Fluctuations in the homogenization of semilinearequations with random potentials

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Communications in Partial Differential Equations

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Fluctuations in the homogenization of semilinear equations with random potentials

Guillaume Bal & Wenjia Jing

To cite this article: Guillaume Bal & Wenjia Jing (2016) Fluctuations in the homogenization of semilinear equations with random potentials, Communications in Partial Differential Equations, 41:12, 1839-1859, DOI: 10.1080/03605302.2016.1238482

To link to this article: http://dx.doi.org/10.1080/03605302.2016.1238482

Accepted author version posted online: 22 Sep 2016.

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2016, VOL. 41, NO. 12, 1839–1859

http://dx.doi.org/10.1080/03605302.2016.1238482

Fluctuations in the homogenization of semilinear equations with random potentials

Guillaume Bal^aand Wenjia Jing^b

aDepartment of Applied Physics and Applied Mathematics, Columbia University, New York, NY, USA;^bDepartment of Mathematics, The University of Chicago, Chicago, IL, USA

ABSTRACT

We study the stochastic homogenization and obtain a random fluctuation theory for semilinear elliptic equations with a rapidly varying random potential. To first order, the effective potential is the average potential and the nonlinearity is not affected by the randomness. We then study the limiting distribution of the properly scaled homogenization error (random fluctuations) in the space of square integrable functions, and prove that the limit is a Gaussian distribution characterized by homogenized solution, the Green’s function of the linearized equation around the homogenized solution, and by the integral of the correlation function of the random potential.

These results enlarge the scope of the framework that we have developed for linear equations to the class of semilinear equations.

ARTICLE HISTORY Received 22 October 2015 Accepted 14 July 2016

KEYWORDS Probability measure in Hilbert spaces; randomfields;

semilinear elliptic equation;

stochastic homogenization;

variational problem

MATHEMATICS SUBJECT CLASSIFICATION 35B27; 35J61; 60F05

1. Introduction

1.1. Motivation and overview

We study the asymptotic stochasticity of solutions to the semilinear elliptic equation with random potential

!−!u^ε+qε(x,ω)u^ε+f(u^ε)=g(x), x∈D,

u^ε=0, x∈∂D, (1.1)

and characterize the limiting distribution of the randomfluctuations (correction to homog- enization). Here,D is an open bounded domain in R^d,d = 2, 3, with smooth (say C²) boundary∂D. The (linear) potentialqε(x,ω)is highly oscillatory and is of the formq(^x_ε,ω), where 0 < ε ≪ 1 denotes the scale of oscillations. We assume thatq(x,ω), the potential function before scaling, is a stationary ergodic randomfield on a probability space(%,F,P).

The nonlinearityf is assumed to satisfy conditions that, essentially, guarantee that for each ω∈%andg∈H⁻¹(D), the above equation admits a unique weak solution.

Under mild conditions, such as, e.g., stationarity and ergodicity, on the random potential q(x,ω)and under natural structural conditions on the nonlinearityf(u), we prove in Theorem 3.3below that the above semilinear problem homogenizes asε → 0 and that the effective potential is in fact given by averaging and the nonlinearity term is not changed. In other words, u^εconverges strongly inL²(D)and weakly inH¹(D), for a.e.ω∈%, to the solutionuof the

CONTACTWenjia Jing wjing@math.uchicago.edu Department of Mathematics, The University of Chicago, 5734 S.

University Avenue, Chicago, IL 60637, USA.

© 2016 Taylor & Francis

effective equation

!−!u+qu+f(u)=g(x), x∈D,

u=0, x∈∂D, (1.2)

whereq = Eqis the average ofq(x,ω). The homogenized equation hence is deterministic and has smooth (non-oscillatory) coefficients.

The main contributions of this paper are an estimate of the size of the homogenization error u^ε−u, say in theL²(%,L²(D))norm, and more importantly, a study of the law (probability distribution) of the randomfluctuationsu^ε −u, viewing them as a random element in the Hilbert spaceL²(D). Such asymptotic estimates of stochasticity beyond the homogenization limitfind applications in uncertainty quantification and Bayesian formulation of PDE-based inverse problems; see e.g. [6,22].

The study of the limiting distribution of the homogenization error goes back to [12], where the Laplace operator with a random potential formed by Poisson bumps was considered.

General random potential with short-range correlations was considered recently in [1], and in [2,3] for other non-oscillatory differential operators with random potential. Long- range correlated random potential was considered in [7]. When random elliptic differential operators are considered, the limiting distribution of the homogenization error was obtained in [9] for short-range correlated elliptic coefficients, and in [8] in the long-range correlated case; all in the one-dimensional setting; we refer to [13–15,20,21] for important recent advances in this direction in higher spatial dimensions under strong structural assumptions on the probability distribution of the coefficients.

The main results of this paper show that the general framework developed by the authors in [1,3,7,18], for linear equations with random potential, essentially apply to the semilinear equation (1.1). Let us briefly explain the main ideas. In the linear framework, e.g., for (1.1) withf =0, let us denote the fundamental solution operator (Green’s operator) of the Dirichlet problem for−!+ qbyG. Then the homogenization error u^ε −u admits the following expansion formula:

u^ε−u=−Gν_εu+Gν_εGν_εu+Gν_εGν_ε(u^ε−u). (1.3) Here,νε = qε−qdenotes the randomfluctuation of the potential. We also haveu^ε−u =

−G^εν_εu, whereG^εis the solution operator associated to−!+q_ε. As long asq_ε >−λ₁, the first eigenvalue of the Laplace operator, we obtain a bound onu^ε−uinL²(%,L²(D))provided that we can estimate the correlation function ofν_ε. The leading term in the right-hand side of (1.3) is thefirst one. The last term is smaller because bothu^ε−uand the operatorGν_εG are small. The middle term is also of smaller order but for a different reason: one can verify thatGν_εGν_εu−E(Gν_εGν_εu)is small, provided we can estimate fourth-order moments ofν_ε, andE(GνεGνεu)is also small in dimensionsd=2, 3.

For the nonlinear equation, (1.3) has to be modified. In fact, as was already pointed out in [12] without technical details, the leading term ofu^ε−uin distribution should be given by

−G_uν_εu, whereG_uis the solution operator associated to the linearization of the homogenized equation around the homogenized solutionu. In this paper, we derive an expansion formula foru^ε −u[see (5.2) below], which plays the role of (1.3). In fact,Gabove is replaced byG_u and there are two more terms involving the nonlinearityf. To overcome the difficulty caused by the nonlinearity, wefirst establish uniformH¹andL^∞estimates on solutions to (1.1) that are independent ofεandω. These uniform bounds, together with regularity assumptions on

f, allow us to estimatef(u^ε)−f(u)asu^ε−uand estimatef(u^ε)−f(u)−f^′(u)(u^ε−u)as (u^ε −u)². As a result, we show that the nonlinear terms in the expansion ofu^ε −udo not contribute to the limiting distribution.

Finally, the picture of the size and the limiting distribution of the homogenization error u^ε −u is as follows; see Theorems 2.2and6.2below. In dimensiond = 2, 3, for short- range correlated random potential,u^ε−uis of orderε^d2 in theL²(%,L²(D))norm, and the distribution in the spaceL²(D)of the normalized errorε^−d2(u^ε − u)converges weakly to a Gaussian distribution onL²(D), with correlation kernel determined by the homogenized solutionu, Green’s function associated toG_u, and the integral of the correlation function ofν(x,ω). In the long-range case, for potentials that are functions of long-range correlated Gaussian randomfields, the size ofu^ε−uis of orderε^α2, whereα∈(0,d)is the decaying rate of the correlation of the underlying Gaussianfield. The limiting distribution ofε^−α2(u^ε−u) is a long-range correlated Gaussian, with correlation kernel depending also onαand certain parameters in the model of the random potential.

The rest of this paper is organized as follows: In Section2, we make precise the main assumptions on the random potential and the nonlinearity in (1.1), and state the main theorems. In Section3, we establish well posedness of (1.1) and more importantly, uniformH¹ andL^∞bounds that are independent ofεandω. These results facilitate the homogenization proof. Sections4and5are devoted to the proof of the main results, where we characterize how the homogenization error scales in energy norm, and determine the limiting distribution of the scaled error. We recall the controls on the terms in the expansion formula that are the same as in the linear setting, without detailing the proof, and concentrate on how to apply the uniformL^∞bound to deal with the nonlinear terms, which are new. Finally, we make some comments in Section6and discuss some directions of further studies.

2. Assumptions and main results 2.1. Assumptions

Wefirst state the main assumptions on the nonlinearityf and the random potentialq(x,ω).

Letλ₁(−!;D), which is henceforth simplified toλ₁(−!)or evenλ₁, be thefirst eigenvalue of the Dirichlet–Laplacian operator onD, that is,

λ₁(−!):=inf

"#

D|Dw|²dx : w∈H₀¹(D),

#

D

w²dx=1

$

. (2.1)

SinceDis bounded, we note thatλ₁(−!) >0. Thefirst set of main assumptions onqandf are:

[A1] q(x,ω),x ∈ R^d, is a stationary ergodic randomfield on(%,F,P). This means, there exists a family ofP-preserving ergodic group of transformations{τ_x : % → %}x∈R^d, i.e.,

P(τ_xA)=P(A)for allx∈R^dandA∈F. τ_yA=Afor ally∈R^dimplies thatP(A)∈{0, 1}, and a random variable%qsuch thatq(x,ω)=%q(τ_xω).

[A2] There existsM≥1 such that

&

&%q(ω)&

&≤M, for allω∈%. (2.2)

Moreover, there existsγ >1 such that|f(s)|≤C(1+ |s|^γ)for alls∈R, and ifd=3, we further assume thatγ <2d/(d−2)−1=5.

[A3] f :R→Ris continuously twice differentiable, and for somec>0,

λ₁+%q(ω)+f^′(s)≥c, for allω∈%ands∈R. (2.3) The assumptions [A2] and [A3] guarantee that for each ω ∈ %, the heterogeneous semilinear equation (1.1) admits a unique weak solution. The stationarity and ergodicity of q(x,ω)further make sure that this problem homogenizes, in the limitε → 0, to the effective equation (1.2). Notice that [A2] and [A3] still hold if we replace%qbyq; hence, the homogenized problem is also uniquely solvable.

To get quantitative estimates on the homogenization erroru^ε−u, and tofind the limiting distribution of this error after proper scaling, we need more information on the random potentialq(x,ω), or equivalently thefluctuation

ν(x,ω):=q(x,ω)−Eq=q(x,ω)−q. (2.4) In most of the paper, we assume that q(x,ω) is a short-range correlated random field satisfying certain fourth-order moment estimates. LetR(x)=E(ν(x,ω)ν(0,ω))=E(ν(x+ y,ω)ν(y,ω))be the autocorrelation function of the stationaryfieldν. The term “short-range correlation” refers to the condition that the correlation functionR(x)is integrable, i.e.,

σ²:=

#

R^d

R(x)dx<∞. (2.5)

Note that by definition, R(x)is a nonnegative definite function in the sense that for any positive integern, for anyn-tuples(x₁,· · · ,x_n), the matrix formed by(R(x_i−x_j))_{i,j=1,···,n}, is nonnegative definite. By Bochner’s theorem, the integral of R, i.e., σ², is nonnegative.

Throughout the paper, we assume also thatσ²>0.

We will need an estimate for (mixed) fourth-order moments ofν later. To simplify the presentation, we impose a stronger condition using the notion of “maximal correlation coefficient”, which quantifies how fast the correlation ofν decays. LetCthe set of compact sets inR^d, and for two setsK₁,K₂inC, the distanced(K₁,K₂)is defined to be

d(K₁,K₂)= min

x∈K1,y∈K2 |x−y|.

Given any compact setK ⊂ C, we denote byF_K theσ-algebra generated by the random variables{ν(x) : x∈K}. The maximal correlation coefficientϱofνis defined as follows: for eachr>0,ϱ(r)is the smallest value such that the bound

E(ϕ₁(ν)ϕ₂(ν))≤ϱ(r) '

E( ϕ₁²(ν))

E( ϕ₂²(ν))

(2.6) holds for any two compact setsK₁,K₂∈Csuch thatd(K₁,K₂)≥rand for any two random variables of the formϕ_i(ν),i=1, 2, such thatϕ_i(ν)isF_Ki-measurable andEϕ_i(ν)=0.

The further assumption we have onq(x,ω)is:

[S] The maximal correlation function satisfiesϱ¹² ∈L¹(R₊,r^d−1dr), that is,

# _∞

0

ϱ¹²(r)r^d−1dr<∞.

Assumptions on the mixing coefficientϱof random media have been used in [1,3,16];

we refer to these papers for explicit examples of randomfields satisfying the assumptions.

Note that the correlation functionR(x)ofνcan be bounded byϱ. For anyx∈R^d,

|R(x)| = |Eν(x,ω)ν(0,ω)|≤ϱ(|x|)Var(q).

By [A2] and [A3],q, and hence its variance are bounded. Since we may assumeϱ ∈ [0, 1] (henceϱ ≤ √ϱ), [S] impliesR ∈ L¹(R^d)andq(x,ω)has short-range correlations. In fact,

|R|¹² ∈L¹(R^d), so roughly speaking,|R(x)|!1/|x|^αwithα>2d. Therefore, [S] is a stronger condition thanqbeing short-range correlated, which would just require the decay rateα>d.

This faster decay of correlation is not necessary for the main results of this paper to hold (see Remark2.4below). But using the assumption [S], we can simplify significantly the following fourth-order moment estimates of the random potentialν(x,ω).

For any four pointsx,y,tandsinR^d, define

._ν(x,y,t,s):=Eν(x)ν(y)ν(t)ν(s)−(Eν(x)ν(y))(Eν(t)ν(s)). (2.7) wereν a Gaussian randomfield, its fourth-order moments would decompose as a sum of products of pairs ofRand the above quantity simplifies to a sum of two products of correlation functions. This property does not hold for general random fields. However, we have the similar estimate:

|._ν(x,t,y,s)|≤ϑ(|x−t|)ϑ(|y−s|)+ϑ(|x−s|)ϑ(|y−t|), (2.8) whereϑ(r)=(Kϱ(r/3))¹²,K =4∥ν∥L^∞(%×D). We refer to [16] for the proof of this lemma.

Estimates of this type based on mixing property already appeared in [1].

Notations.We simplify the notation of Lebesgue spacesL^p(D)onDtoL^pwhen this is not confusing; in particular, if Lebesgue space on the probability space%is concerned, we make the dependence explicit. We useH¹for the Sobolev spaceW^1,2and we useH^s,s∈(0, 1), for the fractional Sobolev spaceW^s,2, which is the closure ofC^∞₀ (D)in the norm

∥u∥²_H^s_(K) :=∥u∥²_L2(K)+

#

K²

|u(x)−u(y)|²

|x−y|^d+2s dxdy.

See [11] for more reference on H^s. Throughout the paper, C denotes various bounding constants that may change line after line and we sayCis universal when it depends only on the parameters in the main assumptions above. For the functionsq,νandR, we use subscript εto denote the scaled function, as inqε(x)=q(_x

ε

).

2.2. Main results

Thefirst main theorem concerns how the homogenization error scales.

Theorem 2.1. Let D⊂R^dbe an open bounded domain with C²boundary∂D, u^εand u be the solutions to(1.1)and(1.2), respectively. Suppose that[A1],[A2],[A3], and[S]hold, g ∈L²(D) and d=2, 3. Then, there exists positive constant C, depending only on the universal parameters and∥g∥L², such that

E∥u^ε−u∥²_L² ≤C(∥g∥L²)ε^d. (2.9)

This theorem providesL²(%,L²(D))estimate ofu^ε−u, and its proof is detailed in Section4.

So roughly speaking, the size of the homogenization error is of orderε^d2. We hence consider the limiting law of the rescaled randomfluctuationε^−d2(u^ε−u), and prove:

Theorem 2.2. Suppose that the assumptions in Theorem2.1hold. Letσ be defined as in(2.5) and G_u(x,y)be Green’s function defined by the linearized equation around u; see(3.6)below.

Let W(y)denote the standard d-parameter Wiener process. Then asε→0, u^ε−u

√ε^d

distribution

−→ σ

#

D

G_u(x,y)u(y)dW(y), in L²(D). (2.10) The proof can be found toward the end of Section5.

Remark 2.3. The integral on the right-hand side of (2.10) is understood, for eachfixedx, as Wiener integral inywith respect to the multiparameter Wiener processW(y). LetXdenote this integral. Ford=2, 3, because Green’s functionG_u(x,y)is square integrable (see below), Xis a random element inL²(D). In particular, for anyϕ ∈L²(D), the inner product⟨ϕ,X⟩ has precisely the Gaussian distributionN(0,σ_ϕ²)with mean zero and variance

σ_ϕ²:=σ²

#

D

(G_uϕ)²u²dy. (2.11)

Remark 2.4. We make some comments on the main assumptions of this paper. In view of the variational formulation of (1.1), assumption [A3] is natural, because it guarantees that the minimizing functional is coercive and convex, as we show in the next section. The full zero-order termf^ε(u):= q_εu+f(u)can be viewed as a reaction–diffusion nonlinearity. As an example, whenf is of bistable type, sayf(u)=u(u−1)(u−θ)for someθ ∈(0, 1), andq is large enough, all of the requirements in [A3] are satisfied.

As pointed out earlier, the assumption [S] imposes stronger decay rate on the correlation functionR than it being integrable. This stronger condition is mainly imposed to simplify the control of the fourth-order moment._νin (2.8). We refer to [3] for an alternative way to control terms like._ν, which allows decay rate|R(x)| ! |x|^−α,α > d, at the cost that there are more pairs of products of controlling functions on the right hand side of (2.8).

We state and prove the main theorems assuming that the random potential q(x,ω) has short-range correlation. Nevertheless, our approach also works for some long-range correlation setting; see the last section of this paper.

3. Homogenization and uniform estimates 3.1. Well-posedness and uniform estimates

We record herefirst the well-posedness result for the heterogeneous semilinear equation (1.1).

Lemma 3.1. Under the assumptions [A2] and [A3], for eachfixedω ∈ %andε ∈ (0, 1), g ∈H⁻¹, there exists a unique weak solution in H¹₀(D)to(1.1). Moreover, there exists C> 0

independent ofεandω, and

∥u^ε∥H₀¹≤C(∥g∥H⁻¹). (3.1) This result is classical; nonetheless, we briefly outline the proof for the sake of completeness.

Proof. Forfixedωandε, one can recast (1.1) as the Euler Lagrange equation associated to the variation problem of minimizing the following (nonlinear) energy functional

I^ε[v]:=

#

D

1

2|∇v|²+1 2q*x

ε,ω+

v²+F(v)−vg dx, (3.2) over the setH¹₀(D), whereFis an antiderivative off, e.g.,F(v)=,_v

0 f(s)ds.

The assumption|f(u)| ≤ C(1+ |u|^γ)implies that|F(u)| ≤ C(1+ |u|^γ+1). By Sobolev embedding, ifd=2,H¹(D)is embedded inL^p(D)for anyp∈[1,∞), and ifd=3,H¹(D)is embedded inL⁶(D)which is included inL^γ+1sinceγ ≤5. In both cases,F(v)is integrable.

So, the integral above is well defined onH₀¹(D).

The assumptionλ₁+%q+f^′(s)≥c>0 for alls∈Rfurther implies thatI^ε, as a functional on H₀¹(D), is strictly convex. Moreover, we can chooseα>0 small so that(1−α)λ₁+%q+f^′(s)≥

c

2 >0 still holds for allsandω. Integrate inson both sides of this inequality; we get 1

2

((1−α)λ₁+qε(x,ω))

s²+F(s)≥ c

4s²+f(0)s.

Using this fact and the Poincaré inequality∥∇v∥²_L2 ≥λ₁∥v∥²_L2, we get

#

D

1

2|∇v|²+ 1

2qε(x,ω)v²+F(v)dx≥

#

D

α

2|∇v|²+ c

4|v|²dx+

#

D

f(0)v dx

≥

#

D

α

2|∇v|²+ c

8|v|²dx− 2|f(0)|²

c |D|, (3.3) where|D|is the volume of the domainD. This shows thatI^εis also bounded from below. As a result, (3.2) and hence (1.1) admits a unique solution for eachεandω.

For the uniformH¹bound, we solve, anyg∈H⁻¹(D), the deterministic Poisson problem

−!u_g=2g, inD,

with boundary conditionu_g = 0 on ∂D. Clearly we have∥u_g∥H¹ ≤ C∥g∥H⁻¹ for some universal constantC. The weak formulation ofu_gthen yields

I^ε[u_g] =

#

D

1

2|∇u_g|²+1 2q*x

ε,ω+

(u_g)²+F(u_g)−gu_gdx= 1 2

#

D

qε(x,ω)u²_gdx+

#

D

F(u_g)dx.

Now we compareI^ε[u_g]andI^ε[u^ε]. Note that (3.3) implies I^ε[u^ε]≥ 1

2α

#

D|∇u^ε|²dx−

#

D

gu^ε−C.

UsingI^ε[u^ε]≤I^ε[u_g]and then Hölder’s and Young’s inequalities, we have 1

2α

#

D|∇u^ε|²≤ 1 2

#

D

q_εu²_gdx+

#

D

F(u_g)dx+4C(c^′,λ₁)∥g∥²_H−1+ 1 4α

#

D|∇u^ε|²+C,

for some big constantC. By [A2], the integral ofqεu²_gis bounded byC∥u_g∥²_L2, which is further dominated byC∥g∥²_H−1. In view of the growth condition onFand the Sobolev embedding as before, we get∥F(u_g)∥L¹ ≤ C(∥u_g∥H¹) = C(∥g∥H⁻¹) for someC depending also on the parameters in the main assumptions but is uniform inεandω. The desired result then follows.

Next, we show that the solutionu^ε ∈ L^∞(D), and the bound on∥u^ε∥L^∞ can be made independent ofεorω.

Lemma 3.2. Assume that [A2] and [A3] hold. For anyfixedε∈(0, 1)andω∈%, let u^ε(·,ω) be the unique solution of (1.1). Then for any g ∈ L²(D), there exists C > 0independent ofε andωsuch that

∥u^ε∥L^∞ ≤C(c,M,∥g∥L²). (3.4) Proof. We use the following results from elliptic regularity theory:

(i) Suppose thatvsolves the linear equation

−!v=h inD,

with Dirichlet boundary conditionv=0 on∂D,h∈L^pwithp>max{1,^d₂}andv∈L^r for somer∈[1,∞), then it holds that∥v∥L^∞ ≤C(∥h∥L^p+∥v∥L^r)withCalso depending onr, p, andD. We refer to [10, Theorem 4.2.2] for the proof of this result.

(ii) Supposevsatisfies−!v+ v = hinD with Dirichlet boundary condition on∂D, h∈H⁻¹(D)andh∈L^p(D)for somep∈(1,^d₂)andd=3. Then there existsC(p,d,D)such that∥v∥

L dp

d−2p ≤C∥h∥L^p. We refer to [10, Theorem 4.2.3] for the proof of this result.

To prove Lemma3.2,first consider the simple situation ofd = 2 andγ > 1 arbitrary in [A2], ord=3 and 1<γ ≤4/(d−2)=4. For notational simplicity, we writeuforu^ε. Recast the semilinear equation (1.1) into the form of−!u=hwithh=g−f(u)−qε(x,ω)u. Now assumption [A2] imposes that∥qεu∥L^p ≤C∥u∥L^puniformly inp,ε, andω. Ifd=2, then by Sobolev embedding and the growth condition onf, we verify thath∈L^pfor allp∈[1,∞).

Ifd = 3 andγ ≤ 4, then by Sobolev embedding,u ∈ L^rwithr = _d^2d₋₂ andr > ^dγ₂ . Then f(u)∈ L^γr and_γ^r > ^d₂. Note also thatg ∈L²,q_εu ∈ L², and 2> ^d₂ ford = 2, 3; we hence verify thath∈L^pforp> ^d₂. Applying regularity theory (i), we conclude that

∥u∥L^∞ ≤C(c,M,∥g∥L²,∥u∥H¹)≤C(c,M,∥g∥L²).

For the more general case,d = 3 and _d^2d₋₂ ≤ ^dγ₂ (i.e.γ ∈ (4, 5)), we need a bootstrap argument. Here, we mimic the proof of Theorem 4.4.1 in Cazenave [10]. The semilinear equation (1.1) can also be recast as−!u+u=bwithb(x)=g(x)+(1−qε(x,ω))u−f(u).

Again the functiongand(1−q_ε)uare inL^pwithp > ^d₂ with uniform bounds, so we only need to take care of the nonlinear termf(u). To start, we setr= _d^2d₋₂and we haveu∈L^rand we note

γ <r< dγ

2 , γ − 2r

d =γ − 4

d−2 < d+2 d−2− 4

d−2 =1,

where we usedγ < ^d+2_d−2 as in [A2]. In particular,θ := _dγ^d_−2r > 1. Sinceu ∈ L^r and

|f(u)| ≤ C(1+ |u|^γ), we check thatb ∈ L^γr and_γ^r ∈ (1,^d₂). By elliptic regularity (ii), we obtain

∥u∥L^θr ≤C∥b∥_Lγ^r, since θr= dr

dγ −2r = d_γ^r d− ^2r_γ .

Ifθr> ^dγ₂, we are back to the simple situation. If not, we repeat the argument using elliptic regularity (ii)kmore times untilθ^kr≤ ^dγ₂ <θ^k+1r. We then getu∈L^θk+1rwithθ^k+1r> ^dγ₂ and hence return to the simple situation, and we can conclude. We further verify that the bounds are uniform inεandωin view of (3.1).

3.2. Homogenization theory

Because the random coefficients appear only in the zeroth-order linear term, i.e., the potential term, the homogenization theory for the equations is relatively straightforward. For the sake of completeness, we present the details here.

Theorem 3.3. Under assumptions [A1], [A2], and [A3], there exists an event%₁∈F with full probability measure, such that for allω∈%₁, u^ε(·,ω)converges strongly in L²(D)and weakly in H₀¹(D), to the deterministic function u∈H₀¹(D)that solves(1.2).

Proof. Owing to Lemmas3.1and3.2, we note that there existsC > 0 uniform inεandω, such that∥u^ε∥H¹(D)+∥u^ε∥L^∞(D) ≤C. Also, by ergodic theorem (see. e.g. [17, Section 7.1]), there exists%₁∈FwithP(%₁)=1 such thatqε(x,ω)converges weakly inL²_loc(R^d)toqfor allω∈%₁. As a consequence, for eachfixedω∈%₁, we canfind a functionu(·,ω)∈H¹₀(D) and extract a subsequenceε(ω)→0, along which we have

∇u^{ε L}

2

⇀∇u, u^ε −→^L2 u, q*· ε,ω

+ _L²

⇀q,

where⇀and→denotes, respectively, weak and strong convergence. Owing to the uniform inεbound onu^εand the regularity assumptionf ∈C²(R),

&

&f(u^ε)−f(u)&&≤C₁|u^ε−u|.

Here,C₁is the Lipchitz constant off inside the domain[−2C, 2C]andCis the uniform bound in (3.4). This implies thatf(u^ε)→f(u)inL²(D). As a result, pass to the limitε →0 in the weak formulation

#

D∇u^ε·∇ϕ+q*x ε,ω

+

u^εϕ+f(u^ε)ϕ−gϕdx=0, for allϕ∈C^∞₀ (D), we get

#

D∇u·∇ϕ+quϕ+f(u)ϕ−gϕdx=0, for allϕ∈C^∞₀ (D).

This shows that the functionu(·,ω)solves the Eq. (1.2). Note that this problem has a unique deterministic solution, and henceu(x,ω) = u(x)is independent ofω. As a result, for all

ω∈%₁and along the full sequenceε →0, the convergenceu^ε →uholds. This completes the proof of the theorem.

3.3. Green’s function estimates for the linearized equation

Letube the homogenized solution. The linearized differential operatorL_u, aroundu, of the homogenized semilinear operatorLu=−!u+qu+f(u)is given by

L_u(v)=−!v+(

q+f^′(u))

v (3.5)

Green’s functionG_u(x,y),x,y∈Dandx̸=y, associated toL_usatisfies

!−!G_u(x;y)+(

q+f^′(u(x)))

G_u(x;y)=δ_y, forx∈D,

G_u(x;y)=0, forx∈∂D. (3.6)

We have the following estimates:

Lemma 3.4. Assume [A2] and [A3], and assume that D ⊂ R^d is an open bounded domain with C²boundary. Then there exists C>0such that,

&

&G_u(x,y)&

&≤

⎧⎨

⎩ C

|x−y|^d−2 for d=3 C(&&log|x−y|&&+1)

for d=2

and &

&∇xG_u(x,y)&

&≤ C

|x−y|^d−1. (3.7) We note that by (3.4), the potential functionq+f^′(u)∈L^∞(D). Moreover, [A3] guarantees that the problem remains elliptic. Thefirst bound in (3.7) is immediate and the second one follows, say, from standard Hölder regularity for gradients. Note that this is the place where regularity of∂Dis used.

4. Quantitative estimates of the homogenization error

In this section, we determine the convergence rate ofu^ε →uinL²(%,L²(D)). Defineξ^ε := u^ε−u. Then, it satisfies

−!ξ^ε+qεξ^ε+f(u^ε)−f(u)=−ν_ε(x,ω)u.

Formally, the nonlinear termf(u^ε)−f(u)is approximated byf^′(u)(u^ε−u). So we expect that the leading term inξ^εis given by the solution of the following equation:

!L_uχ^ε =−!χ^ε+qχ^ε+f^′(u)χ^ε =−ν_εu, inD,

χ^ε=0, on∂D. (4.1)

Let G_u be the inverse operator, i.e., the fundamental solution operator of the Dirichlet problem, andG_u(x,y)is Green’s function. In view of [A3], these notations are well defined, and we have

χ^ε =−G_uνεu. (4.2)

Our goal is to estimate the quantity ∥ξ^ε∥L²(%,L²(D)). First, an estimate for χ^ε in L²(%,L²(D))is easily obtained from the linear theory. Next, by applying the mean value theorem to the nonlinear term in the equation ofξ^ε, we show that the remainderz^ε :=ξ^ε−χ^ε

satisfies a linear equation with coefficients that depend onu^εandu, and withχ^εon the right- hand side. Therefore, by linear theory again, wefinally obtain estimates forz^ε and hence for ξ^ε. To use the linear theory forz^ε, however, we need the uniform (inεandω) bound onu^ε because the coefficient of the equation forz^εdepends onu^ε.

Wefirst present the estimates for the correctorχ^εand briefly recall the proof. For detailed argument, we refer to [18, Lemma 4.1].

Lemma 4.1. Let d=2, 3. Assume that [A1], [A2], [A3], and [S] hold. Then there exists C>0 such that

E∥χ^ε∥²_L²≤Cε^d. (4.3)

Proof. The mean square of theL²(D)norm of the functionχ^ε(·,ω)is given by E∥χ^ε∥²_L2(D)=E∥G_uν_εu∥²_L2(D) =E

#

D³

G_u(x,y)G_u(x,z)νε(y)νε(z)u(y)u(z)dydzdx.

Note thatEν_ε(y)νε(z)=R^ε(y−z)and use the bounds on Green’s function. We get, ford=3, E∥χ^ε∥²_L²_(D)≤

#

D³

C

|x−y|^d−2|x−z|^d−2

&

&

&

&R

0y−z

ε 1&&&

&|u(y)u(z)|dydz.

In view of the bound (3.7), the product ofG_u(·,y)andG_u(·,z)is integrable onDwith bounds independent ofyorz. Hence, we can integrate overxfirst and then change variable in the remaining integral, which yields a factor ofε^dand verifies the desired result. The situation of d=2 can be treated in the same manner.

Next, we move on the remainderz^ε = u^ε−u−χ^ε. In view of the equations satisfied by u^ε,uandχ^ε, we have

(−!+q_ε)z^ε+f(u^ε)−f(u)=−ν_εχ^ε+f^′(u)χ^ε. Lethεbe the function

f(u^ε)−f(u)

u^ε−u 1_{|uε−u|>0}

where1denotes the indicator function. Then, in view off ∈C¹(R)and the bound (3.4), we haveh_ε ∈L^∞(D). We verify thatz^εsolves the Dirichlet problem

−!z^ε+q_εz^ε+h_εz^ε=ν_εχ^ε+(f^′(u)−h_ε)χ^ε, (4.4) with zero boundary condition. Indeed, this problem has a unique solution and we verify easily thatu^ε−u−χ^εis the solution. We have the following result.

Lemma 4.2. Assume that the conditions in Lemma4.1hold. Then there exists C>0such that

E∥z^ε∥²_L² ≤Cε^d. (4.5)

Proof. LetL^ε,ωdenote the random linear differential operator−!+(qε+hε). In view of [A3], the uniform bound onu^ε and the definition ofh_ε, we observe that for eachfixedε ∈ (0, 1)

andω∈%, the potential termp^ε :=qε+hεsatisfies

−λ₁+c≤p^ε≤M^′.

By standard elliptic theory,L^ε,ω :H₀¹→H⁻¹is invertible. In particular, the inverse operator (L^ε,ω)⁻¹ is also a bounded transformation on L²(D). In fact, ∥(L^ε,ω)⁻¹∥L²→L² ≤ c⁻¹. Applying this fact, we have

∥z^ε∥²_L²≤2c⁻²(

∥νεχ^ε∥²_L²+∥(f^′(u)−hε)χ^ε∥²_L²) .

By [A2] and [A3],∥ν_ε∥L^∞ ≤ Cuniformly inεandω. By the uniform bounds onu^ε,u, and by the smoothness off, we confirm that∥f^′(u)−h_ε∥L^∞≤Calso uniformly inεandω. As a result, we have∥z^ε∥²_L2 ≤C∥χ^ε∥²_L2. The desired result follows.

The following control ofu^ε−ufollows immediately.

Corollary 4.3. Under the same conditions of Lemma4.1, there exists C>0such that

E∥u^ε−u∥²_L²≤Cε^d. (4.6)

5. Limiting distributions of the homogenization error

In this section, we identify the limiting distribution of the scaled homogenization error, i.e., the limit of the law ofε^−d2(u^ε−u)in the space ofL²(D)functions.

5.1. Expansion formula and overview of proof

To characterize the limiting distribution ofε^−d2(u^ε −u), we extend to nonlinear equations the framework developed in [1,3,18] for linear equations. Thefirst step is tofind a suitable expansion formula similar to (1.3) for the homogenization error.

We observe thatz^ε =u^ε−u−χ^εsatisfies the linear equation,

−!z^ε+qz^ε+f^′(u)z^ε =−ν_εξ^ε−(f(u^ε)−f(u)−f^′(u)ξ^ε).

Hencez^εhas the following representation z^ε =−G_uν_εξ^ε−G_u(

f(u^ε)−f(u)−f^′(u)ξ^ε)

. (5.1)

Substituting this formula tou^ε−u=χ^ε+z^εand using the formulaχ^ε=−G_uνεu, we obtain u^ε−u=−G_uν_εu−G_uν_ε(u^ε−u)−G_u(f(u^ε)−f(u)−f^′(u)ξ^ε)

=−G_uν_εu+G_uν_εG_uν_εu+G_uν_εG_uν_ε(u^ε−u)

−G_u(f(u^ε)−f(u)−f^′(u)ξ^ε)+G_uν_εG_u(f(u^ε)−f(u)−f^′(u)ξ^ε). (5.2) With this expansion formula at hand, we willfind the limiting distribution ofε^−d2(u^ε−u) by examining the terms on the right-hand side one by one. In particular, we note that the first three terms are the same as those obtained for linear equations, while the last two terms involve the nonlinearity.

We first recall the standard criterion for establishing weak convergence of probability

measures determined by random processes that areL²(D)functions.

Theorem 5.1. Let{X^ε},ε ∈ (0, 1), be a family of random processes on the probability space (%,F,P)and{X^ε}⊂L²(D). Then X^εconverges in distribution in L²(D)to the random process X in L²(D)if

(i) (Convergence of inner products with test functions). For anyϕ ∈ L²(D), the random variable⟨ϕ,X^ε⟩converges in distribution inRto⟨ϕ,X⟩.

(ii) (Tightness of distributions). The family of distributions in L²(D)determined by{X^ε}is tight.

This Prohorov-type result is well known and we refer to [23, Chapter VI, Lemma 2.1] for a proof in the general Hilbert space setting. When the tightness of the distribution of{X^ε} is concerned, the following result becomes handy, provided that{X^ε}in fact admits higher regularity.

Lemma 5.2. Let {X^ε}be as in Theorem 5.1. Suppose further that{X^ε} ⊂ H^s(D), for some s ∈(0, 1). Then the family of distributions in L²(D)determined by{X^ε}is tight if there exists some constant C>0such that

E∥X^ε∥H^s(D)≤C. (5.3)

We refer to [18, Theorem A.1] for a proof. SinceX^εisε^−d2(u^ε−u)in this paper, it indeed belongs to the more regular spaceH¹(D). Nevertheless, to obtain the control (5.3), which is uniform inε, it requires some work.

5.2. Liming distribution for the homogenization error

Thefirst three terms in the expansion formula (5.2) are precisely those encountered in the linear setting; compare with (1.3). We recall the following results.

Lemma 5.3. Assume that the conditions of Theorem2.1hold. Then asε→0, we have

− G_uν_εu

√ε^d

distribution

−→ σ

#

D

G_u(x,y)u(y)dW_y, (5.4) with convergence in distribution in the space L²(D). Moreover,G_uν_εG_uν_ε(u^ε−u)converges in L¹(%,L²(D))andG_uν_εG_uν_εu converges in L²(%,L²(D))to the zero function.

We briefly explain how these results are proved and refer to [18, Lemmas 4.2, 4.4, 4.5, 5.1, and 5.3] for detailed proofs.

(1) To show that−ε^−d2G_uν_εuhas the correct limit, according to the criterion Theorem5.1, Remark2.3, and Lemma5.2, it suffices to show, for allϕ∈L²(D),

√1

ε^d⟨−G_uν_εu,ϕ⟩= 1

√ε^d

#

D

ν*x ε,ω+

u(x)m(x)dxdistribution

−→ε→0 N( 0,σ_ϕ²)

, (5.5) wherem:=G_uϕandσ_ϕ²=σ²∥mu∥²_L2(D), and for someC>0 ands∈(0, 1)independent ofε,

E 22

2ε^−d2G_uν_εu 22 2²

H^s ≤C. (5.6)

We recall that (5.5) follows from a generalized central limit theorem for oscillatory integrals with short range correlated randomfields; see [1, Theorem 3.7]. The uniform H^s estimate (5.6) forscan be found in [18, Lemma 5.1]; see Lemma 5.4 below where such a control is needed to estimate thefirst nonlinear term in (5.2).

(2) To show thatG_uν_εG_uν_ε(u^ε −u)converges to the zero function inL¹(%,L²(D)), we need the fact that ford=2, 3,

E∥G_uνεG_u∥²_L²_→L² ≤Cε^d, (5.7) where∥G_uν_εG_u∥L²→L² is the operator norm. This can be proved by using Green’s function bound (3.7) and assumption [S]; see [18, Lemma 4.5]. We then get

∥G_uν_εG_uν_ε(u^ε−u)∥L²≤ ∥G_uν_εG_u∥L²→L²∥ν_ε∥L^∞∥u^ε−u∥L².

Take expectation, apply Hölder inequality and then (3.7) and Corollary4.3to get the desired estimate.

(3) To show thatG_uν_εG_uν_εuconverges to zero inL¹(%,L²(D)), the argument above is not valid because∥ν_εu∥L²is possibly of order one. To circumvent this lack of control, we note that

∥G_uν_εG_uν_εu∥²_L²≤2(

∥G_uν_εG_uν_εu−E(G_uν_εG_uν_εu)∥²_L²+∥E(G_uν_εG_uν_εu)∥²_L²) . For the mean functionE(G_uν_εG_uν_εu), we have

∥E(G_uν_εG_uν_εu)∥²_L²=

#

D

0#

D²

G_u(x,y)G_u(y,z)R

0y−z

ε 1

u(z)dzdy 12

dx

=

#

D⁵

G_u(x,y)G_u(x,ξ)G_u(y,z)G_u(ξ,η)R

0y−z

ε 1

R

0ξ−η

ε 1

u(z)u(η)dξdηdydzdx.

Integrate overxfirst and note that, ford=3,∥u∥L^∞ ≤CandG_uis square integrable and

#

D|G_u(x,y)G_u(x,ξ)|dx≤C

#

D

1

|x−y|^d−2|x−ξ|^d−2dx≤C.

We get

∥E(G_uν_εG_uν_εu)∥²_L2≤C

#

D⁴

1

|y−z|^d−2|ξ−η|^d−2

&

&

&

&R

0y−z

ε 1

R

0ξ−η

ε 1&&&

&dξdηdydz.

After a change of variables and using the fact thatR(·)/| · |^d−2is integrable overR^d, we verify that the above integral is of orderε⁴ ≪ε^d. It follows that∥ε^−d2E(G_uν_εG_uν_εu)∥L²converges to zero. The case ofd=2 is similar.

For the variationI₂:=G_uν_εG_uν_εu−E(G_uν_εG_uν_εu), we have ford=3, E∥I₂∥²_L²=E

#

D

0#

D²

G_u(x,y)G_u(y,z)3

ν_ε(y)ν_ε(z)−Eν_ε(y)ν_ε(z)4

u(z)dzdy 12

dx

=

#

D⁵

G_u(x,y)G_u(x,y^′)G_u(y,z)G_u(y^′,z^′)u(z)u(z^′)._ν

0y

ε,z ε,y^′

ε,z^′ ε

1

dz^′dy^′dzdydx.

Now, using the estimates for ._ν given by (2.8), and by standard techniques for potential integrals, we getE∥I₂∥²_L2 ≤ Cε^2d ≪ ε^d, which shows that ε^−d2I₂ converges to the zero function inL²(%,L²(D)). Ford=2, we have the same result.

Now we deal with the two nonlinear terms in (5.2).

Lemma 5.4. Under the assumptions of Theorem2.1, asε→0, we have G_u(f(u^ε)−f(u)−f^′(u)(u^ε−u))

√ε^d

distribution

−→ 0, (5.8)

with convergence in distribution in L²(D)and0denotes the constant zero function. Moreover, the term ε^−d2G_uν_εG_u(f(u^ε) − f(u) − f^′(u)(u^ε − u)) converges to the zero function in L¹(%,L²(D)).

Proof. Part I: The convergence ofG_uνεG_u(f(u^ε)−f(u)−f^′(u)(u^ε−u)). This part is simpler and is similar to the control of the remainder termG_uν_εG_uν_ε(u^ε−u)in the linear setting.

First, by Taylor’s theorem, the regularity off and the uniform (inε) bound ofu^εandu, we have

|f(u^ε)−f(u)−f^′(u)(u^ε−u)|≤C^′|u^ε−u|², (5.9) whereC^′=∥f∥C²([−2C,2C])andCis the bound in (3.4). This shows

∥f(u^ε)−f(u)−f^′(u)(u^ε−u)∥L²≤C∥u^ε−u∥L². Applying the uniform bounds on the operator norm ofG_uν_εG_u, we have again

∥G_uν_εG_u(f(u^ε)−f(u)−f^′(u)(u^ε−u))∥L²≤C∥G_uν_εG_u∥L²→L²∥u^ε−u∥L².

The desired result then follows by taking expectation and applying the Hölder inequality, the bound (5.7) and (4.6).

Part II: The limiting distribution ofε^−d2G_u(f(u^ε)−f(u)−f^′(u)(u^ε−u)). We follow again the criterion in Theorem5.1.

Convergence in distribution of inner products. Fix an arbitraryϕ ∈ L²(D). We need to consider the distribution of ε^−d2⟨G_u(f(u^ε) − f(u)− f^′(u)ξ^ε,ϕ⟩ = ε^−d2⟨f(u^ε)− f(u)− f^′(u)ξ^ε,m⟩, wherem=G_uϕ. Note that∥m∥L^∞ ≤C∥ϕ∥L². Using (5.9), we get

&

&⟨m,f(u^ε)−f(u)−f^′(u)(u^ε−u)⟩&&≤C∥ϕ∥L²∥u^ε−u∥²_L². As a result, we get

E

&

&

&ε^−d2⟨m,f(u^ε)−f(u)−f^′(u)(u^ε−u)⟩&&

&≤Cε^−d2E∥u^ε−u∥²_L²≤Cε^d2. (5.10) This shows that⟨ε^d2G_u(f(u^ε)−f(u)−f^′(u)(u^ε−u)),ϕ⟩converges, inL¹(%)and hence in distribution, to 0, agreeing with the trivial distribution of⟨ϕ, 0⟩.

Tightness.Letr^ε = f(u^ε)−f(u)−f^′(u)ξ^ε. The function we are considering isG_ur^ε. To show tightness of the distributions of{G_ur^ε}, we control the expectation of theH^snorm of