure zero. This is a dyadic version of the Calderón-Zygmund decomposition. Theorem Let B 0 be a ball in R d and F L 2 (4B 0 ). Let q > 2 and f L p (4B 0 ) for some 2 < p < q. Suppose that for each ball B 2B 0 with B c 1 B 0, there exist two measurable functions F B and R B on 2B, such that F F B + R B on 2B, ( 2B ( 2B ) 1/q ( ) 1/2 ( ) 1/2 R B q C 1 F 2 + sup f 2, c 2 B 4B 0 B B B ) 1/2 ( ) 1/2 F B 2 C 2 sup f 2, 4B 0 B B B where C 1, C 2 > 0, 0 < c 1 < 1, and c 2 > 2. Then F L p (B 0 ) and ( ) 1/p F p C B 0 ( F 2 4B 0 where C depends only on C 1, C 2, c 1, c 2, p, and q. (2.3.4) ) 1/2 ( ) 1/p + f p, (2.3.5) 4B 0 Proof. Let Q 0 be a cube such that 2Q 0 2B 0 and Q 0 B 0. We will show that ( ) 1/p F p C Q 0 ( F 2 4B 0 ) 1/2 ( ) 1/p + f p, (2.3.6) 4B 0 where C depends only on C 1, C 2, c 1, c 2, p, q, and Q 0 / B 0. Estimate (2.3.5) follows from (2.3.6) by covering B 0 with a finite number of non-overlapping cubes Q 0 of same size such that 2Q 0 2B 0.

36 36 CHAPTER 2. INTERIOR ESTIMATES To prove (2.3.6), let E(t) = x Q 0 : M 4B0 ( F 2 )(x) > t for t > 0. We claim that it is possible to choose three constants δ, γ (0, 1) and C 0 > 0, depending only on C 1, C 2, c 1, c 2, p, and q, such that for all t > t 0, where E(αt) δ E(t) + x Q 0 : M 4B0 ( f 2 )(x) > γt (2.3.7) α = (2δ) 2/p and t 0 = C 0 F 2. 4B 0 (2.3.8) Assume the claim (2.3.7) for a moment. We multiply both sides of (2.3.7) by t p 2 1 and then integrate the resulting inequality in t over the interval (t 0, T ). This leads to T t 0 T t p 2 1 E(αt) dt δ t p 2 1 E(t) dt + C γ f t 0 4B p dx, (2.3.9) 0 where we have used the boundedness of M 4B0 on L p/2 (4B 0 ). By a change of variables in the left hand side of (2.3.9), we may deduce that for any T > 0, α p 2 (1 δα p 2 ) T 0 t p 2 1 E(t) dt C Q 0 t p C γ 4B 0 f p dx. (2.3.10) Note that δα p/2 = (1/2). By letting T in (2.3.10) we obtain Q 0 F p dx C Q 0 t p C 4B 0 f p dx, (2.3.11) which, in view of (2.3.8), gives (2.3.6). It remains to prove (2.3.7). To this end we first note that by the weak (1, 1) estimate for M 4B0, E(t) C d t 4B 0 F 2 dx, where C d depends only on d. It follows that E(t) < δ Q 0 for any t > t 0, if we choose C 0 = 2δ 1 C d 4B 0 / Q 0 in (2.3.8) with δ (0, 1) to be determined. We now fix t > t 0. Since E(t) is open in Q 0, by the (dyadic) Calderón-Zygmund decomposition, E(t) \ k Q k = 0,