東北大学機関リポジトリTOUR

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An Introduction to the Qualitative

and Quantitative Theory of Homogenization

Stefan NEUKAMM

Faculty of Mathematics, Technische Universita¨t Dresden

We present an introduction to periodic and stochastic homogenization of elliptic partial differential equations. The first part is concerned with the qualitative theory, which we present for equations with periodic and random coefficients in a unified approach based on Tartar’s method of oscillating test functions. In particular, we present a self-contained and elementary argument for the construction of the sublinear corrector of stochastic homogenization. (The argument also applies to elliptic systems and in particular to linear elasticity). In the second part we briefly discuss the representation of the homogenization error by means of a two-scale expansion. In the last part we discuss some results of quantitative stochastic homogenization in a discrete setting. In particular, we discuss the quantification of ergodicity via concentration inequalities, and we illustrate that the latter in combination with elliptic regularity theory leads to a quantification of the growth of the sublinear corrector and the homogenization error.

KEYWORDS: stochastic homogenization, quantitative stochastic homogenization, corrector, two-scale expansion

Preface and Acknowledgments The present notes originate from a one week mini-course given by the author during the GSIS International Winter School 2017 on ‘‘Stochastic Homogenization and its Applications’’ at the Tohoku University, Sendai, Japan. The author would like to thank the organizers of that workshop, especially Reika Fukuizumi, Jun Masamune and Shigeru Sakaguchi for their very kind hospitality. The present notes are devoted to graduate students and young researchers with a basic knowledge in PDE theory and functional analysis. The first three chapters are rather self-contained and offer an introduction to the basic theory of periodic homogenization and its extension to homogenization of elliptic operators with random coefficients. The last chapter, which is in parts based on an extended preprint to the paper [13] by Antoine Gloria, Felix Otto and the author, is a bit more advanced, since it invokes some input from elliptic regularity theory (in a discrete setting) that we do not develop in this manuscript. The author would like to thank Mathias Scha¨ffner and Helmer Hoppe for proofreading the original manuscript, and Andreas Kunze for providing the illustrations and numerical results, which were obtained in his master thesis [23]. The author was supported by the DFG in the context of TU Dresden’s Institutional Strategy ‘‘The Synergetic University’’.

1. Introduction — A One-dimensional Example

Consider a heat conducting body that occupies some domain O Rd, where d ¼ 1; 2; . . . denotes the dimension. Suppose that the body is exposed to a heat source/sink that does not vary in time, and suppose that the body is cooled at its boundary, such that its temperature is zero at the boundary. If time evolves the temperature of the body will converge to a steady state, which can be described by the elliptic boundary value problem

r ðaruÞ ¼ f in O; u ¼ 0 on @O: In this equation

. u : O ! R denotes the (sought for) temperature ﬁeld, . f : O ! R is given and describes the heat source.

The ability of the material to conduct heat is described by a material parameter a 2 ð0; 1Þ, called the conductivity. The material is homogeneous, if a does not depend on x. The material is called heterogeneous, if aðxÞ varies in x 2 O. In this lecture we are interested in heterogeneous materials with microstructure, which means that the heterogeneity varies on a length scale, called the microscale, that is much smaller than a macroscopic length scale of the problem, e.g., the diameter of the domain O or the length scale of the right-hand side f .

To ﬁx ideas, suppose that aðxÞ ¼ a0ð_‘xÞwith a0periodic, i.e., the conductivity is periodic with the period ‘. If the ratio " :¼_macroscalemicroscale¼‘_L

is a small number, e.g., " . 103, then we are in the regime of a microstructured material. The goal of homogenization is to derive a simpliﬁed PDE by studying the limit " # 0, i.e., when the micro- and macroscale separate.

In the rest of the introduction we treat the following one-dimensional example: Let O ¼ ð0; LÞ R, " > 0 and let u": O ! R be a solution to the equation

@xðaðx_"Þ@xu"ðxÞÞ ¼ f in O; ð1:1Þ

u"¼0 on @O: ð1:2Þ

We suppose that a : R ! R is 1-periodic and uniformly elliptic, i.e., there exists > 0 such that aðxÞ 2 ð; 1Þ for all x 2 R. For simplicity, we assume that f and a are smooth.

We are going to prove the following homogenization result:

. For all " > 0 equations ð1.1Þ, ð1.2Þ admit a unique smooth solution u". . As " # 0, u"converges to a smooth function u0.

. The limit u0 is the unique solution to the equation

@xða0@xu0Þ ¼ f in O; ð1:3Þ

u0¼0 on @O; ð1:4Þ

where a02 Rdenotes the harmonic mean of a, i.e.,

a0¼ Z1 0 a1ðyÞ dy 1 :

Problem 1. Show that ð1.1Þ and ð1.2Þ admit a unique, smooth solution.

The solution to this and all subsequent problems in this introduction can be found in Appendix A. We have an explicit presentation for the solution:

u"ðxÞ ¼ Z x 0 a1_" ðx0Þ c" Z x0 0 f ðx00Þdx00 dx0; ð1:5Þ where c"¼ ZL 0 a1_" ðx0Þdx0 1Z_L 0 Z x0 0 a1_" ðx0Þf ðx00Þdx00dx0:

In order to pass to the limit " # 0 in the representation ð1.5Þ, we need to understand the limit of functions of the form

x 7! 1 aðx_"Þ

Z x 0

f ðx0Þdx0:

This function rapidly oscillates on scale " and the amplitude of the oscillations is of unit order. Hence, the expression does not converge uniformly (or in a strong sense). Nevertheless, we have the following result:

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Lemma 1.1. Let Fðy; xÞ be a smooth function that is 1-periodic in y 2 R and assume that F and @xF are bounded. Then lim "#0 Z b a Fðx_"; xÞ dx ¼ Zb a FðxÞ dx; FðxÞ ¼ Z1 0 Fðy; xÞ dy:

Furthermore, there exists a constant C (only depending on F) such that Zb a ðFðx_"; xÞ FðxÞÞ dx Cðjb aj þ 1Þ": Proof. Consider the functions

Gðy; xÞ ¼ Zy

0

ðFðy0; xÞ FðxÞÞ dy0; g"ðxÞ :¼ "Gðx"; xÞ:

Note that Gðy; xÞ and @xGðy; xÞ are 1-periodic in y; indeed, we have

Gðy þ 1; xÞ Gðy; xÞ ¼ Zyþ1

y

ðFðy0; xÞ FðxÞÞ dy0¼ FðxÞ FðxÞ ¼ 0;

and the same is true for @xG. Furthermore, G and @xG are smooth and bounded, and we have @xg"ðxÞ ¼ "@xGðx_"; xÞ þ @yGðx_"; xÞ ¼ "@xGðx_"; xÞ þ ðFðx_"; xÞ FðxÞÞ; and thus Zb a ðFðx_"; xÞ FðxÞÞ dx ¼ Zb a ð@xg"ðxÞ "@xGðx;x_"ÞÞdx ¼" Gðb_"; bÞ Gða_"; aÞ Zb a @xGðx_"; xÞ dx : (a) (b) (c) (d) (e) (f)

Fig. 1. (a)–(c) show the rapidly oscillating coeﬃcient ﬁeld aðx_"Þ ¼2 þ sinð2x_"Þfor " 2 f1₄;₁₆1;₃₂1g. (d)–(f) show the solutions to ð1.1Þ and ð1.2Þ with f ðxÞ ¼ 3ð2x 1Þ.

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The expression in the brackets is bounded uniformly in " (by smoothness and periodicity of G and @xG), and thus the

statement follows.

Problem 2. Show that maxx2Oju"ðxÞ u0ðxÞj C" where C only depends on O, f and a.

The physical interpretation of the result of Problem 2 is the following: While the initial problem ð1.1Þ & ð1.2Þ describes a heterogeneous, microstructured material (a periodic composite with period "), the limiting equation ð1.3Þ & ð1.4Þ describes a homogeneous material with conductivity a0. Hence, Problem 2 states that if we observe a material with a rapidly oscillating conductivity að

"Þon a macroscopic length scale, then it behaves like a homogeneous material with eﬀective conductivity given by a0. We therefore call ð1.3Þ & ð1.4Þ the homogenized problem. It is much simpler than the heterogeneous initial problem ð1.1Þ & ð1.2Þ:

Problem 3. Let f 1. Show that a solution to

@xða@xuÞ ¼ 1 in O; u ¼ 0 on @O:

is a quadratic function, if and only if the material is homogeneous, i.e., iﬀ a does not depend on x.

The homogenization result shows that u"!u0 as " # 0. Hence, for " 1 the function u0 is a consistent approximation to the solution to ð1.1Þ & ð1.2Þ. We even have a rate: u"¼u0þOð"Þ. Thanks to the homogenization result certain properties of the diﬃcult equation ð1.1Þ & ð1.2Þ can be studied by analyzing the simpler problem ð1.3Þ & ð1.4Þ:

Problem 4. Let f 1 and O ¼ ð0; 1Þ. Show that M":¼ max_Ou"¼_8a1₀þOð"Þ. What can be said about the convergence of the gradient @xu"?

Problem 5. Show that lim supR_Oj@xu"@xu0j2 > 0 (unless the initial material is homogeneous). Show on the other hand, that for all smooth functions ’ : R ! R we have

Z O u0_"ðxÞ’ðxÞ dx ! Z O u0₀ðxÞ’ðxÞ dx

i.e., we have weak convergence, but not strong convergence.

Yet, we can modify u0 by adding oscillations, such that the gradient of the modiﬁed functions converges: Lemma 1.2(Two-scale expansion). Let a; f be smooth, O ¼ ð0; 1Þ. Let : R ! R denote a 1-periodic solution to

@yðaðyÞð@yðyÞ þ 1ÞÞ ¼ 0 ð1:6Þ

with ð0Þ ¼ 0. Let u0 and u"be as above. Consider

v"ðxÞ :¼ u0ðxÞ þ "ðx_"Þ@xu0ðxÞ: Then there exists a constant C > 0 such that for all " > 0 with 1

"2 Nwe have Z O ju"v"j2þ j@xu"@xv"j2 4 2max jj 2 "2 Z O j@2_xu0j2:

Proof. To ease notation we write

a"ðxÞ :¼ aðx_"Þ; "ðxÞ :¼ ðx_"Þ: Step 1.

It can be easily checked (by direct calculations) that

ðyÞ :¼ Zy 0 a0 aðtÞ1 dt

and that is smooth and bounded. Note that

a0¼aðyÞð@yðyÞ þ 1Þ for all y 2 R:

Indeed, by the corrector equation ð1.6Þ and the deﬁnition of a0the diﬀerence of both functions is constant and has zero mean. (This is only true in the one-dimensional case!)

Step 2.

Set z":¼ u"v". Since 1_"2 Nwe have ð1_"Þ ¼0. Combined with the boundary conditions imposed on u"and "we conclude that z"ð0Þ ¼ z"ð1Þ ¼ 0. We claim that

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Z O jz"j2 Z O j@xz"j2:

Indeed, since O ¼ ð0; 1Þ and z"¼0 on @O, this follows by Poincare´’s inequality: Z1 0 jz"j2 ¼ Z 1 0 Z x 0 @xz" 2 Z1 0 j@xz"j2: Hence, Z O jz"j2þ j@xz"j22 Z O j@xz"j22 Z O j@xz"j2a";

where we used that a" by assumption. Since z"¼0 on @O, we may integrate by parts and get Z O jz"j2þ j@xz"j22 Z O z"ð@xða"@xz"ÞÞ:

Step 3. We compute ð@xða"@xz"ÞÞ:

@xz"¼@xu" ð@yðx_"Þ þ1Þ@xu0""@2_xu0 use a0¼a"ð@yð_"Þ þ1Þ

a"@xz"¼a"@xu"a0@xu0"a""@2xu0

@xða"@xz"Þ ¼ @xða"@xu"Þ þ@xða0@xu0Þ þ@xð"a""@2_xu0Þ:

The ﬁrst two terms on the right-hand side are equal to the left-hand side of the PDEs for u"and u0. Hence, these two terms evaluate to f f ¼ 0:

@xða"@xz"Þ ¼@xð"a""@2xu0Þ: Combined with the estimate of Step 2 we deduce that

Z O jz"j2þ j@xz"j22 Z O z"@xð"a""@2xu0Þ integration by parts ¼ 2 Z O @xz"ð""a"@2xu0Þ

Cauchy{Schwarz and Young’s inequality in the form ab ₂a2þ₂1 b2 with ¼₂ 1₂ Z O j@xz"j2þ22" 2 Z O j"j2ja"j2j@2_xu0j2; and thus Z O jz"j2þ j@xz"j2 4 2" 2 Z O j"j2j@2_xu0j2: In this lecture we extend the previous one-dimensional results to

. higher dimensions — the argument presented above heavily relies on the fact that we have an explicit representation for the solutions. In higher dimensions such a representation is not available and the argument will be more involved. In particular, we require some input from the theory of partial diﬀerential equations and functional analysis such as the notion of distributional solutions, the existence theory for elliptic equations in divergence form in Sobolev spaces, the theorem of Lax–Milgram, Poincare´’s inequality, the notion of weak convergence in L2_{-spaces, and the theorem of Rellich–Kondrachov, e.g., see the textbook on functional analysis} by Brezis [8].

. periodic and random coeﬃcients — to treat the later we require some input from ergodic & probability theory. Moreover, we discuss

. the two-scale expansion in higher dimension and in the stochastic case, and explain . quantitative results for stochastic homogenization in a discrete setting.

2. Qualitative Homogenization of Elliptic Equations

In this section we discuss the homogenization theory for elliptic operators of the form r ðarÞ with uniformly elliptic coeﬃcients. We say that a : Rd ! Rdd is uniformly elliptic with ellipticity constant > 0, and write a 2 MðRd; Þ, if a is measurable, and for a.e. x 2 Rd we have

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8 2 Rd : aðxÞ jj2 and jaðxÞj jj: ð2:1Þ A standard result (that invokes the Lax–Milgram Theorem) yields existence of weak solutions to the associated elliptic boundary value problem.

Problem 6. Let a 2 MðRd; Þ, O Rdopen and bounded, f 2 L2ðOÞ, F 2 L2ðO; RdÞ. Show that there exists a unique solution u 2 H₀1ðOÞ to the equation

r ðaruÞ ¼ f r F inD0ðOÞ: ð2:2Þ

It satisﬁes the a priori estimate

kukH1_ðOÞCð; d; diamðOÞÞðk f k_L2_ðOÞþ kFk_L2_ðOÞÞ: ð2:3Þ

In this section we study a classical problem of elliptic homogenization: Given a family of coeﬃcient ﬁelds ða"Þ MðRd; Þ, consider the weak solution u"2H1

0ðOÞ to the equation r ða"ru"Þ ¼f r F inD

0_{ðOÞ. A prototypical} homogenization result states that under appropriate conditions on ða"Þ,

. u" weakly converges to a limit u0 in H1

0ðOÞ as " # 0.

. The limit u0 can be characterized as the unique weak solution in H1

0ðOÞ to a homogenized equation r ðahomru0Þ ¼ f r F.

. The homogenized coeﬃcient ﬁeld ahomcan be computed from ða"Þby a homogenization formula.

We discuss two types of structural conditions on the coefficient fields ða"Þthat allow to prove such a result. In the first case, which usually is referred to as periodic homogenization, the coefficient fields are assumed to be periodic, i.e., a"ðÞ ¼a0ð_"Þ, where a0 is periodic in the following sense:

Deﬁnition 2.1. We call a measurable function f deﬁned on Rd L-periodic, if for all z 2 Zd we have f ð þ LzÞ ¼ f ðÞ a.e. in Rd:

In the second case, called stochastic homogenization, the coefficient fields are supposed to be stationary and ergodic random coefficients. We discuss the stochastic case in more detail in Sect. 2.2.

Both cases (the periodic and the stochastic case) can be analyzed by a common approach that relies on Tartar’s method of oscillating test function, see [26]. In the following we present the approach in the periodic case in a form that easily adapts to the stochastic case.

2.1 Periodic homogenization

In this section we prove the following classical and prototypical result of periodic homogenization.

Theorem 2.2 (e.g., see textbook Bensoussan, Lions and Papanicolaou [6]). Let > 0 and a 2 MðRd; Þ be 1-periodic. Then there exists a constant, uniformly elliptic coeﬃcient matrix ahomsuch that:

For all O Rd open and bounded, for all f 2 L2ðOÞ and F 2 L2ðO; RdÞ, and " > 0, the unique weak solution u"2H10ðOÞ to

r ðaðx_"Þru"Þ ¼ f r F inD0ðOÞ weakly converges in H1ðOÞ to the unique weak solution u02H1₀ðOÞ to

r ðahomru0Þ ¼f r F inD0ðOÞ:

Above and throughout the paper we write r F ¼ f inD0ðOÞ to express that the identity holds in the distributional sense, i.e.,R_OF r’ ¼R_Of ’ for all ’ 2 C1

c ðOÞ. A numerical illustration of the theorem is depicted in Fig. 2. The main diﬃculty in the proof of the theorem is to pass to the limit in expressions of the form

Z O

aðx_"Þru"ðxÞ ðxÞeidx ð 2 C_c1ðOÞÞ;

since the integrand is a product of weakly convergent terms. In a nutshell Tartar’s method relies on the idea to approximate the test ﬁeld ðxÞeiby some gradient ﬁeld rðgi;"Þ, where gi;"denotes an oscillating test function with the property that r at_ð

"Þrgi;"! r a t

homei in H1ðOÞ. We can then pass to the limit by appealing to the following special form of Murat & Tartar’s celebrated div-curl lemma, see [26]:

Lemma 2.3. Consider ðu"Þ H₀1ðOÞ and ðF"Þ L2ðO; RdÞ. Suppose that . u"* u0 weakly in H₀1ðOÞ,

. F"* F0weakly in L2ðO; RdÞandR_OF" r"!R_OF r for any sequence ð"Þ H1₀ðOÞ with "* weakly in H1ðOÞ.

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Z O ðru"F"Þ ! Z O ðru0F0Þ: Proof. Z O ðru"F"Þ ¼ Z O rðu"Þ F" Z O u"r F":

Since u" * u0 weakly in H1

0ðOÞ, and u"r ! u0r strongly in L2ðOÞ (by the Rellich–Kondrachov Theorem), we ﬁnd that the right-hand side converges to

Z O rðu0Þ F0 Z O u0r F0 ¼ Z O ðru0F0Þ: It turns out that the homogenization result holds, whenever we are able to construct an oscillating test function gi;". This motivates the following deﬁnition:

Definition 2.4. We say that ða"Þ MðRd; Þ admits homogenization if there exists an elliptic, constant coefficient matrix ahom, called ‘‘the homogenized coefficients’’, such that the following properties hold: For i ¼ 1; . . . ; d there exist oscillating test functions ðgi;"Þ H1

locðR

d_Þ_{such that}

r at_"rgi;"¼0 inD0ðRdÞ; ðC1Þ

gi;"* xi weakly in H1locðR

d_Þ; _ðC2Þ

at_"rgi;"* at_homei weakly in L2_locðRdÞ: ðC3Þ

(a) (b)

(c) (d)

Fig. 2. Illustration of Theorem 2.2 in the periodic, two-dimensional case. (a) shows a periodic checkerboard-like coeﬃcient ﬁeld aðÞ. (b)–(c) show the solution to the equation r ðað

"Þru"Þ ¼1 on the unit cube with homogeneous Dirichlet boundary values

for " 2 f1 2; 1 8; 1 32g.

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Based on (C1)–(C3) and the div-curl lemma we obtain the following general homogenization result:

Lemma 2.5. Suppose ða"Þ MðRd; Þ admits homogenization with homogenized coeﬃcients ahom. Then for all O Rd open and bounded, for all f 2 L2ðOÞ and F 2 L2ðO; RdÞ, and " > 0, the unique weak solution u"2H1₀ðOÞ to

r ða"ru"Þ ¼ f r F inD0ðOÞ weakly converges in H1_{ðOÞ to the unique weak solution u0}₂_H1

0ðOÞ to r ðahomru0Þ ¼f r F inD0ðOÞ: Moreover, we have

a"ru"* ahomru0 weakly in L2_{ðO; R}d_Þ: Proof. Step 1. Compactness.

We denote the ﬂux by

j":¼ a"ru": By the a priori estimates of Problem 6 we have

Z O ju"j2þ jru"j2þ jj"j2C Z O jf j2þ jFj2

where C does not depend on ". Since bounded sets in L2_{ðO; R}d

Þand H1_{ðOÞ are precompact in the weak topology, and} since H1_{ðOÞ b L}2

locðOÞ is compactly embedded (by the Rellich–Kondrachov Theorem), there exist u02H01ðOÞ and j02L2ðOÞ such that, for a subsequence (that we do not relabel), we have

u"* u0 weakly in H1ðOÞ; u"!u0 in L2_locðOÞ; j"* j0 weakly in L2ðOÞ: We claim that

r j0¼ f r F inD0ðOÞ: ð2:4Þ

Indeed, for all ’ 2 C1

c ðOÞ we have Z O j0 r’ Z O j" r’ ¼ Z O a"ru" r’ ¼ Z O f ’ þ F r’: Step 2. Identiﬁcation of j0.

We ﬁrst argue that it suﬃces to prove the identity

j0¼ahomru0: ð2:5Þ

Indeed, the combination of ð2.5Þ and ð2.4Þ shows that

r ðahomru0Þ ¼f r F inD0_ðOÞ:

Since this equation has a unique solution (recall that ahomis assumed to be elliptic), we deduce that u0and j0 (which were originally obtained as a weak limits of ðu"Þand ð j"Þalong a subsequence), are independent of the subsequence. Hence, we get u"* u0weakly in H1ðOÞ and j"* ahomru0weakly in L2ðO; RdÞalong the entire sequence, and thus the claimed statement follows.

It remains to prove ð2.5Þ. By the fundamental lemma of the calculus of variations, it suﬃces to show: For all 2 C1

c ðOÞ and i ¼ 1; . . . ; d we have

Z O

ð j0ahomru0Þ ei¼0: ð2:6Þ

For the argument let gi;" denote the oscillating test function of Deﬁnition 2.4, and note that for any sequence "* 0 weakly in H1 0ðOÞ we have Z O j" r"¼ Z O f "þF r"! Z O f 0þF 0¼ Z O j r0:

Hence, an application of the div-curl lemma, see Lemma 2.3, and property (C2) yield Z O ð j" rgi;"Þ ! Z O ð j0eiÞ:

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Z Q ð j" rgi;"Þ ¼ Z Q ðru"at_"rgi;"Þ ! Z Q

ðru0at_homeiÞ;

and thus ð2.6Þ.

With Lemma 2.5 at hand, the proof of Theorem 2.2 reduces to the construction of the oscillating test functions gi;". In the periodic case the construction is based on the notion of the periodic corrector. Before we come to its deﬁnition we introduce a Sobolev space of periodic functions: Let :¼ ð1₂;1₂Þd denote the unit box in Rd. For L > 0 set

H1_#ðLÞ :¼ fu 2 H_loc1 ðRdÞ: u is L-periodic:g:

Problem 7. Show that . H1

#ðLÞ with the inner product of H

1_{ðLÞ is a Hilbert space (and can be identiﬁed with a closed linear subspace} of H1_ðLÞ).

. The space of smooth, L-periodic functions on Rd is dense in H1 #ðLÞ. . For any F 2 H1

#ðL; R

d_Þ_{we have the integration by parts formula} Z

LðzþÞ

r F ¼ 0 for all L 2 N and z 2 Rd:

Lemma 2.6(Periodic corrector). Let a 2 MðRd; Þ be 1-periodic. (a) For i ¼ 1; . . . ; d there exists a unique i2H1

#ðÞwith

>

i¼0 s.t. ?

aðriþeiÞ r ¼ 0 for all 2 H#1ðÞ: ð2:7Þ

(b) ican be characterized as the unique, 1-periodic function i2H_loc1 ðRdÞwith>i¼0 and

r ðaðriþeiÞÞ ¼0 inD0ðRdÞ: ð2:8Þ

Proof of Lemma 2.6 part (a). Throughout the proof ‘ denotes a non-negative integer. We set ‘:¼ ð2‘þ1₂ ;2‘þ1₂ Þdfor ‘ 2 N0 and note that

_‘_¼ [

x2Zd_\ ‘

ðx þ Þ up to a null-set;

where the union on the right-hand side is disjoint. We ﬁrst remark that the problem Z

_‘

aðr‘þeiÞ r ¼ 0 for all 2 H1

#ð‘Þ; ð2:9Þ

admits a unique solution ‘₂_H1

#ð‘Þsatisfying R

_‘

‘_¼_{0, as follows from the Lax–Milgram theorem and Poincare´’s} inequality. In particular, for ‘ ¼ 0 this proves (a). We claim that ‘_¼0 _{for any ‘ 2 N. Indeed, let denote a test} function in H1

#ð‘Þ. Then by 1-periodicity of aðr0þeiÞwe have Z _‘ aðr0þeiÞ r ¼ X x2Zd_\ ‘ Z xþ aðr0þeiÞ r ¼ X x2Zd_\ ‘ Z aðr0þeiÞ rð þ xÞ ¼ Z aðr0þeiÞ r;~ where ~ :¼P_x2Zd_\ ‘ð þ xÞ. By construction we have ~ 2 H 1

#ðÞ, and thus the right-hand side is zero (by appealing to the equation for 0). Hence, we deduce that 0solves ð2.9Þ and the conditionR_‘

0_¼_{0. Since ð2.9Þ admits a unique} solution, we deduce that 0 ¼‘.

We are now in position to prove the equivalence of the problems ð2.7Þ and ð2.8Þ. For the direction ‘‘)’’ it suﬃces to show that for arbitrary 2 C1

c ðR d

Þwe have Z

aðr0þeiÞ r ¼ 0:

Note that here and throughout the paper we simply writeR instead ofRRd. For the argument, choose ‘ 2 N suﬃciently large such that ¼ 0 outside ‘. Then can be extended to a periodic function ‘₂_H1

#ð‘Þ, and we conclude that (since 0¼‘), Z aðr0þeiÞ r ¼ Z _‘ aðr‘þeiÞ r‘¼0:

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For the other direction let denote the solution to ð2.8Þ. It suﬃces to show that for arbitrary 2 H1

#ðÞwe have ?

aðr þ eiÞ r ¼ 0: ð2:10Þ

By periodicity, we have for any ‘2C1_c ð‘Þ, ? aðr þ eiÞ r ¼ ? _‘ aðr þ eiÞ r ¼ ? _‘ aðr þ eiÞ rð‘Þ þ ? _‘ aðr þ eiÞ ðrð1 ‘Þ r‘Þ ¼ ? _‘ aðr þ eiÞ ðrð1 ‘Þ r‘Þ;

where the last identity holds thanks to ð2.8Þ. Since distð‘1; Rdn ‘Þ ¼1, we can ﬁnd a cut-oﬀ function ‘2C1 c ð‘Þ such that 0 ‘1, ‘¼1 on ‘1 and jr‘j C with C independent of ‘. We thus conclude that

? _‘ aðr þ eiÞ ðrð1 ‘Þ r‘Þ ðC þ 1Þj‘n ‘1j j‘j ? _‘n‘1 jaðr þ eiÞjðjrj þ jjÞ ¼ ðC þ 1Þj‘n ‘1j j‘j ? jaðr þ eiÞjðjrj þ jjÞ;

where the last identity holds by 1-periodicity of the integrand. In the limit ‘ ! 1, the right-hand side converges to 0,

and thus ð2.10Þ follows.

Definition 2.7(Periodic corrector and homogenized coefficient). Let a 2 MðRd; Þ be 1-periodic. The solution ito ð2.7Þ is called the (periodic) corrector in direction ei (associated with a). The matrix ahom2 Rdd defined by

ahomei:¼ ?

aðriþeiÞ ði ¼ 1; . . . ; dÞ

is called the homogenized coeﬃcient (associated with a).

Lemma 2.8(Properties of the homogenized coeﬃcients). Let a 2 MðRd; Þ be 1-periodic and denote by ahom the associated homogenized coeﬃcients.

(a) (ellipticity). For any 2 Rd we have

ahom jj2:

(b) (invariance under transposition). Let t_i denote the corrector associated with the transposed matrix at. Then

ðahomÞtei¼ ?

atðrt_iþeiÞ:

(c) (symmetry). If a is symmetric (a.e. in Rd), then ahomis symmetric. Proof. For 2 Rd set :¼ iiand note that is the unique solution in H1

#ðÞwith

>

¼0 to ?

aðrþÞ r ¼ 0 for all 2 H1_#ðÞ:

Hence, 1 ahom ¼ 1 ? ð þ rÞ að þ rÞ ? j þ rj2¼ jj2þ ? jrj2;

where we used that>r¼0 by periodicity. This proves the ellipticity of ahom. For (b) note that

ðahomÞtei ¼ eiahom ¼ ei ? aðrþÞ ¼ ? ðrt_iþeiÞ aðrþÞ ¼ ? atðrt_iþeiÞ ðrþÞ ¼ ? atðrt_iþeiÞ :

Since this is true for arbitrary 2 Rd, (b) follows. If a is symmetric, then t_i¼i, and thus atðrt_iþeiÞ ¼aðriþeiÞ. In this case (b) simpliﬁes to

ðahomÞtei¼ ?

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and thus ahom is symmetric. We ﬁnally give the construction of the oscillating test function and establish the properties (C1)–(C3):

Lemma 2.9(Construction of the oscillating test function). Let a 2 MðRd; Þ be 1-periodic, let ahom denote the associated homogenized coefficient and denote by t₁; . . . t_d the periodic correctors associated with the transposed matrix at. Then ðað_"ÞÞadmits homogenization with homogenized coefficients ahom, and the oscillating test function can be defined as

gi;"ðxÞ :¼ xiþ"iðx_"Þ:

For the proof we need to pass to the limit in sequences of rapidly oscillating functions:

Proposition 2.10(Rapidly oscillating functions). Let g 2 L2_locðRdÞbe 1-periodic. Consider g"ðxÞ :¼ gðx_"Þ, " > 0. Then

g"*g :¼ ?

g weakly in L2_ðOÞ;

for any O Rd open and bounded.

Proof. Fix O Rd open and bounded. It suﬃces to prove:

lim sup "#0 kg"k2L2_ðOÞdiamðOÞd Z jgj2; ð2:11Þ 8Q Rd cube : ? Q g"!g: ð2:12Þ

Indeed, this is suﬃcient, since ð2.11Þ yields boundedness of ðg"Þ, and for weak convergence in L2_{ðOÞ it suﬃces to test} with a class of test functions that is dense in L2_{ðOÞ, e.g., D :¼ spanf1Q} _{indicator function of a cube Q Og. Now, by} linearity of the integral and by ð2.12Þ, we have

Z O g"v ! Z O gv for all v 2 D:

Step 1. Argument for ð2.11Þ

W.l.o.g. let O be a cube. Set Qz;" :¼ z þ ", and set Z":¼ fz 2 "Zd: Qz;"\O 6¼ ;g. Then

kg"k2L2_ðOÞ X z2Z" Z Qz;" g x " 2 dx |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} ¼"dR jgj 2 ¼"d#Z"kgk2_L2_ðÞ;

and the conclusion follows since "d_#Z

"! jOj diamðOÞd. Step 2. Argument for ð2.12Þ

Let Q denote a cube, set Z":¼ fz 2 "Zd : Qz;"Qg, and Q":¼ [z2Z_"Qz;", so that jQ"j ! jQj. Then Z Q g" Z Q" g" Z QnQ" jg"j jQ n Q"j12_kg_"_k_L2_ðQÞ!0; andR_Q z;"g"¼" dR g, and thus Z Q" g"¼X z2Z" Z Qz;" g"¼"d#Z" Z g ¼ jQ"jg ! jQj g: Proof of Lemma 2.9. To ease notation we simply write g", t _{and e instead of gi;",}t

i and ei. We only need to check (C1)–(C3).

Step 1. Argument for (C1) and (C3)

Consider the periodic function j :¼ atðrtþeÞ 2 L2_locðRdÞ, and note that we have r j ¼ 0 inD0ðRdÞby Lemma 2.6 (b). Scaling yields r jð_"Þ ¼0 inD0ðRdÞ, and thus (C1). Since j is periodic, Proposition 2.10 yields jð_"Þ*>j ¼ at_home weakly in L2_locðRdÞ, and thus (C3).

Step 2. Argument for (C2).

Since t_{and r}t_{are periodic functions, we conclude from Proposition 2.10 that}t_ð

"Þand r t_ð "Þweakly converge in L2 locðR d_{Þ, and thus g}_{"ðxÞ ¼ x} iþ"ðx_"Þ* xi in Hloc1 ðR d_Þ. 2.2 Stochastic homogenization

Description of Random Coefficients In stochastic homogenization we only have ‘‘uncertain’’ or ‘‘statistical’’ information about the coefficient matrix a (which models the microstructure of the material). Hence, faðxÞg_x2Rd has to be considered as a family of matrix valued random variables. For stochastic homogenization the random field a is

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required to be stationary in the sense that for any finite number of points x1; . . . ; xkand shift z 2 Rdthe random variable ðaðx1þzÞ; . . . ; aðxkþzÞÞ has a distribution independent of z, i.e., the coefficients are statistically homogeneous. In addition, for homogenization towards a deterministic limit, the random field a is required to be ergodic in the sense that spatial averages of a over cubes of size R converge to a deterministic constant as R " 1; one could interpret this by saying that a typical sample of the coefficient field already carries all information about the statistics of the random coefficients. For our purpose it is convenient to work within the following mathematical framework:

. We introduce a configuration space of admissible coefficient fields

:¼ fa : Rd! Rdd_sym is measurable and uniformly elliptic in the sense of (2.1)g

. We introduce a probability measure P on (which we equip with a canonical -algebra). We write hi for the associated expectation.

The measure P describes ‘‘the frequency of seeing a certain microstructure in our random material’’. The assumption of stationarity and ergodicity can be phrased as follows:

Assumption (S). Let ð;F ; PÞ denote a probability space equipped with the (spatial) ‘‘shift operator’’ : Rd ! ; ðz; aÞ :¼ að þ zÞ ð¼: zaÞ;

which we assume to be measurable. We assume that the following properties are satisﬁed: . (Stationarity). For all z 2 Rd _{and any random variable f 2 L}1_{ð; PÞ we have}

hf zi ¼ hf i: . (Ergodicity). For any f 2 L1_{ð; PÞ we have}

lim R"1

? R

f ðzaÞ dz ¼ h f i for P-a.e. a 2 : ð2:13Þ

Remark 2.11. Assumption (S) can be rephrased by saying that ð;F ; P; Þ forms a d-dimensional ergodic, measure-preserving dynamical system. Ergodicity is usually deﬁned as follows: For any E (measurable) we have

E is shift-invariant ) PðEÞ 2 f0; 1g:

Here, a (measurable) set E is called shift invariant, if zE ¼ E for all z 2 Rd. The fact that this deﬁnition of ergodicity implies ð2.13Þ is due to {Birkhoﬀ’s pointwise ergodic theorem}, e.g., see Ackoglu & Krengel [2] for reference that covers the multidimensional case.

Example 2.12(Random Checkerboard). Let z 2 ð0; 1Þd denote a random vector with uniform distribution, and fakg_k2Zd a family of independent, identically distributed random matrices in 0 :¼ fa02 Rdd: a0 satisfies (2.1)g. Then

a : Rd! Rdd; aðxÞ :¼X k2Zd

1kþzþðxÞak;

(a) independent and identically distributed tiles (b) correlated tiles

Fig. 3. Typical sample of a stationary, ergodic random checkerboard type coeﬃcient ﬁeld that takes two values with the same probability.

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defines random field in whose distribution is stationary and ergodic. Figures 3(a) shows an example for a sample of such a random field in the case, when a only takes two values, say awhiteand ablack. More precisely, the construction of P is as follows: We start with the probability space ð0;F0; P0ÞwhereF0denotes the Borel- -algebra on 0 Rdd, and P0 describes the distribution on a single tile. Then consider the product space

ð0;F0; P0Þ:¼ ðZ₀d ;FZ₀ dBðÞ; PZ₀ dLÞ; whereL denotes the Lebesgue measure on , and the map

: 0!; ða; zÞ :¼X k2Zd

1kþzþðÞak:

The probability measure P is then obtained as the push-forward of P0 _{under and yields a stationary and ergodic} measure. Note that the associated coeﬃcients have a ﬁnite range of dependence, in the sense that if we take x; x0_{2 R}d with jx x0_j_{> diamðÞ, then the random variables aðxÞ and aðx}0_Þ_{are independent, and ð2.13Þ is a consequence of the} law of large numbers. We might vary the example by considering the convolution

’: Z₀d !Z₀d; ’ðaÞk:¼ X j2Zd

’ð j kÞaj;

with some non-negative convolution kernel ’ : Zd! R0 satisfying P_k2Zd’ðkÞ ¼ 1. If we deﬁne P as the push-forward of P0 _{under the mapping}

’: 0!; ða; zÞ :¼X k2Zd

1kþzþðÞ’ðaÞk;

we obtain again a stationary and ergodic measure. If ’ is not compactly supported, then aðxÞ and aðx0Þare always correlated (even for jx x0j 1), yet they decorrelate on large distances, i.e., for x; x02 Zd we have

hðaðxÞ haiÞðaðx0Þ haiÞi ¼ X j; j0_2Zd ’ð j xÞ’ð j0xÞ Covðaj; aj0Þ Varða0ÞX j2Zd ’ð j xÞ’ð j x0Þ ¼X j2Zd ’ð jÞ’ð j þ x x0Þ !0 as jx x0_{j ! 1:}

Figure 3(b) shows a typical sample of a coeﬃcient ﬁeld obtained in this way (with a kernel ’ that exponentially decays).

Example 2.13(Gaussian random fields). Let fðxÞg_x2Rd denote a centered, stationary Gaussian random field with covariance function CðxÞ ¼ CovððxÞ; ð0ÞÞ. Roughly speaking this means that for any x1; . . . ; xN 2 Rd the random vector ððx1Þ; . . . ; ðxNÞÞ has the distribution of a multivariate Gaussian with mean zero and covariance matrix ij¼CðxixjÞ. Suppose that jCðxÞj ðjxj þ 1Þfor some > 0 (i.e., at least some algebraic decay of correlations). Let : R ! 0 denote a Lipschitz function. Then aðxÞ :¼ ððxÞÞ defines a stationary and ergodic ensemble of coefficient fields.

Problem 8(Periodic coefficients). Let a#denote a periodic coefficient field in . Show that there exists a stationary and ergodic measure P on , s.t. for any open set O we have

Pðfa#ð þzÞ : z 2 OgÞ ¼ jOj:

(Hence, with full probability a sample a is a translation of a#. In this sense periodic coeﬃcients can be recast into the stochastic framework).

Homogenization in the stochastic case. The analogue to Theorem 2.2 in the stochastic case is the following: Theorem 2.14(Papanicolaou & Varadhan ’79 [27], Kozlov ’79 [20]). Suppose Assumption (S). There exists a (uniformly elliptic) constant coeﬃcient tensor ahomsuch that for P-a.e. a 2 we have:

For all O Rdopen and bounded, for all f 2 L2ðOÞ, F 2 L2ðO; RdÞ, and " > 0, the unique weak solution u"2H₀1ðOÞ to

r ðaðx_"Þru"Þ ¼f r F in O weakly converges in H1_{ðOÞ to the weak solution u0} ₂_H1

0ðOÞ to

r ðahomru0Þ ¼ f r F in O; and we have

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Except for the assumption on a, the statement is similar to Theorem 2.2. Since a 2 is random, the solutions u"2H1₀ðOÞ (which depend in a nonlinear way on a) are random quantities. In contrast, the homogenized coefficient matrix ahomis deterministic and only depends on P, but not on the individual sample a, the domain O or the right-hand side. Therefore, the limiting equation and thus u0 is deterministic. Hence, in the theorem we pass from an elliptic equation with random, rapidly oscillating coefficients to a deterministic equation with constant coefficients, which is a huge reduction of complexity. A numerical illustration of the result is given in Fig. 4.

As in the periodic case, the core of the proof is the construction of a corrector (which is then used to deﬁne the oscillating test functions in Deﬁnition 2.4).

Proposition 2.15(The corrector in stochastic homogenization). Suppose that Assumption (S) is satisﬁed. For any 2 Rd there exists a unique random ﬁeld : Rd! R, called the corrector associated with , such that:

(a) For P-a.e. a 2 the function ða; Þ 2 H1_locðRdÞis a distributional solution to

r aðrða; Þ þ Þ ¼ 0 inD0_ðRd_Þ; _ð2:14Þ

with sublinear growth in the sense that

lim sup R!1 1 R2 ? R jða; Þj2¼0; ð2:15Þ

and ða; Þ is anchored in the sense that>ða; yÞ dy ¼ 0. (b) r is stationary in the sense of Deﬁnition 2.21 below. (c) h>jrj2i 122jj

2_{, and h}>_{ri ¼ 0.}

Let us remark that the arguments that we are going to present extend verbatim to the case of systems, see Remark 2.32. Before we discuss the proof of Proposition 2.15, we note that in combination with Lemma 2.9, Proposition 2.15 yields a proof of Theorem 2.14. In fact, we only need to show:

Lemma 2.16. Suppose that Assumption (S) is satisfied. For i ¼ 1; . . . ; ei let i (resp. ti) denote the corrector associated with ei and a (resp. the transposed coefficient field at_{) from Proposition 2.15 and consider the matrix} ahom2 Rdd defined by

ahomei:¼ ?

aðyÞðriða; yÞ þ eiÞdy

:

Then ahomis elliptic and for P-a.e. a 2 the family ðað

"ÞÞadmits homogenization with homogenized coeﬃcients given by ahomand oscillating test functions given by

gi;"ðxÞ :¼ xiþ"t_iða;x_"Þ:

The proof is a rather direct consequence of the properties of the corrector and ergodicity in form of ð2.13Þ. We present it in Sect. 2.2.1.

(a) (b)

Fig. 4. Illustration of Theorem 2.14 in the case of a random checkerboard-like coeﬃcient ﬁeld with independent and identically

distributed tiles. (a) and (b) show realizations of the solutions to r ðað

"ru"Þ ¼1 in H01ðð0; 1Þ

2_Þ_{for " 2 f}1 8;

1 32g.

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The main part of this section is devoted to the proof of Proposition 2.15. We first remark that the sublinearity condition ð2.15Þ is a natural ‘‘boundary condition at infinity’’. Indeed, if the coefficient field a is constant, then sublinearity implies that the solution to ð2.14Þ is unique up to an additive constant.

Lemma 2.17(A priori estimate for sublinear solutions). Let a 2 . Suppose u 2 H_loc1 ðRdÞhas sublinear growth in the sense of ð2.15Þ and solves

r aru ¼ 0 inD0_ðRd_Þ: Then lim sup R!1 ? R jruj2¼0: ð2:16Þ

In particular, if the coeﬃcient ﬁeld a is constant, then u is constant. Proof. Let 2 C1

c ðR

d_{Þ. By Leibniz’ rule we have}

rðuÞ arðuÞ ¼ rðu2Þ aru þ u2r ar þ uðrðuÞ ar r arðuÞÞ Note that by Young’s inequality we have

juðrðuÞ ar r arðuÞÞj 2jujjrðuÞjjrj 2jrðuÞj

2 þ2

u 2_jrj2_;

and thus by ellipticity, and the equation for u,

Z jrðuÞj2 Z rðuÞ arðuÞ 1 þ2 Z u2jrj2þ 2 Z jrðuÞj2: We conclude that Z jrðuÞj2CðdÞ Z u2jrj2:

We now specify the cut-oﬀ function: Let 12C1_c ð2Þ satisfy 1¼1 on . Then the above estimate applied with 1ð_RÞyields ? R jruj2Rd Z jrðu1ð_RÞÞj2Cðd; ÞRd2 Z u2jðr1Þð_RÞj2 Cðd; ; 1Þð2RÞ2 ? 2R juj2:

By sublinearity, for R ! 1 the right-hand side converges to 0, which yields the ﬁrst claim. If a is constant, then by a standard interior estimate we have

kruk2_L1_ðRÞC ?

2R jruj2;

for a constant C that is independent of R. We conclude that ru ¼ 0 a.e. in Rd and thus u is constant. Let us anticipate that the above estimate also yields uniqueness in the case of stationary and ergodic random coefficients for solutions with sublinear growth and stationary gradients, see Corollary 2.26 below. On the other hand it is not clear at all that the equation r ðarÞ ¼ r ðaÞ admits a sublinear solution. In fact, this is only true for ‘‘generic’’ coefficient fields a 2 , in particular, we shall see that this is true for P-a.e. a 2 when P is stationary and ergodic. Our strategy is the following:

. Instead of the equation r ðarÞ ¼ r ðaÞ we consider the modiﬁed corrector equation 1

TT r aðrTÞ ¼ r ðaÞ in R

d _{ðT
1Þ;} _ð2:17Þ

which turns out to be well-posed for all a 2 and yields an a priori estimate of the form

8R T : ? ffiffiffi R p 1 TjTj 2_{þ jr} Tj2Cðd; Þjj2: ð2:18Þ

. By stationarity of P we can turn ð2.18Þ into an averaged estimate that on the level of rT is uniform in T, ?

jrTj2

Cðd; Þjj2:

. This allows us to pass to the weak limit (for T " 1) in an appropriate subspace of random ﬁelds. The limit is a solution to the corrector equation, its gradient is stationary, i.e., rða; x þ zÞ ¼ rðxa; yÞ and satisﬁes

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? jrj2 < 1 and ? r ¼0:

. Finally, by exploiting ergodicity and the property that h>rTi ¼0 we deduce sublinearity.

We start with the argument that establishes sublinearity, since the latter is the most interesting property of the corrector. In fact, the argument can be split into a purely deterministic argument (that we state next), and a non-deterministic part that exploits ergodicity, see proof of Corollary 2.27 below.

Lemma 2.18(sublinearity). Let u 2 H_loc1 ðRdÞsatisfy>u ¼ 0,

lim sup R!1 ? R jruj2< 1; and ð2:19Þ lim sup R!1 ? R

ruðyÞ FðRxÞ ¼ 0 for all F 2 L2_{ð; R}d_Þ: _ð2:20Þ Then we have lim sup R!1 1 R2 ? R u ? R u 2 !1 2 ¼0; ð2:21Þ lim sup R!1 1 R2 ? R juj2 1 2 ¼0: ð2:22Þ

Proof. Step 1. Proof of ð2.21Þ.

We appeal to a scaling argument: Consider uRðxÞ :¼1

RðuðRxÞ

>

RuÞ and note that ruRðxÞ ¼ ruðRxÞ; Z juRj2 ¼ 1 R2 ? R u ? R u 2 :

Hence, it suﬃces to show that uR!0 strongly in L2ðÞ. Since

>

uR ¼0, Poincare´’s inequality yields Z u2_R Z jruRj2 ¼ ? R jruj2;

and ð2.19Þ implies that ðuRÞ is bounded in H1_{ðÞ. By weak compactness of bounded sequences in H}1_{ðÞ, we ﬁnd} u12H1_ðÞ_{such that uR} _{* u1}_{weakly in H}1_ðÞ_{(for a subsequence that we do not relabel). Since H}1_ðÞ_L2_ðÞ_is compactly embedded (by the theorem of Rellich–Kondrachov), we may assume w.l.o.g. that we also have uR!u1 strongly in L2ðÞ. We claim that u1¼0 (which then also implies that the convergence holds for the entire sequence). Indeed, from ð2.20Þ we deduce that

? ru1F ¼ lim R!1 ? ruRF ¼ lim R!1 ? ruðRxÞ FðxÞ ¼ 0:

Hence, ru1¼0, and thus u1 is constant. Since

>

u1¼0, u1¼0 follows. Step 2. Proof of ð2.22Þ.

Set JðtÞ ¼>_tu ¼>uðtxÞdx. We have

@tJðtÞ ¼ ? ruðtxÞ xdx: Let R T 1. Then ? R u ? u ¼ ZR 1 @tJðtÞdt ZT 1 j@tJðtÞjdt þ ZR T j@tJðtÞjdt ¼ ZT 1 ? ruðtxÞ x dt þ Z R T ? ruðtxÞ x dt CðdÞ ZT 1 ? jruðtxÞj2dx 1 2 þ ðR TÞ sup tT ? ruðtxÞ x : By ð2.19Þ ZT 1 ? jruðtxÞj2dx 1 2 ¼ ZT 1 ? t jruðxÞj2dx 1 2 < 1:

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lim sup R!1 1 R ? R u ? u sup_tT ? ruðtxÞ x :

By ð2.20Þ (applied with FðxÞ ¼ x), in the limit T ! 1, the last expression converges to 0. We conclude

1 R2 ? R juj2 1 2 1 R2 ? R u ? R u 2! 1 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} !0 þ1 R ? R u ? u |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} !0 þ1 R ? u |fflfflfflffl{zfflfflfflffl} ¼0 : Remark 2.19. Let us anticipate that in the proof of Proposition 2.15 we apply Lemma 2.18 in the special situation where u is a realization of a random field u : Rd! R, whose gradient is stationary and satisfies h>jruj2_i_{< 1} and h>rui ¼ 0. Then, properties ð2.19Þ and ð2.20Þ hold for uða; Þ for P-a.e. a 2 as we will prove by appealing to ergodicity ð2.13Þ.

Another argument that is purely deterministic is the existence theory for the modiﬁed corrector. Note that the right-hand side of ð2.17Þ is a divergence of a vector ﬁeld F : Rd ! Rd that is not integrable (yet bounded). For the deterministic a priori estimate it is convenient to consider the weighted norm

kFk2:¼ Z

jFðxÞj2ðxÞ dx; ð2:23Þ

where : Rd! R denotes a positive, exponentially decaying weight to be speciﬁed below. In the following various estimates are localized on cubes. We use the notation

Q :¼ fQ ¼ x þ r : x 2 Rd_{; r > 0g;} _:¼ 1 2; 1 2 d :

Lemma 2.20. There exists a positive, exponentially decaying weight withR_Rd ¼ 1 (that only depends on d and ) and a constant C ¼ Cðd; Þ, such that the following properties hold: Let a 2 , T > 0, F 2 L2

locðR d_{; R}d

Þwith kFk< 1(cf. ð2.23Þ). Then there exists a unique solution u 2 H1

locðR d Þto 1 Tu r aru ¼ r F inD 0_ðRd_Þ; _ð2:24Þ

such that R 7!>_Rjuj2grows at most polynomially for R ! 1. The solution satisﬁes the a priori estimate

8R T : ? ffiffiffi R p 1 Tjuj 2 þ jruj2Cðd; ÞkFðpffiffiffiRÞk2: ð2:25Þ Proof. Step 1. Proof of the a priori estimate.

We claim that u satisﬁes ð2.25Þ. For the argument let 2 C1 c ðR

d_{Þ. By Leibniz’ rule we have} rðuÞ arðuÞ ¼ rðu2Þ aru þ u2r ar þ uðrðuÞ ar r arðuÞÞ: Note that

juðrðuÞ ar r arðuÞÞj 2jujjrðuÞjjrj: Thus, integration, ellipticity, and ð2.24Þ yield

Z ₁ Tjuj 2_þ_jrðuÞj2 Z ₁ Tjuj 2_{þ rðuÞ arðuÞ} Z ₁ Tðu

2_{Þu þ rð}2_{uÞ aru þ} Z u2jrj2 þ2 Z jrðuÞjjujjrj ¼ Z F rðu2Þ þ Z u2jrj2þ2 Z jrðuÞjjujjrj Z jFjjjjrðuÞj þ jFjjujjjjrj þ Z u2jrj2 þ2 Z jrðuÞjjujjrj:

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jFjjujjjjrj 1 2jFj 22_þ1 2u 2_jrj2_; jFjjjjrðuÞj 1 jFj 2 2þ 4jrðuÞj 2 ; 2jrðuÞjjujjrj 4 u 2_jrj2_þ 4jrðuÞj 2_; we get 1 T Z juj2þ 2jrðuÞj 2 _c Z jFj22þ Z u2jrj2 ; c :¼ 4 þ 3 2 :

Let R T. By an approximation argument (that exploits that R 7!>Rjuj2grows at most polynomially), this estimate extends to the exponential cut-off function ðxÞ ¼ expðc0pjxjffiffiffi_R_{Þ, where c0}_:¼ 1

2 ffiffiffiffiffidc p _{to the effect of} cjrj2 2 dcc20 R 1 2R 1 2T: We conclude that Z ₁ 2Tjuj 2_þ 2jrðuÞj 2 _c Z jFj22; and thus Z ₁ Tjuj 2_{þ jruj}2 2Cðd; Þ Z jFj22: ð2:26Þ Since min ffiffiffip_R

2 _expð2c0Þ_{> 0, we deduce that} ? ffiffiffi R p 1 Tjuj 2_{þ jruj}2 Cðd; ÞRd2 Z jFj22:

On the other hand, with

ðxÞ :¼ Z

expð2c0jyjÞ dy

1

expð2c0jxjÞ;

we may estimate the right-hand side of the previous estimate by Z

jFj22 Cðd; ÞRd2

Z

jFðpffiffiffiRÞj2ðÞ; and thus obtain ð2.25Þ.

Step 2. Conclusion.

Consider a general right-hand side F with kFk< 1. For k 2 N set FkðxÞ :¼ 1ðjxj < kÞFðxÞ, which is a vector ﬁeld in L2_ðRd_{; R}d_{Þ. Therefore, by the theorem of Lax–Milgram we ﬁnd uk}₂_H1_ðRd_Þ_{that solves}

1

Tuk r aruk¼ r Fk; and satisﬁes the standard a priori estimate,

1 T Z u2_kþ 2jrukj 2 2 Z jFkj2: ð2:27Þ

In particular, R 7!>Rjukj2 is bounded and thus uk satisﬁes the a priori estimate of Step 1,

8R T : ? ffiffiffi R p 1 Tjukj 2_{þ jrukj}2 Cðd; ÞkFkk2Cðd; ÞkFk2;

which is uniform in k. Consider the nested sequence of cubes Q‘:¼ 2‘pffiffiffiffiT_{, ‘ 2 N0. By the a priori estimate we} conclude that ðukÞis bounded in H1ðQ‘Þfor any ‘ 2 N0. Since Q‘Q‘þ1and Q‘" Rd, we conclude that there exists u 2 H_loc1 ðRdÞsuch that uk* u weakly in H1ðQ‘Þfor any ‘ 2 N0. Consequently u solves ð2.24Þ in a distributional sense. Thanks to the lower-semicontinuity of the norm, we deduce that u satisfies the a priori estimate ð2.25Þ. This proves the existence of the solution. Uniqueness of u is a consequence of the a priori estimate. As already mentioned, in order to obtain an estimate that is uniform in T, we need to exploit stationarity of P and random fields in the following sense:

Definition 2.21(Stationary random field). A measurable function u : Rd! Ris called a stationary L1_-random field (or short: stationary), if h>Qjuji < 1 for all cubes Q Q and if for P-a.e. a 2 ,

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Z xþQ

uða; yÞ dy ¼ Z

Q

uðxa; yÞ dy for all cubes Q 2Q and x 2 Rd: ð2:28Þ A prototypical example of a stationary random variable is as follows: Take u02L1ðÞ and consider uða; xÞ :¼ u0ðxaÞ. Then u is a stationary L1-random ﬁeld, called the stationary extension of u0. One can easily check that for any A Rd open and bounded we have

? A uða; yÞ dy ¼ ? A hu0ðyaÞi ¼ hu0i;

where the last identity holds by stationarity of P. In particular, we deduce that the value of h>Quða; yÞ dyi, Q 2Q, is independent of Q. The same properties are true for general stationary L1-random fields (except for the difference that we need to invoke an average w.r.t. Rd-component to obtain well-defined quantities):

Lemma 2.22. Suppose P is stationary. Let f denote a stationary L1_{-random ﬁeld.} (a) For any A Rd open and bounded we have

? A f ¼ ? f :

(b) Let > 0 and set fðaÞ :¼>f ða; yÞ dy. Then f2L1ðÞ and for P-a.e. a 2 , ?

f ða; x þ yÞ dy ¼ fðxaÞ for all x 2 Rd:

We postpone the proof to Sect. 2.2.1. As a consequence of ð2.13Þ (i.e., Birkhoﬀ’s ergodic theorem), we obtain the following variant for stationary ﬁelds:

Lemma 2.23(Variant of Birkhoﬀ’s ergodic theorem). Let f denote a stationary L1_{-random ﬁeld, then for P-a.e.} a 2 we have lim R!1 ? R f ða; xÞ dx ¼ ? f :

Moreover, if additionally h>jf j2i< 1, then

lim R!1 ? f ða; RxÞðxÞ dx ¼ ? f ? dx for all 2 L2ðÞ:

We postpone the proof to Sect. 2.2.1.

We turn back to the modiﬁed corrector equation which corresponds to the equation ð2.24Þ with right-hand side Fða; xÞ :¼ aðxÞ. This random ﬁeld (and by uniqueness the associated solution) is stationary. Hence, as a corollary of Lemma 2.20 and Lemma 2.22 we obtain:

Corollary 2.24. Suppose P is stationary. Let T 1 and 2 Rd. Then there exists a unique stationary random ﬁeld T that solves the modiﬁed corrector equation

1

TT r ðaðrTþÞÞ ¼ 0 inD

0_ðRd_{Þ, P-a.s.;} _ð2:29Þ

and which satisﬁes the a priori estimate ? 1 T 2 Tþ jrTj2 Cðd; Þjj2: Moreover, we have ? rT ¼0:

Proof. Step 1. Existence and a priori estimate.

For a 2 let Tða; Þ denote the solution to ð2.24Þ with F ¼ Fða; Þ ¼ aðÞ of Lemma 2.20. By uniqueness of the solution we deduce that T is stationary. Hence, by Lemma 2.22,

? 1 T 2 Tþ jrTj 2 ? ffiffiffi T p 1 T 2 Tþ jrTj 2 Cðd; ÞhkFðpffiffiffiffiTÞk2i; and the estimate follows, since

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hkFðpffiffiffiffiTÞk2i ¼ Z

hjaðpffiffiffiffiTxÞj2iðxÞ dx jj2: Step 2. Zero expectation of the gradient.

This is in fact a general property of stationary random ﬁelds u satisfying h>u2_{þ jruj}2_i_{< 1. Indeed, by stationarity} and the divergence theorem we have

I :¼ ? @iuða; xÞ dx ¼ ? ? @iuða; x þ yÞ dx dy ¼ 1 jj ? Z @ uða; x þ yÞ iðxÞ dSðxÞ dy;

where ðxÞ denotes the outer unit normal at x 2 @. By Fubini we may switch the order of the integration and get

I ¼ 1 jj Z @ ? uða; x þ yÞ dy iðxÞ dSðxÞ ¼ 1 jj ? uða; yÞ dy Z @ iðxÞ dSðxÞ ¼0;

where in the second last step we used stationarity, and in the last step R_@ iðxÞ dSðxÞ ¼ 0.

The estimate on rT of Corollary 2.24 is uniform T. Motivated by this we introduce a suitable function space in which we can pass to the limit T ! 1. Since we can only pass to the limit on the level of the gradient, it is convenient to consider uT ¼T>T, which satisﬁes>uT¼0, and thus is uniquely determined by ruT ¼ rT.

Lemma 2.25. Suppose P is stationary. Consider the linear space H :¼ u : Rd ! R: ? juj2þ jruj2 < 1; ? u ¼0; ru is stationary : Then,

(a) for any cube Q 2Q we have

? Q juj2þ jruj2 Cðd; QÞ ? jruj2 :

(b) H equipped with the inner product

ðu; vÞ_H:¼ ? ru rv is a Hilbert space. Proof. Step 1. Proof of (a).

We start with a deterministic estimate. Consider the dyadic family of cubes Qn¼2n_{, n ¼ 0; 1; . . . . We claim that} ? Qn juj2 1 2 CðdÞX n ‘¼1 2‘ ? Q‘ jruj2 1 2 : ð2:30Þ Indeed, ? Qn juj2 1 2 ? Qn u ? Qn1 u 2!12 þ ? Qn1 u ? Qn u ? Qn1 u 2! 1 2 þ ? Qn1 juj2 1 2 X n ‘¼1 ? Q‘ u ? Q‘1 u 2! 1 2 þ ? u |fflffl{zfflffl} ¼0 CðdÞX n ‘¼1 2‘ ? Q‘ jruj2 1 2 :

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? Q‘ jruj2 ¼ ? jruj2 ; yields ? Qn juj2 CðdÞnX n ‘¼1 22‘ ? Q‘ jruj2 Cðd; nÞ ? jruj2 : ð2:31Þ

Now, let Q denote an arbitrary cube. Then we have Q Qn for some n 2 N, and thus ? Q juj2þ jruj2 ¼ ? Q juj2 þ ? jruj2 C0ðd; nÞ ? jruj2 : Step 2.H is Hilbert.

Obviously ð; Þ_H turnsH into an inner product space and the deﬁniteness of the norm follows from (a). We argue that ðH; k k_HÞis complete. First note that by stationarity of ru, we have for all n 2 N,

? Qn jruj2 ¼ ? jruj2 ¼ kuk2_H; ð2:32Þ

and thus by Step 1,

? Qn

juj2þ jruj2

Cðd; nÞkuk2_H:

Let ðukÞdenote a Cauchy sequence inH. Then the previous estimate implies that ðukÞis Cauchy in any of the spaces L2_{ð; H}1_{ðQnÞÞ, n 2 N. Thus, uk}_!_uðnÞ _{in L}2_{ð; H}1_ðQnÞÞ _{for all n 2 N0. For ‘ n, we have Q‘}_{Qn, and thus} uð‘Þ_¼_uðnÞ_{on Q‘. We conclude that there exists a random ﬁeld u with u 2 L}2_{ð; H}1_{ðQÞÞ for all cubes Q 2}_{Q, and} uk!u in L2_{ð; H}1_{ðQÞÞ for any Q 2}_{Q. This in particular implies that hj}>_{uji ¼ 0. To conclude u 2}_{H it remains to} argue that ru is stationary. It suﬃces to show for any ’ 2 L2ðÞ, Q 2Q and x 2 Rd,

? xþQ

@iuða; yÞ dy’ðaÞ

¼ ?

Q

@iuðxa; yÞ dy’ðaÞ

:

Since @iukis stationary, this identity is satisﬁed for u replaced by uk. Since @iuk!@iu in L2_{ð QÞ for any Q 2}_{Q, the}

identity also holds for @iu.

As a corollary of Lemma 2.25, Corollary 2.24 and Lemma 2.18 we obtain the existence and uniqueness of the sublinear corrector:

Corollary 2.26(Uniqueness of the sublinear corrector). Suppose P is stationary and ergodic. Then there exists at most one 2H satisfying the corrector equation ð2.14Þ and the sublinear growth condition ð2.15Þ P-a.s.

Proof. Let ; 02H be two sublinear solutions to the corrector equation and consider u :¼ 0. Then P-a.s. u satisﬁes the assumptions of Lemma 2.17 and we conclude

lim R!1

? R

jruj2¼0:

On the other hand, by stationarity of ru and ergodicity we have ? jruj2 ¼ lim R!1 ? R jruj2¼0;

and thus u is constant P-a.s. Since >u ¼ 0, we conclude that u ¼ 0.

Corollary 2.27(Existence of the sublinear corrector). Suppose P is stationary and ergodic. Let T denote the solution to the modiﬁed corrector equation ð2.29Þ of Corollary 2.24. Then there exists 2H such that uT :¼ T>T * weakly in H (for T ! 1), and is the unique solution to the corrector equation in the sense of Corollary 2.26.

Proof. By Corollary 2.24 ðuTÞis a bounded sequence inH. Since H is Hilbert, we may pass to a subsequence (not relabeled) such that uT* weakly inH. We claim that solves the corrector equation ð2.14Þ. Let 2 C1

c ðR d_Þ_and ’ 2 L2ðÞ denote test functions and let Q 2Q denote a cube centered at 0 with supp Q. From uT * weakly in H, we infer that rT * r weakly in L2ð QÞ, and thus

’ Z aðr þ Þ r ¼ lim T!1 ’ Z aðrTþÞ r ¼ lim T!1 ’ Z ₁ TT : Note that

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Z ₁ TT jQj 1 T ? Q 2_T 1 2 ₁ T ? Q 2 1 2 : ð2:33Þ

By stationarity and the a priori estimate of Corollary 2.24 we have

’ Z ₁ TT T12_jQj ? Q 2 1 2 h’2i12 1 T ? 2_T 1 2 !0;

and thus we deduce with ð2.33Þ that

’ Z

aðr þ Þ r

¼0:

Since the test functions are arbitrary, ð2.14Þ follows. Since 2H, we have h>jrj2i< 1, h>ri ¼ 0, and hj>ji ¼ 0. By ergodicity, which we use in form of Lemma 2.23, we ﬁnd that the assumptions of Lemma 2.18 are satisﬁed P-a.s. Hence, ða; Þ is sublinear in the sense of ð2.15Þ P-a.s., and a solution to ð2.14Þ. By uniqueness of the solution (cf. Corollary 2.26) we conclude that is independent of the subsequence, and we deduce that uT * inH

for the entire sequence.

Note that Corollary 2.27 proves Proposition 2.15 except for the a priori estimate ? jrj2 1 2 2 jj 2 ; ð2:34Þ

whose argument we postpone to the end of this section. In fact, the estimate h>jrj2i Cðd; Þjj2(for some constant Cðd; Þ < 1) follows (by lower semicontinuity) directly from the a priori estimate in Corollary 2.24. The sublinear corrector of Proposition 2.15 can alternatively be characterized as the unique solution to an abstract variational problem in the Hilbert spaceH. (This formulation also entails a short argument for ð2.34Þ). In the rest of this section, we discuss this alternative formulation. We start with the observation that the space of stationary H1_{-random ﬁelds} forms a Hilbert space:

Lemma 2.28. Suppose P is stationary. Consider the linear space S :¼ u is a stationary random field with

? juj2þ jruj2 < 1 :

ThenS with inner product

ðu; vÞ_S:¼ ?

uv þ ru rv

is a Hilbert space. Moreover, for any u 2S we have h>rui ¼ 0.

Proof. Obviously ð; Þ_SturnsS into an inner product space. We argue that ðS; k k_SÞis complete and ﬁrst note that for any u 2S, the stationarity of u implies stationarity of ru, and thus for all Qn:¼ 2n_{, n 2 N, we have}

? Qn u2þ jruj2 ¼ ? u2þ jruj2 ¼ kuk2_S: ð2:35Þ

The remaining argument for completeness is similar to the proof of Lemma 2.25. The fact that gradients of stationary random ﬁelds are mean-free has already been proven in Step 2 in the proof of Corollary 2.24. Next we observe that on the level of the gradient any function u 2H can be approximated by functions in S. With help of this observation we can pass from distributional equations on Rd to problems inH (and vice versa): Lemma 2.29. Suppose P is stationary and ergodic.

(a) For any u 2H we can ﬁnd a sequence uT2S such that uT>uT * u weakly inH.

(b) Let F be a stationary random vector ﬁeld with h>jFj2i< 1. Then the following are equivalent ? F r’ ¼0 for all ’ 2H; ð2:36Þ r F ¼ 0 inD0ðRdÞ, P-a.s. ð2:37Þ

Proof of Lemma 2.29. Step 1.

Let F denote a stationary vector ﬁeld with h>jFj2_i_{< 1, let T 1. We claim that there exists a unique u}

T2S such that

1

TuT 4uT¼ r F inD

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and that uT is characterized by the weak equation ? 1 TuT’ þ ruT r’ ¼ ? F r’ for all ’ 2S: ð2:39Þ

We first argue that a solution uT2S to ð2.38Þ exists. Note that by stationarity we have for all R 1, hkFðpffiffiffiRÞk2i ¼ ? jFj2 :

Thus, by Lemma 2.20, there exists a unique random ﬁeld uT that satisﬁes ð2.38Þ and the a priori bound ð2.25Þ P-a.s. Since F is stationary, uT and ruT are stationary, and thus the a priori bound turns into

? 1 Tu 2 Tþ jruTj 2 Cðd; Þ ? jFj2 : ð2:40Þ

On the other hand, the Lax–Milgram Theorem yields a unique solution vT 2S to the weak formulation ð2.39Þ. In order to conclude that both formulations are equivalent, it suﬃces to show that uT solves ð2.39Þ. For the argument let ’ 2S and 2 C1

c ðÞbe arbitrary test functions. It suﬃces to show I :¼ ? 1 TuT’ þ ðruTþFÞ r’ ¼0:

For R 1 set ’R :¼_R1’ðRÞ, uT;R:¼1_RuT;RðRÞ, and FR :¼ FðRÞ. Then by stationarity and scaling we have

I ¼ ? R2 T uT;R’Rþ ðruT;RþFRÞ r’R ; and by ð2.38Þ, ? R2 T uT;Rð’RÞ þ ðruT;RþFRÞ rð’RÞ ¼0:

The diﬀerence of the previous two equations is given by ? R2 T uT ;R’Rð1 Þ þ ðruT;R þFRÞ ðr’R rð’RÞ¼: II þ III: By Cauchy–Schwarz, stationarity, and the a priori estimate ð2.40Þ,

jIIj 1ffiffiffiffi T p ? 1 TjuTðRÞj 2 1 2 ? j’ðRÞj2j1 j2 1 2 CðdÞ 1_ffiffiffiffi T p ? jFj2 1 2 ? ’2 1 2 Z j1 j2 1 2 Cðd; T; F; ’Þk1 kL2_ðÞ: Regarding III we note that

jIIIj ?

jruRþFRjðjr’Rjj1 j þ j’RjjjrjÞ

Arguing as above, we deduce that

jIIIj Cðd; F; ’Þðk1 kL2_ðÞþ krk_L1_ðÞhk’Rk2_L2_ðÞi 1 2_Þ:

Note that by stationarity we have

hk’Rk2_L2_ðÞi ¼R 2 ? R j’j2 ¼R2 ? j’j2 !0:

In conclusion we deduce that

jIj lim sup R!1

ðjIIj þ jIIIjÞ Cðd; T; F; ’Þk1 kL2_ðÞ:

Since is arbitrary, the right-hand side can be made arbitrarily small, and thus I ¼ 0. Step 2. Proof of (a).

Let u 2H, set Fða; xÞ :¼ ruða; xÞ, and let uT denote the unique solution inS to ð2.38Þ. From ð2.39Þ we obtain the a priori estimate

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? 1 TjuTj 2_þ1 2jruTj 2 1 2 ? jruj2 ; ð2:41Þ

which for the gradient is uniform in T 1. We conclude that vT :¼ uT>uT deﬁnes a bounded sequence inH. Let v 2H denote a weak limit of ðvTÞalong a subsequence T ! 1 (that we do not relabel). We claim that v ¼ u (which implies that the convergence holds for the entire sequence). First notice that it suﬃces to show that for all ’ 2 L2_ðÞ and 2 C1 c ðR d_Þ_{we have} ’ Z ðrv ruÞ r ¼0: ð2:42Þ

Indeed, this implies that w ¼ v u satisﬁes 4w ¼ 0 in D0ðRdÞ, P-a.s. Since w 2H has sublinear growth, we conclude with Lemma 2.17 that w is constant. Since >w ¼ 0 by construction, we deduce that w ¼ v u ¼ 0. We prove ð2.42Þ. Since rv is a weak limit of rvT¼ ruT, it suﬃces to show that

I :¼ ’ ? Q ðruT ruÞ r !0 for T ! 1;

where Q 2Q is a cube that contains the support of . Since uT solves ð2.38Þ with F ¼ ru, we have I ¼ ’ ? Q 1 TuT ;

which for T ! 1 converges to 0, thanks to the a priori estimate ð2.41Þ and stationarity. Step 4. Proof (b).

First note that ð2.36Þ, thanks to (a), is equivalent to ?

F r’

¼0 for all ’ 2S: ð2:43Þ

Let uT2S denote the unique solution to 1

TuT 4uT¼ r F inD

0_ðRd_{Þ, P-a.s.;}

which exists thanks to Step 2, and is equivalent to ? 1 TuT’ þ ðruTFÞ r’ ¼0 for all ’ 2S: ð2:44Þ

Then for all T 1, ð2.37Þ is equivalent to uT¼0. On the other hand, in view of ð2.44Þ, uT ¼0 implies ð2.43Þ, and ð2.43Þ implies ? 1 TuT’ ¼0 for all ’ 2S; and thus uT ¼0.

Finally, we present the characterization of and T by means of variational problems in the Hilbert spacesH and S: Lemma 2.30. Suppose P is stationary and ergodic. Let denote the sublinear corrector associated with 2 Rd of Proposition 2.15, and T the unique modiﬁed corrector associated with 2 Rd of Corollary 2.24. Then 2H and T2S are uniquely characterized by

? aðr þ Þ r’ ¼0 for all ’ 2H; ð2:45Þ ? 1 TT’ þ aðrTþÞ r’ ¼0 for all ’ 2S; ð2:46Þ and we have lim T !1 ? jrT rj2 ¼0: Moreover, ð2.34Þ holds.

Proof. First note that the variational equations for and T inH and S, respectively, admit a unique solution by the theorem of Lax–Milgram. The equivalence of the formulations for follows from Lemma 2.29 (b). The equivalence of the formulation for Tfollows by the argument in Step 1 in the proof of Lemma 2.29. (We only need to replace 4 by r ðarÞ and F by a). For the convergence statement it is convenient to work with the variational equations:

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? ðr rTÞ aðr rTÞ ¼ ? ðr rTÞ aðr þ Þ ? ðr aðrTþÞ þ ? ðrTaðrTþÞ ¼ ? ðr aðrTþÞ 1 T ? 2_T : Hence, lim sup T!1 ? ðr rTÞ aðr rTÞ lim T !1 ? ðr aðrTþÞ ¼ ? r að þ rÞ ¼0;

and the claim follows by ellipticity of a. The a priori estimate for r easily follows from the variational formulation of the corrector equation: We ﬁrst note that by ellipticity and ð2.45Þ, we have

? jr þ j2 ? ðr þ Þ aðrTþÞ ¼ ? aðrTþÞ 1 2jj 2_þ 2 ? jrTþj2 ; and thus 2 ? jr þ j2 1 2jj 2_:

On the other hand,

? jr þ j2 ¼ ? jrj2 þ jj2;

since the cross-term h>r i ¼ 0, thanks to h>ri ¼ 0. Thus, ð2.34Þ follows from the combination of these

estimates.

Note that Corollary 2.27 combined with ð2.34Þ, which follows from the previous lemma, completes the proof of Proposition 2.15. As a corollary of the previous lemma, and in analogy to Lemma 2.8, we have:

Lemma 2.31(Properties of the homogenized coeﬃcients). Suppose Assumption (S) is satisﬁed and let 1; . . . ; d denote the correctors associated with e1; . . . ; ed. Set

ahomei:¼ ? aðriþeiÞ : Then:

(a) (ellipticity). For any 2 Rd we have

ahom jj2: (b) (invariance under transposition). Let t

i denote the corrector associated with the transposed matrix at. Then ðahomÞtei¼ ? atðrt_iþeiÞ :

(c) (symmetry). If a is symmetric (a.e. in Rd and P-a.s.), then ahom is symmetric. The proof is similar to the proof of Lemma 2.8. We leave it to the reader.

Remark 2.32(Systems). The arguments that we presented in this section (in particular the construction of the sublinear corrector and the proof of Theorem 2.14 extend to systems of the form

r aru ¼ F;

with u : Rd!H taking values in a ﬁnite dimensional Euclidean space H. The matrix ﬁeld a : Rd!LinðHd_{; H}d_Þ_is required to be bounded and uniformly elliptic in the integrated form of

Z r ar Z jDj2; for all 2 C1 c ðR d_{; HÞ:}

東北大学機関リポジトリTOUR

An Introduction to the Qualitative

and Quantitative Theory of Homogenization

Contents

1.

Introduction — A One-dimensional Example

2.

Qualitative Homogenization of Elliptic Equations