HOMOGENIZATION, LINEARIZATION AND LARGE-SCALE …HOMOGENIZATION, LINEARIZATION AND LARGE-SCALE...

HOMOGENIZATION, LINEARIZATION AND LARGE-SCALEREGULARITY FOR NONLINEAR ELLIPTIC EQUATIONS

SCOTT ARMSTRONG, SAMUEL J. FERGUSON, AND TUOMO KUUSI

Abstract. We consider nonlinear, uniformly elliptic equations with random,highly oscillating coefficients satisfying a finite range of dependence. We provethat homogenization and linearization commute in the sense that the linearizedequation (linearized around an arbitrary solution) homogenizes to the linearizationof the homogenized equation (linearized around the corresponding solution of thehomogenized equation). We also obtain a quantitative estimate on the rate ofthis homogenization. These results lead to a better understanding of differencesof solutions to the nonlinear equation, which is of fundamental importance inquantitative homogenization. In particular, we obtain a large-scale C0,1 estimatefor differences of solutions—with optimal stochastic integrability. Using thisestimate, we prove a large-scale C1,1 estimate for solutions, also with optimalstochastic integrability. Each of these regularity estimates are new even inthe periodic setting. As a second consequence of the large-scale regularity fordifferences, we improve the smoothness of the homogenized Lagrangian by showingthat it has the same regularity as the heterogeneous Lagrangian, up to C2,1.

1. Introduction

1.1. Motivation and informal summary of main results. We are motivatedby the goal of developing a quantitative theory of stochastic homogenization fornonlinear elliptic equations in divergence form. Such a theory has been developedin recent years for linear equations which is by now rather satisfactory: see forinstance [3, 18] and the references therein. In the linear case, the phenomenon ofimproved regularity of solutions on large-scales plays an important role, for instanceby providing control of “small errors” in a sufficiently strong norm. In the nonlinearsetting, such small errors are typically not solutions of the equation but rather thedifference of two solutions which satisfy a linearized equation. In this paper, weobtain quantitative homogenization estimates for such linearized equations andobtain a large-scale C0,1–type estimate for differences of solutions.

The equations we analyze take the form

(1.1) −∇ ⋅ (DpL(∇u(x), x)) = 0 in U ⊆ Rd, d ≥ 2,

where the Lagrangian L = L(p, x) is assumed to be uniformly convex and C2,γ inthe variable p. Of course, this equation is variational: u ∈ H1(U) is a solutionof (1.1) if and only if it is a local minimizer of the integral functional

v ↦ ∫UL(∇v(x), x)dx.

Date: September 26, 2019.2010 Mathematics Subject Classification. 35B27, 35B45, 60K37, 60F05.Key words and phrases. stochastic homogenization, large-scale regularity, nonlinear equation,

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2 S. ARMSTRONG, S. J. FERGUSON, AND T. KUUSI

We further assume that L is a stochastic object and that its law P is Zd–stationaryand has a unit range of dependence (with respect to the variable x). The goal is tounderstand the statistics of the solutions, under the probability measure P, and onlarge length scales, that is, the (“macroscopic”) domain U is very large relative to(“microscopic”) unit scale, which is the correlation length scale of the coefficients.

The principle of homogenization asserts that, in the regime in which the ratio ofthese two length scales is large, a solution of (1.1) is, with probability approachingone, close in a strong norm (relative to its size in the same norm) to a solution of adeterministic equation of the form

(1.2) −∇ ⋅ (DpL (∇uhom)) = 0 in U.

Dal Maso and Modica [9, 10] were the first to prove such a result for the equation (1.1)in this (and actually a much more general) setting. They realized that the variationalstructure of the equation provides a natural subadditive quantity, which has a P–almost sure limit by the subadditive ergodic theorem, and that this limit implies ageneral homogenization result for the equation.

We are interested here in quantitative results, in particular the speed of conver-gence to the homogenization limit. There has been a lot of recent interest in buildinga quantitative theory of homogenization for linear, uniformly elliptic equations, andthere is now an essentially complete and optimal theory available in this simplestof settings: see [3, 2, 16, 18]. While there has been some success in extending thistheory to degenerate linear equations (see [1, 11]), quantitative homogenizationestimates for nonlinear equations are relatively sparse: the only previous resultsappeared in [4, 5], which quantified the subadditive argument of [9, 10] to obtainan estimate for the homogenization error which is at most a power of the ratio ofthe length scales (see Theorem 2.1 below for the precise statement). The paper [5]also introduced the concept of a large-scale regularity theory for random ellipticoperators and, in particular, proved a large-scale C0,1–type estimate for solutionsof (1.1) (see Theorem 2.3 below). Subsequently, some variations and extensions ofthis result were developed in [17, 4, 13, 2, 3] in the linear setting.

To develop a more precise quantitative theory for nonlinear equations, extendingthe results known in the linear setting—such as sharp exponents for the scaling ofthe homogenization error and a characterization of the scaling limit of solutions—what is needed is finer estimates on solutions and, more importantly, on differencesof two solutions (which are typically very close to each other), on all length scaleslarger than a multiple of the microscopic scale. For linear equations, since thedifferences of solutions are also solutions, the large-scale regularity already givesexactly the sort of information which is required. For a nonlinear equation suchas (1.1), the difference of two solutions u and v is the solution of the linear equation

−∇ ⋅ (a(x)∇(u − v)) = 0,

with coefficients a(x) given by

a(x) = ∫1

0D2pL(∇u + t∇(v − u), x)dt.

HOMOGENIZATION, LINEARIZATION AND REGULARITY 3

If v is a small perturbation of u, then this equation is very close to the linearizationof (1.1) around u, namely

(1.3) −∇ ⋅ (D2pL(∇u(x), x)∇w(x)) = 0.

It is therefore very natural to consider the large-scale behavior of solutions oflinearized equations of the form (1.3), where u is a solution of (1.1).

In this paper, we show that the linearized equation (1.3) around an arbitrarysolution u of (1.1) homogenizes, with an algebraic rate of convergence, and thatthe homogenized equation for this linearized equation is the linearization of thehomogenized equation (1.2) around the corresponding homogenized solution uhom(with the same boundary conditions as u), namely

(1.4) −∇ ⋅ (D2L(∇uhom)∇whom) = 0.

In other words, homogenization and linearization commute. The precise statementcan be found in Theorem 1.1 below. This result yields much finer informationregarding the differences of solutions of the original nonlinear equation and weexpect it to play a crucial role in the future development of a quantitative theoryof stochastic homogenization for nonlinear equations.

As a first consequence, we show that it provides sufficient information aboutthe differences of solutions to improve their regularity. Recall that, since thedifference of two solutions solves a linear equation, it satisfies a C0,β estimate for atiny exponent β(d,Λ) > 0 as a consequence of the De Giorgi-Nash estimate. Onthe other hand, differences of solutions of the homogenized equation (1.2) possessmuch better regularity: they satisfy at least a C1,β estimate in our setting, by theSchauder estimates. However, the quantitative estimate on the homogenization ofthe linearized equation implies that differences of solutions of (1.1) can be well-approximated, on large scales, by differences of solutions of (1.2). This allows usto obtain a large scale C0,1–type estimate for differences of solutions of (1.1) by“borrowing” the better regularity of the homogenized equation. On a technical level,this is achieved via a Campanato-type excess decay argument very similar to theone introduced in [5]. See Theorem 1.2 below for the statement.

We expect the large-scale C0,1 estimate for differences to play a fundamental rolein the development of quantitative homogenization estimates which are optimalin the scaling of the error. This estimate implies, for instance, a bound on thedifference of two correctors (with sharp stochastic integrability): see (1.9), below.Essentially, this is the key nonlinear ingredient that makes it possible to develop acomplete quantitative theory analogous to the linear setting, as we will show in aforthcoming article. In fact, such results have appeared in a very recent preprintof Fischer and Neukamm [12] (which was posted more than a year after the firstversion of the present paper), who relied crucially on a version of the large-scale C0,1

estimate for differences.

Our third result, stated in Theorem 1.3, is a large-scale C1,1 estimate for solutionsof the nonlinear heterogeneous equation. It characterizes the set L1 of solutions withat-most linear growth (it coincides with the set of first-order correctors) and assertsthat an arbitrary solution on a finite domain can be approximated, up to a quadraticerror in the ratio of the scales, by elements of L1. That is, a solution of the nonlinearsolution can be approximated by an element of L1 with the same precision that a


harmonic function can be approximated by an affine function. This result can becompared to that of Moser and Struwe [20] in the case of periodic case, who provedthe qualitative characterization of L1 but not the quantitative statement of C1,1

regularity. The proof of the latter, even in the periodic case, relies crucially on theanalysis of the linearized equation and the approximation of differences developedhere and can thus be considered the first “truly nonlinear” large-scale regularityresult in homogenization. We remark that obtaining a C1,δ–type statement, where

the factor ( rR)2 in (1.11) is replaced, for a small exponent δ > 0, by ( r

R)1+δ, is much

less difficult to obtain. The optimal quadratic scaling is a much more subtle issue.

The last main result we state concerns the regularity of L itself. It is easy tosee from the definition of L that it satisfies the same upper and lower bounds ofuniform convexity that is assumed for L in the variable p, and therefore L ∈ C1,1.It is natural to expect that L is as smooth as L is in the variable p. It turns outhowever that proving more smoothness for L is subtle and intractably tied to thelarge-scale regularity theory for differences of solutions. As a consequence of thelarge-scale C0,1–type estimate we are able to show that, for each γ ∈ (0,1], andunder an additional assumption on the spatial regularity of L (see (6.1)), we haveroughly that

L(⋅, x) ∈ C2,γ Ô⇒ L ∈ C2,γ.

See Theorem 1.4 for the precise statement.

The case of (non-random) Lagrangians L = L(p, x) which are periodic in x is aspecial case of our assumptions and the large-scale regularity results in this paper,even in this much simpler situation and in their qualitative form, are new.

While the equation (1.1) considered here is variational (in the sense that thecoefficients are the gradient DpL of a convex Lagrangian L), our arguments can beextended, with only minor, mostly notational changes, to the more general equation

−∇ ⋅ (a (∇u(x), x)) = 0

where p↦ a(p, x) is a uniformly monotone map (but not necessarily the gradient ofa uniformly convex function) which belongs to C2,γ . This is because, contrary to awidely-held belief, every such divergence-form, uniformly monotone operator has avariational structure (see [4, Section 2] and [3, Chapter 10]). Indeed, we note thatthe quantitative homogenization results of [5] were extended in [4] to the setting ofgeneral uniformly monotone maps without changing the variational structure of thearguments.

The “commutability of homogenization and linearization” has been previouslyconsidered in the works of Muller and Neukamm [21] and Gloria and Neukamm [15].The results in these papers are, however, not directly related to ours as the notionof linearization considered there is very different from ours. In particular, theylinearize around the zero function in the direction of a fixed function which issmoothly varying on the macroscopic scale (and which is not necessarily a solution),rather than linearize around a solution oscillating on the microscopic scale. Inparticular, their results do not give information on the differences of solutions.While making a revision of this article, we became aware of the recent work ofNeukamm and Schaffner [22] who study the commutativity of homogenization andlinearization in the periodic setting, under a perturbative assumption, for possibly


nonconvex energy functionals arising in linear elasticity. Their results are qualitativein nature and do not provide information regarding the differences of solutions ortheir regularity, but are definitely in the spirit of Theorem 1.1.

1.2. Statement of the main results. The parameters d ∈ N with d ≥ 2, γ ∈ (0, 1]and M0,Λ ∈ [1,∞) are fixed throughout the paper. For short we denote

data ∶= (d, γ,M0,Λ).We assume the Lagrangians L satisfy the following conditions:

(L1) L ∶ Rd × Rd → R is a Caratheodory function which is measurable in xand C2,γ in p. It is assumed to satisfy the bound

ess supx∈Rd

(∣DpL(0, x)∣ + ∥D2pL(⋅, x)∥C0,γ(Rd)) ≤M0.

(L2) L is uniformly convex in p: for every p ∈ Rd and Lebesgue-a.e. x ∈ Rd,

Id ≤D2pL(p, x) ≤ ΛId.

We define Ω to be the set of all such functions:

Ω ∶= L ∶ L satisfies (L1) and (L2) .Note that Ω depends on the fixed parameters (d, γ,Λ,M0). We endow Ω with thefollowing family of σ–algebras: for each Borel U ⊆ Rd, define

F(U) ∶= the σ–algebra generated by the family of random variables

L↦ ∫UL(p, x)φ(x)dx, p ∈ Rd, φ ∈ C∞

c (Rd).

The largest of these is denoted by F ∶= F(Rd). We also denote by Ω(γ,M0) the setof Lagrangians L which satisfy (L1) and (L2) and do not depend on the variable x.

We assume that the law of the “canonical Lagrangian” L is a probability measure Pon (Ω,F) satisfying the following two assumptions:

(P1) P has a unit range of dependence: for all Borel subsets U,V ⊆ Rd such thatdist(U,V ) ≥ 1,

F(U) and F(V ) are P–independent.

(P2) P is stationary with respect to Zd–translations: for every z ∈ Zd and E ∈ F ,

P [E] = P [TzE] ,where the translation group Tzz∈Zd acts on Ω by (TzL)(p, x) = L(p, x + z).

The expectation with respect to P is denoted by E.

Since we are often concerned with showing that the fluctuations of our randomvariables are small, the following notation is convenient: for every σ ∈ (0,∞), θ > 0and random variable X on Ω, we write

X ≤ Oσ (θ) ⇐⇒ E [exp((X+

θ)σ

)] ≤ 2.

The result of Dal Maso and Modica [9, 10], under more general assumptions thanthe ones here, implies that local minimizers of the energy functional for L converge,on large scales, P–a.s., to those of the energy functional for L, for some deterministicand constant L. This qualitative homogenization result was quantified in [5], under


the finite range of dependence assumption, a version of which we recall below inTheorem 2.1. Our assumptions are still stronger than the ones in [5] because werequire that L be C2,γ in the variable p, uniformly in x, which is necessary tostudy the linearized equations. Even without this assumption, it is fairly easyto show that the homogenized Lagrangian L must inherit the uniform convexitycondition (L2) and is therefore C1,1. It is less obvious that L is necessarily C2,even under the uniform C2,γ assumption. We show in Proposition 2.5 below that infact L ∈ C2,β for an exponent β(data) > 0 (which a priori may be smaller than γ).In fact, L ∈ Ω(β,C) for a constant C(data) <∞ (which may be larger than M0).

We next present the first main result of the paper, which is a quantitativestatement concerning the commutability of linearization and homogenization. Thestatement should be compared with that of Theorem 2.1, below. We remark thatits statement, like those of Theorems 1.2 and 1.3 below, is new even in the case ofdeterministic equations with periodic coefficients.

Theorem 1.1 (Quantitative homogenization of linearized equations).Let σ ∈ (0, d), δ ∈ (0, 12], M ∈ [1,∞) and U ⊆ B1 be a bounded Lipschitz domain.Then there exist an exponent α(U,data) > 0, a constant C(σ, δ,M, U,data) <∞ anda random variable X satisfying the bound

(1.5) X = O1 (C)

such that the following statement holds. For each ε ∈ (0, 1], pair uε, uhom ∈W 1,2+δ(U)satisfying

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

−∇ ⋅ (DpL (∇uε, xε)) = 0 in U,

−∇ ⋅ (DpL (∇uhom)) = 0 in U,

uε − uhom ∈H10(U),

∥∇uhom∥L2+δ(U) ≤M,

function f ∈ W 1,2+δ(U) and pair wε,whom ∈ H1(U) satisfying the correspondingDirichlet problems for the linearized equations

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

−∇ ⋅ (D2pL (∇uε, xε)∇w

ε) = 0 in U,

−∇ ⋅ (D2pL (∇uhom)∇whom) = 0 in U,

wε,whom ∈ f +H10(U),

we have the estimate

(1.6) ∥∇wε −∇whom∥H−1(U) + ∥D2pL (∇uε, ⋅ε)∇w

ε −D2pL (∇uhom)∇whom∥

H−1(U)

≤ C ∥∇f∥L2+δ(U) (εα(d−σ) +X εσ) .

Recall that H−1(U) is defined as the dual space to H10(U) and that a se-

quence of L2(U) functions converges weakly in L2(U) if and only if they convergestrongly in H−1(U) (see Section 2.1). Therefore the inequality (1.6) should beregarded as a quantification of the weak convergence of the gradient ∇wε andflux D2

pL(∇uε, x)∇wε to their homogenized limits. Of course, the left side of (1.6)also controls the strong L2 norm of the homogenization error, in view of the follow-ing functional inequality (see [3, Lemma 1.13]): there exists C(U,d) <∞ such that,


for every v ∈H10(U),

∥v∥L2(U) ≤ C ∥∇v∥H−1(U) .

This L2 estimate for the homogenization error can be upgraded to an estimatein L∞ using the De Giorgi-Nash Holder estimate and an interpolation argument(see the proof of [5, Corollary 4.2]).

The estimate (1.6) can be expressed in a more familiar way by using Chebyshev’sinequality in combination with (1.5) to obtain that, for each σ < d, there exist αand C <∞, as in the statement of the theorem, such that

(1.7) P [∥∇wε −∇whom∥H−1(U) > Cεα(d−σ) ∥∇f∥L2+δ(U)] ≤ C exp (−C−1ε−σ) ,

with a similar bound holding of course for the fluxes. While the small exponent α isnot explicit and thus this estimate is evidently not optimal in terms of the scaling ofthe homogenization error, it is optimal in terms of stochastic integrability. Indeed,it is not possible to prove an estimate like (1.7) for any exponent σ > d. One reasonfor writing the estimate as we have in (1.6), with the explicit random variable X , isthat it emphasizes its uniformity in u, uε, f , which is important in view applications.In particular, we are able to linearize around an arbitrary solution and solve thelinearized equation with an arbitrary Dirichlet condition. While we are interestedhere in quantitative statements, we remark that the proof of Theorem 1.1 can bemodified to give a qualitative homogenization result for the linearized equationunder more general, qualitative assumptions (e.g., P is only stationary and ergodic).

One of the difficulties encountered in proving Theorem 1.1 is due to the fact thatthe coefficients in the linearized equation are not stationary and do not have afinite range of dependence, since ∇uε has neither of these properties. It is thereforenecessary to first establish that the D2

pL (∇uε(x), xε) can be approximated by amatrix-valued random field which is locally stationary and has a finite (mesoscopic)range of dependence. We then show that the corresponding (local) homogenizedmatrix is close to D2L(∇u(x)) and adapt the classical two-scale expansion argumentto obtain the theorem. The proof of Theorem 1.1 is given in Section 3.

Our second main result is a large-scale C0,1-type estimate for differences ofsolutions. This can be compared to [5, Theorem 1.2] which proved a similar boundsfor solutions. Since the difference of two solutions is the solution of a linear equation,we therefore have a priori C0,α bounds for differences as a consequence of the DeGiorgi-Nash estimate. This Holder regularity with a small exponent is the bestdeterministic bound we can expect to hold on large scales, due to the oscillatorynature of our Lagrangians. However, we show that this estimates can be improvedto a C0,1-type bound on scales larger than a random scale X which is finite P–a.s.In fact, its stochastic moments with respect to P are very strongly controlled.

In the following statement and throughout the rest of the paper, for each U ⊆ Rd

with ∣U ∣ <∞, we denote

∥f∥Lq(U) ∶= (⨏U∣f(x)∣q dx)

1q

= ∣U ∣−1q ∥f∥Lq(U) .

Theorem 1.2 (Large-scale C0,1 estimate for differences of solutions).Fix σ ∈ (0, d) and M ∈ [1,∞). There exist C(σ,M,data) < ∞ and a random


variable X satisfyingX ≤ Oσ (C)

such that the following holds. For every R ≥ 2X and u, v ∈H1(BR) satisfying

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

−∇ ⋅ (DpL(∇u,x)) = 0 in BR,

−∇ ⋅ (DpL(∇v, x)) = 0 in BR,

∥∇u∥L2(BR) , ∥∇v∥L2(BR) ≤M,

and every r ∈ [X , 12R], we have the estimate

∥∇(u − v)∥L2(Br) ≤C

R∥u − v∥L2(BR) .

An immediate but very important corollary of Theorem 1.2 is a gradient estimateon the difference of two first-order correctors. Recall that, for each ξ ∈ Rd, thefirst-order correctors φξ is characterized uniquely (up to an additive constant) asthe solution of

⎧⎪⎪⎪⎨⎪⎪⎪⎩

−∇ ⋅ (DpL(ξ +∇φξ(x), x)) = 0 in Rd,

∇φξ is Zd–stationary and E [⨏[0,1]d

∇φξ] = 0.

For each ξ1, ξ2 ∈ Rd, the ergodic theorem gives that, P–a.s., we have

lim supR→∞

1

R2 ⨏BR∣(ξ1 ⋅ x + φξ1(x)) − (ξ2 ⋅ x + φξ2(x))∣

2dx = ⨏

B1

∣(ξ1 − ξ2) ⋅ x∣2

≤ ∣ξ1 − ξ2∣2 .Applying Theorem 1.2 to u(x) ∶= ξ1 ⋅ x + φξ1(x) and v(x) ∶= ξ2 ⋅ x + φξ2(x) yields

∥∇φξ1 −∇φξ2∥L2(BX ) ≤ ∣ξ1 − ξ2∣ + ∥u − v∥L2(BX ) ≤ C infR≥X

1

R∥u − v∥L2(BR) ≤ C ∣ξ1 − ξ2∣ .

Giving up the volume factor, we then get

(1.8) ∥∇φξ1 −∇φξ2∥L2([0,1]d) ≤ CXd2 ∥∇φξ1 −∇φξ2∥L2(BX ) .

This is a very strong estimate. Note that if the coefficients are assumed to beHolder continuous on the unit scale, then one can upgrade this L2-type bound to apointwise bound with the same right side. For each s ∈ (0,2), the minimal scalesatisfies X = O sd

2(C), and we deduce in particular that

(1.9) E [exp (∥∇φξ1 −∇φξ2∥sL2([0,1]d))] ≤ E [exp (X ds

2 )] ≤ C.

This kind of estimate on the difference of first-order correctors is exactly what isneeded to adapt the optimal homogenization estimates proved for linear equations tothe nonlinear setting. Note that the (almost) Gaussian-type stochastic integrabilityof the estimate (1.9) is (almost) optimal.

The proof of Theorem 1.2, which is presented in Section 4, follows a similar ideato the one of [5, Theorem 1.2]. We first obtain an algebraic error estimate fordifferences of solutions—showing that they can be well-approximated by differencesof solutions to the homogenized equation—by interpolating the homogenizationerror estimates for linearized equations (Theorem 1.1 above) with the ones forthe original nonlinear equation ([5, Theorem 1.2], restated below in Theorem 2.1).


Since the difference of solutions to the homogenized equation satisfies C1,α estimate,this allows us to transfer the higher regularity to the heterogeneous difference viathe excess decay argument introduced in [5]. The latter is a quantitative versionof an idea originating in the work of Avellaneda and Lin [6, 8] in the periodiccase. We remark that, by a similar argument, Theorem 1.1 also yields a large-scaleC0,1–type estimate for the linearized equation. Since this result is very close inspirit to Theorem 1.2, we postpone the statement to Section 4. As with previouslarge-scale regularity estimates proved in [5, 4, 17, 3], the stochastic integrability ofthe minimal scale X is optimal in the sense that X ≤ Oσ(C) is false, in general, forany exponent σ > d. See [3, Section 3.6] for details.

The next main result that we state is the large-scale C1,1 estimate, which canbe compared to the case k = 1 of [3, Theorem 3.8]. Parts (i) and (ii) consist ofa first-order Liouville theorem, characterizing the set of solutions with at-mostlinear growth. This is then made quantitative in part (iii) in the form of thelarge-scale C1,1-type estimate. This generalizes the estimate in the linear setting,which can be found in [3, 17].

To state the theorem, we must first introduce some additional notation. Given adomain U ⊆ Rd, we denote the set of solutions in U by

L(U) ∶= u ∈H1loc(U) ∶ −∇ ⋅DpL(∇u,x) = 0 in U .

We also define L1 to be the set of global solutions of the nonlinear equation whichexhibit at most linear growth at infinity:

L1 ∶= u ∈ L(Rd) ∶ lim supr→∞

r−1 ∥u∥L2(Br) <∞ .

Note that L1 is a random object, as it depends on L. For each p ∈ Rd, we let `pdenote the affine function `p(x) ∶= p ⋅ x. Since the difference of two elements ofL(Rd) which exhibits strictly sublinear growth at infinity must be constant, byTheorem 1.2, the following theorem gives a complete classification of L1.

Theorem 1.3 (Large-scale C1,1-type estimate). Fix σ ∈ (0, d) and M ∈ [1,∞).There exist δ(σ, d,Λ) ∈ (0, 12], C(M, σ,data) <∞ and a random variable Xσ whichsatisfies the estimate

(1.10) Xσ ≤ Oσ (C)such that the following statements are valid.

(i) For every u ∈ L1 satisfying lim supr→∞1r ∥u − (u)Br∥L2(Br) ≤ M, there exist

an affine function ` such that, for every R ≥ Xσ,

∥u − `∥L2(BR) ≤ CR1−δ.

(ii) For every p ∈ BM, there exists u ∈ L1 satisfying, for every R ≥ Xσ,

∥u − `p∥L2(BR) ≤ CR1−δ.

(iii) For every R ≥ Xs and u ∈ L(BR) satisfying 1R ∥u − (u)BR∥L2(BR) ≤M, there

exists φ ∈ L1 such that, for every r ∈ [Xσ,R],

(1.11) ∥u − φ∥L2(Br) ≤ C ( rR

)2

infψ∈L1

∥u − ψ∥L2(BR) .


In the case of deterministic, periodic coefficient fields, the Liouville theorem(parts (i) and (ii) of Theorem 1.3) was obtained by Avellaneda and Lin [7] for linearequations and subsequently generalized by Moser and Struwe [20] to the nonlinearsetting. Obtaining a C1,α–type version of statement (iii), for a tiny α(d,Λ) > 0, isnot difficult, and in fact we believe it can be proved even in the random case witha qualitative argument. Obtaining the full C1,1 regularity is more difficult (evenin the periodic setting), requires a quantitative argument and crucially relies onTheorem 1.2.

We turn to the last main result of the paper concerning the improved regularityof L. As we show in Section 2.3, under the assumption (L1) that L(⋅, x) ∈ C2,γ,one can use deterministic regularity estimates (either of the De Giorgi-Nash orMeyers estimates will do) to obtain a tiny bit better regularity for the homogenizedLagrangian, namely L ∈ C2,β for a tiny exponent 0 < β ≪ γ. (Note that thisobservation is necessary even to ensure that the statement of Theorem 1.1 iscoherent.) However, once we have proved Theorem 1.2, we can upgrade the Holderexponent of D2L all the way up to the exponent γ, confirming that L is as regularas L(⋅, x), at least up to C2,γ . This result requires an additional assumption on thespatial regularity of L, which is stated in Section 6.

Theorem 1.4. In addition to the standing assumptions, suppose that (6.1) holdsfor some exponent γ ∈ (0,1]. Then L ∈ C2,γ

loc (Rd) and, for every M ∈ [1,∞),

(1.12) [D2L]C0,γ(BM) ≤ C(M, γ,data) <∞.

The proof of Theorem 1.4 appears in Section 6.

We expect the higher regularity for L (i.e., Ck,β for k ≥ 3), under appropriateadditional regularity assumptions on L, to be a natural consequence of a large-scaleregularity theory for higher-order linearized equations. This will be elaborated in afuture paper.

1.3. Outline of the paper. In the next section we give some notation, recallsome previous results and show that L ∈ C2,β for a tiny β > 0. The proofs ofTheorems 1.1, 1.2, 1.3 and 1.4 are given in Sections 3, 4, 5 and 6, respectively. InAppendix B we recall some homogenization estimates from [5] which are needed inSection 3.

2. Preliminaries

In this section we introduce some notation and state some previous quantitativehomogenization results which are used throughout the paper. We also prove somepreliminary deterministic estimates regarding the approximation of differences ofsolutions by the solutions of linearized equations. As a consequence, we show that Lbelongs to C2,β for a small exponent β > 0.

2.1. Notation. If E ⊆ Rd, the Lebesgue measure of E is denoted by ∣E∣. If U ⊆ Rd

is a domain with ∣U ∣ <∞ we define the normalized domain

U0 ∶= ∣U ∣−1d U.


Various of the parameters in our statements depend on U , but in most cases thisdependence is only on U0 and therefore will be invariant under changes of scale.We denote the cube of side length 3n centered at the origin by

◻n ∶= (−1

23n,

1

23n)

d

.

If U ⊆ Rd is a domain, we define the norm

∥u∥H1(U) ∶= (∥u∥2L2(U) + ∥∇u∥2L2(U))12

and define H1(U) and H10(U), respectively, to be the completion of C∞(U) and

C∞c (U), respectively, with respect to the norm ∥⋅∥H1(U). We also define the space

H−1(U) to be the completion of C∞(U) with respect to the norm

∥u∥H−1(U) ∶= sup∣∫Uuv∣ ∶ v ∈H1

0(U), ∥v∥H1(U) ≤ 1 .

If ∣U ∣ <∞, we define

⨏Uf ∶= 1

∣U ∣ ∫Uf.

We also sometimes write (f)U ∶= ⨏U f . It is often convenient to work with thefollowing scale-invariant norms, defined for domains U with ∣U ∣ <∞:

∥u∥Lp(U) ∶= (⨏U∣u∣p)

1p

= ∣U ∣−1p ∥u∥Lp(U) ,

∥u∥H1(U) ∶= (∣U ∣− 2d ∥u∥2L2(U) + ∥∇u∥2L2(U))

12,

∥u∥H−1(U) ∶= sup∣⨏Uuv∣ ∶ v ∈H1

0(U), ∥v∥H1(U) ≤ 1 .

It is useful to note these “underlined” norms have the following scaling properties:if ur(x) ∶= ru(x/r) for r > 0, then

(2.1)

⎧⎪⎪⎪⎨⎪⎪⎪⎩

∥ur∥Lp(rU) = r ∥u∥Lp(U) , ∥ur∥H1(rU) = ∥u∥H1(U) , ∥ur∥H−1(rU) = ∥u∥H−1(U) ,

∥∇ur∥Lp(rU) = ∥∇u∥Lp(U) , ∥∇u∥H−1(U) =1

r∥∇ur∥H−1(rU) .

We also require a notion of H−1 norm defined by testing against any H1 function,not simply H1

0 functions. It is defined (in its normalized version) by:

(2.2) ∥u∥H−1(U) ∶= sup∣⨏Uuv∣ ∶ v ∈H1(U), ∥v∥H1(U) ≤ 1 .

Directly from its definition we find that the H−1(U) norm has the following usefulsubadditivity property: if U is the disjoint union of Vii∈1,...,N (up to a set ofmeasure zero), then

(2.3) ∥u∥H−1(U) ≤N

∑i=1

∣Vi∣∣U ∣

∥u∥H−1(Vi)

It is clear that

(2.4) ∥u∥H−1(U) ≤ ∥u∥H−1(U) ≤ ∣U ∣ 1d ∥u∥L2(U) .


We will also need that the ∥⋅∥H−1(U) obeys the following product rule in everybounded Lipschitz domain U , which can be checked directly from the definitionof ∥⋅∥H−1 and the Poincare inequality: there exists C(U0, d) <∞ such that, for every

f ∈W 1,∞(U) and g ∈H−1(U),

(2.5) ∥fg∥H−1(U) ≤ C (∥f∥L∞ + ∣U ∣1d ∥∇f∥L∞(U)) ∥g∥H−1(U) ,

and the inequality holds also if H−1(U) replaces H−1(U).As mentioned above, if X is a random variable and σ, θ ∈ (0,∞), then we use

X ≤ Oσ(θ)as shorthand notation for the statement that

(2.6) E [exp((X+

θ)σ

)] ≤ 2.

It roughly means that “X+ is at most of order θ with stretched exponential tailswith exponent σ.” Indeed, by Chebyshev’s inequality,

(2.7) X ≤ Oσ(θ) Ô⇒ ∀λ > 0, P [X > λθ] ≤ 2 exp (−λσ) .The converse of this statement is almost true: for every θ ≥ 0,

(2.8) ∀λ ≥ 0, P [X ≥ λθ] ≤ exp (−λσ) Ô⇒ X ≤ Oσ (21σ θ) .

This can be obtained by integration. We also use the notation

X = Oσ(θ) ⇐⇒ X ≤ Oσ(θ) and −X ≤ Oσ(θ).We also write X ≤ Y +Oσ(θ) to mean that X −Y ≤ Oσ(θ) as well as X = Y +Oσ(θ)to mean that X − Y = Oσ(θ). If σ ∈ [1,∞), then Jensen’s inequality gives us atriangle inequality for Oσ(⋅) in the following sense: for any measure space (E,S, µ),measurable function K ∶ E → (0,∞) and jointly measurable family X(z)z∈E ofnonnegative random variables, we have

(2.9) ∀z ∈ E, X(z) ≤ Oσ(K(z)) Ô⇒ ∫EX dµ ≤ Oσ (∫

EK dµ) .

If σ ∈ (0,1], then the statement is true after adding a prefactor constant C(σ) ∈[1,∞) to the right side. We refer to [3, Appendix A] for proofs of these facts.

2.2. Previous quantitative homogenization results. As far as we are aware,the papers [5, 4] contain the only previous quantitative stochastic homogenizationresults for nonlinear equations. In this subsection, we recall several of the mainresults of [5] which are needed in this paper, particular in the next section.

The first we present is essentially the same as [5, Theorem 1.1], although itsstatement is slightly differently than the latter and can be found in [3, Chapter 11].Note that the proof given in [3] follows the same high-level outline of the one in [5],but is much more efficient.

Theorem 2.1 (Quantitative homogenization [3, Theorem 11.10]).Fix σ ∈ (0, d), δ ∈ (0, 12], M ∈ [1,∞) and a Lipschitz domain U ⊆ B1. Thereexist α(U,data) > 0, C(σ,U, δ,data) <∞ and a random variable X (σ, δ,M, U,data)satisfying the bound

(2.10) X = O1 (C)


such that the following holds. Fix ε ∈ (0,1], a pair u,uε ∈W 1,2+δ(U) satisfying

(2.11)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

−∇ ⋅ (DpL (∇uε, xε)) = 0 in U,

−∇ ⋅ (DpL (∇u)) = 0 in U,

uε − u ∈H10(U),

∥∇u∥L2+δ(U) ≤M.

Then we have the estimate(2.12)

∥∇uε −∇u∥H−1(U) + ∥DpL (∇uε, ⋅ε) −DpL (∇u)∥H−1(U) ≤ CM (εα(d−σ) +X εσ) .

The proof of Theorem 2.1 is based on an analysis of the subadditive quantity,introduced previously in [9], defined by

ν(U, ξ) ∶= infu∈ξ⋅x+H1

0(U)⨏UL (∇u(x), x) dx.

The effective Lagrangian is defined through the limit

L(ξ) ∶= limn→∞

E [ν(◻n, ξ)] ,

and this limit exists since the sequence n ↦ E [ν(◻n, ξ)] is nonincreasing by thesubadditivity of ν(⋅, ξ). It was shown by an iterative argument in [5, Theorem 3.1]that for every σ ∈ (0, d), there exists α(d,Λ) ∈ (0, 12] and C(σ, d,Λ) <∞ such that,for every n ∈ N,

(2.13) ∣ν(◻n, ξ) −L(ξ)∣ ≤ C3−nα(d−σ) +O1 (C3−nσ) .

This estimate then implies Theorem 2.1 by a quantitative two-scale expansionargument, as demonstrated in [5].

It is sometimes useful to state Theorem 2.1 in a slightly different way, byindicating a random scale above which homogenization holds with a deterministicestimate, rather that giving an estimate with a random right-hand side as in (2.12).We present such a statement in the following corollary, which is an immediateconsequence of the previous theorem.

Corollary 2.2. Let σ ∈ (0, d), δ ∈ (0, 12] and M ∈ [1,∞). There exist α(δ,data) ∈(0, 12], C(σ, δ,M,data) <∞ and a random variable Xσ, satisfying the bound

(2.14) Xσ = Oσ (C)

such that the following statement holds. For every r ∈ [Xσ,∞) and f ∈W 1,2+δ(Br)satisfying the bound ∥∇f∥L2+δ(Br) ≤M and every pair u,u ∈H1(Br) satisfying

(2.15)

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

−∇ ⋅ (DpL (∇u,x)) = 0 in Br,

−∇ ⋅ (DpL (∇u)) = 0 in Br,

u, u ∈ f +H10(Br),

we have the estimate1

r∥u − u∥L2(Br) ≤ Cr

−α(d−σ).


Proof. The passage from Theorem 2.1 to the statement of the corollary is essentiallyidentical to the argument of [3, Proposition 3.2] or the first paragraph of the proofof [5, Theorem 1.2].

The final result from [5] we need is the following large-scale C0,1–type estimate(see [5, Theorem 1.2]). We denote by P1 the linear space of affine functions.

Theorem 2.3 (Large-scale C0,1-type estimate). Fix σ ∈ (0, d) and M ∈ [1,∞).There exist constants C(σ,M,data) < ∞, α(d,Λ) ∈ (0, 12] and a random variableX (σ,M,data) satisfying X ≤ Oσ (C) such that the following holds. For every R ≥ 2Xand u ∈H1(BR) satisfying

⎧⎪⎪⎪⎨⎪⎪⎪⎩

−∇ ⋅ (DpL(∇u,x)) = 0 in BR,

1

R∥u − (u)BR∥L2(BR) ≤M,

and every r ∈ [X , 12R], we have the estimates

∥∇u∥L2(Br) ≤C

R∥u − (u)BR∥L2(BR)

and

inf`∈P1

∥u − `∥L2(Br) ≤ C ( rR

)1+α

inf`∈P1

∥u − `∥L2(BR) +Cr1−α(d−σ).

2.3. Deterministic linearization estimates. In this subsection we prove somedeterministic estimates which measure the error in approximating the difference oftwo solutions to the nonlinear equation by the linearized equation. As a consequence,we obtain the C2 regularity of the homogenized Lagrangian L.

Lemma 2.4 (Approximation of differences by linearization). Fix δ > 0, a boundedLipschitz domain U and f ∈W 1,2+δ(U). There exist constants β(δ,U0,data) ∈ (0, 12]and C(δ,U0,data) <∞ such that the following statement is valid. If u, v ∈ H1(U)solve

−∇ ⋅ (DpL(∇u,x)) = −∇ ⋅ (DpL(∇v, x)) in U,

u − v = f on ∂U

and w ∈H1(U) is the solution of the linearized problem

−∇ ⋅ (D2

pL(∇u,x)∇w) = 0 in U,

w = f on ∂U,

then∥∇u −∇v −∇w∥L2(U) ≤ C ∥∇f∥1+β

L2+δ(U) .

Proof. We first observe that the difference w ∶= u − v satisfies the linear equation

−∇ ⋅ (a∇w) = 0 in U,

where the coefficients a are given by

a(x) ∶= ∫1

0D2pL (t∇u(x) + (1 − t)∇v(x), x) dt.

In particular, by the Meyers estimate–and shrinking δ, if necessary, so that it is atmost the Meyer exponent, which we note depend on (δ,U0,data)–we have that

(2.16) ∥∇w∥L2+δ(U) ≤ C ∥∇f∥L2+δ(U) .


We compare this to w by defining z ∶= w − w and observing that z ∈H10(U) satisfies

−∇ ⋅ (D2pL(∇u,x)∇z) = −∇ ⋅ ((a −D2

pL(∇u,x))∇w) in U.

Thus the energy estimate gives us

(2.17) ∥∇u −∇v −∇w∥L2(U) = ∥∇z∥L2(U) ≤ C ∥(a −D2pL(∇u, ⋅))∇w∥

L2(U) .

To estimate the term on the right side, observe that

∣a(x) −D2pL(∇u(x), x)∣ ≤ [D2

pL(⋅, x)]C0,γ(Rd) ∣∇u(x) −∇v(x)∣γ

≤ C ∣∇u(x) −∇v(x)∣γ .Therefore we have the L1 bound

∥a −D2pL(∇u, ⋅)∥L1(U) ≤ C ∥∇u −∇v∥γ

L2(U) .

We also have the trivial L∞ bound

∥a −D2pL(∇u, ⋅)∥L∞(U) ≤ ∥a∥L∞(U) + ∥D2

pL(∇u, ⋅)∥L∞(U) ≤ C.

By interpolation, we therefore get, for every q ∈ [1,∞),

∥a −D2pL(∇u, ⋅)∥Lq(U) ≤ C ∥∇u −∇v∥

γq

L2(U) = C ∥∇w∥γq

L2(U) .

Therefore, by the Holder inequality,

∥(a −D2pL(∇u, ⋅))∇w∥

L2(U) ≤ ∥a −D2pL(∇u, ⋅)∥L 4+2δ

δ (U)∥∇w∥L2+δ(U)

≤ C ∥∇w∥1+γδ/(4+2δ)L2+δ(U) .

Combining this with (2.16) and (2.17) completes the proof.

A consequence of the previous lemma is the C2 regularity of the homogenizedLagrangian L.

Proposition 2.5. Let U ⊆ Rd be a bounded Lipschitz domain. There exist anexponent β(U0,data) ∈ (0, 12] and a constant C(U0,data) <∞ such that the mappingξ ↦ ν(U, ξ) belongs to C2,β(Rd) and

(2.18) ∥D2ξν(U, ⋅)∥C0,β(Rd) ≤ C.

Moreover, there exist β(data) ∈ (0, 12] and C(data) <∞ such that L ∈ C2,β(Rd) and

(2.19) [D2L]C0,β(Rd) ≤ C.

Proof. Fix a bounded Lipschiz domain U ⊆ Rd and ξ ∈ Rd. Let v(⋅, U, ξ) denote thesolution of the Dirichlet problem (recall here that `ξ(x) ∶= ξ ⋅ x)

−∇ ⋅ (DpL(∇v(⋅, U, ξ), x) = 0 in U,

v(⋅, U, ξ) = `ξ on ∂U.

Defineaξ(x) ∶=D2

pL(∇v(x,U, ξ), x)and, for each e ∈ Rd, let we(⋅, U, ξ) solve the linearized problem

−∇ ⋅ (aξ(x)∇we(⋅, U, ξ)) = 0 in U,

we(⋅, U, ξ) = è on ∂U.


For C(U0, d,Λ) <∞, we have

(2.20) ∥∇we(⋅, U, ξ)∥L2(U) ≤ C.

According to Lemma 2.4, there exist β(U0,data) > 0 and C(U0,data) < ∞ suchthat, for every ξ, ξ′ ∈ Rd,

(2.21) ∥∇v(⋅, U, ξ′) −∇v(⋅, U, ξ) −d

∑i=1

(ξ′i − ξi)∇wei(⋅, U, ξ)∥L2(U)

≤ C ∣ξ − ξ′∣1+β .

In particular, ξ ↦ ∇v(⋅, U, ξ) is C1,β mapping from Rd into L2(U) and

∇wei(⋅, U, ξ) = ∂ξi∇v(⋅, U, ξ).

In fact, in view of the formula

Dξν(U, ξ) = ⨏UDpL (∇v(x,U, ξ), x) dx

and the regularity assumption on L, the estimate (2.21) implies (2.18) with β ∧ γin place of β. Indeed, differentiating the previous display yields

(2.22) D2ξν(U, ξ) = ⨏

UD2pL (∇v(x,U, ξ), x)Dξ∇v(x,U, ξ)dx.

It is clear from (2.20) and this expression that ∣D2ξν(U, ⋅)∣ is bounded on Rd and,

for each ξ, ξ′ ∈ Rd,

∣D2ξν(U, ξ) −D2

ξν(U, ξ′)∣

≤ ⨏U∣D2

pL (∇v(x,U, ξ), x) −D2pL (∇v(x,U, ξ′), x)∣ ∣Dξ∇v(x,U, ξ)∣ dx

+ ⨏U∣D2

pL (∇v(x,U, ξ′), x)∣ ∣Dξ∇v(x,U, ξ) −Dξ∇v(x,U, ξ′)∣ dx

≤ CM0⨏U∣∇v(x,U, ξ) −∇v(x,U, ξ′)∣γ ∣Dξ∇v(x,U, ξ)∣ dx

+C ⨏U∣Dξ∇v(x,U, ξ) −Dξ∇v(x,U, ξ′)∣ dx

≤ C (∣ξ − ξ′∣γ + ∣ξ − ξ′∣β) .

This proves the desired bound on [D2ν(U, ⋅)]C0,β(Rd). Since ν(U, ⋅) is bounded

in C1,1(Rd) by a constant C(Λ) < ∞ (see for instance [3, Lemma 11.2]), weobtain (2.18).

The last statement concerning the C2,β regularity of L follows immediatelyfrom (2.18) and the pointwise, deterministic limit L(ξ) = limn→∞E [ν(◻n, ξ)].

We next combine Proposition 2.5 with (2.13) to obtain the following estimate onthe convergence of D2ν(◻n, ⋅) to D2L.

Lemma 2.6. Let σ ∈ (0, d) and M ∈ [1,∞). There exist α(d,Λ, γ) ∈ (0, 12] andconstant C(σ,M,data) <∞ such that, for every n ∈ N,

(2.23) sup∣ξ∣≤M

∣D2L(ξ) −D2pν(◻n, ξ)∣ ≤ C3−nα(d−σ) +O1 (C3−nσ) .


Proof. The argument is based on the elementary fact (by the Arzela-Ascoli theorem,for instance) that if fkk∈N is a sequence of functions uniformly bounded in C2,β(B1)and converges pointwise to zero, then D2fk → 0 uniformly in B1. This can be seenfrom a more quantitative perspective as a consequence of the following interpolationinequality: for each β ∈ (0,1], there exists a constant C(β, d) <∞ such that

∥D2f∥L∞(B1)

≤ C ∥f∥β

2+βL1(B1) ∥f∥

22+βC2,β(B1)

.

Applying this to the difference of ν(◻n, ⋅) − L in the ball BM yields, in viewof (2.13), (2.18) and (2.19), an exponent β(data) ∈ (0, 12] and C(σ′,M,data) <∞such that

sup∣ξ∣≤M

∣D2L(ξ) −D2pν(◻n, ξ)∣

≤ C (⨏BM

∣L(ξ) − ν(◻n, ξ)∣ dξ)β

2+β(∥L∥

C2,β(Rd) + ∥ν(◻n, ⋅)∥C2,β(Rd))2

2+β

≤ C (C3−nα(d−σ′) +O1 (C3−nσ

′))β

2+β ,

where α(d,Λ) ∈ (0,1) is as in (2.13), σ′ ∈ (σ, d) will be chosen below and dependonly on (σ, d), and we can justify the derivation of the last line with the aid of (2.9).Finally, we use the elementary inequality

(a + b)ε ≤ aε + εaε−1b ∀a, b ∈ (0,∞), ε ∈ (0,1]

to obtain

sup∣ξ∣≤M

∣D2L(ξ) −D2pν(◻m, ξ)∣ ≤ C3−nαβ(d−σ

′)/(2+β) +O1 (C3nα(d−σ′)−nσ′)

Taking σ′ = d+σ2 and relabeling αβ

2(2+β) as α, we obtain (2.23) since α ∈ (0,1].

3. Quantitative homogenization of the linearized equation

This section is devoted to the proof of the first main result, Theorem 1.1. Asmentioned in the introduction, the main difficulty is that the linearized equationaround an arbitrary solution does not possess nice statistical properties since thecoefficients depend on the solution. The first step in the proof therefore is toapproximate the solution and thus the linear coefficients by gluing together localsolutions defined on a mesoscopic scale. Theorem 2.1 ensures that the error resultingfrom this approximation is sufficiently small, and the resulting equation is locallystationarity and has a finite range of dependence property. We then homogenize itusing the techniques from [5, 3].

3.1. Setup. It is convenient to rescale the statement of Theorem 1.1. We select σ ∈(0, d), a reference Lipschitz domain U0 ⊆ B1, take r ≥ 1, δ ∈ (0, 12] and set U ∶= rU0.We also fix u,uhom ∈W 1,2+δ(U) satisfying

(3.1)

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

−∇ ⋅ (DpL(∇u,x)) = 0 = −∇ ⋅ (DpL (∇uhom)) in U,

u − uhom ∈H10(U),

∥∇uhom∥L2+δ(U) ≤M.


We then select another function f ∈W 1,2+δ(U) and denote by w,whom ∈H1(U) thesolutions of

(3.2)

⎧⎪⎪⎨⎪⎪⎩

−∇ ⋅ (D2pL (∇u,x)∇w) = 0 = −∇ ⋅ (D2

pL (∇uhom)∇whom) in U,

w,whom ∈ f +H10(U).

The goal is to prove the following estimate, for an exponent α(δ,U0,data) > 0, aconstant C(σ, δ,M, U0,data) <∞ and random variable X satisfying the bound (1.5):

(3.3)1

r∥∇w −∇whom∥H−1(U) +

1

r∥D2

pL (∇u, ⋅)∇w −D2pL (∇uhom)∇whom∥

H−1(U)

≤ ∥∇f∥L2+δ(U) (Cr−α +X r−σ) .

This estimate is equivalent to Theorem 1.1 by a rescaling (cf. (2.1)).

We may suppose without loss of generality that

(3.4) r ≥ C(σ, δ,M, U0,data)for any particular constant C(σ, δ,M, U0,data) < ∞ of our choosing. Indeed, forr ≤ C, we obtain the estimate (3.3), after suitably enlargening the constant onthe right side, from the fact that the quantities on the left side are boundedby C ∥∇f∥L2(U) with C(U0, d,Λ) <∞.

Throughout the rest of the section, and unless otherwise stated to the contrary, Cdenotes a large constant belonging to the interval [1,∞) which depends only on theparameters (σ, δ,M, U0,data) which may vary from line to line (or even betweendifferent occurrences in the same line). Similarly, unless otherwise indicated, αand β denote small exponents belonging to (0, 12] which depend only on (δ,data)(actually they will depend only on (δ, d,Λ, γ)) which may vary in each occurrence.Finally, X denotes a random variable satisfying the bound X = O1(C) which is alsoallowed to vary from line to line.

3.2. Definition of mesoscopic scales. To begin the proof of (3.2), we take n ∈ Nsuch that r ∈ (3n−1,3n]. In addition, we will work with three mesoscopic scalesrepresented by k, l,m ∈ N with

(3.5) k < l <m < nAmong these, l and m will be a very large mesoscopic scales close to the macroscopicscale with m much closer than l (n−m≪ n− l≪ n), and k a very small mesoscopicscale close to the microscopic scale (k ≪ n). We select k to be the largest integerand l and m the smallest integers satisfying

(3.6) k ≤ %(d − σ)n, n − l ≤ %(d − σ)n, 1

2≤ n −m%2(d − σ)n

≤ 1,

where %(σ, δ, d,Λ, γ), ∈ (0, 1100

] is minimum of various exponents appearing below(each of which depends on the appropriate parameters). We will select % near theconclusion of the proof. We can ensure that the condition (3.5) is enforced bychoosing C large enough in (3.4). This choice of C will depend of course on theconstant % in (3.6).

For each x ∈ Rd and j ∈ N, we define [x]j to be the closest element of 3jZd to x(in the case this closest point is not unique, we make any choice which preserves


measurability, lexicographical ordering for instance). We also define open sets U i

with i ∈ 1,2,3,4 and V such that

U 4 ⊆ U

3 ⊆ U 2 ⊆ U

1 ⊆ U ⊆ Vby

U i ∶= x ∈ U ∶ [x]m +◻m+2i ⊆ U and V ∶= ⋃

x∈U([x]m +◻m+2) .

By taking the C in (3.4) sufficiently large, depending on U0, we can ensure that U 4

is nonempty. We have that, for each i ∈ 1,2,3,

3m ≤ dist (U i+1, ∂U

i ) , dist (U

1 , ∂U) , dist (∂V,U) ≤ C3m

and there exists C(U0, d) <∞ such that

(3.7) ∣V ∖U 4 ∣ ≤ C3−(n−m) ∣U ∣ .

3.3. Deterministic regularity estimates. We next record some deterministicregularity estimates which will be used many times in the forthcoming argument.

By the global Meyers estimates in Lipschitz domains (see [4, Appendix B] forinstance), under the assumption that the exponent δ is sufficiently small, dependingonly on (U0, d,Λ), there exists a constant C(U0, d,Λ) <∞ such that

(3.8) ∥∇u∥L2+δ(U) ≤ CM ≤ C

and

(3.9) ∥∇w∥L2+δ(U) + ∥∇whom∥L2+δ(U) ≤ C ∥∇f∥L2+δ(U) .

Without loss of generality (by shrinking δ, if necessary) we may assume that thesebounds hold. Next, by subtracting a constant from u and uhom as well as from f ,w and whom and applying the Sobolev extension theorem, we may suppose as wellthat these functions are globally defined, belong to W 1,2+δ(Rd) and satisfy

(3.10) ∣U ∣− 12+δ (∥∇u∥L2+δ(Rd) + ∥∇uhom∥L2+δ(Rd)) ≤ CM ≤ C

and

(3.11) ∥∇w∥L2+δ(Rd) + ∥∇whom∥L2+δ(Rd) ≤ C ∥∇f∥L2+δ(Rd) ≤ C ∥∇f∥L2+δ(U) .

We next recall some pointwise bounds for the solutions uhom and whom of thehomogenized equations. By the De Giorgi-Nash estimate, there exist an expo-nent β(d,Λ) > 0 and constant C(U0, d,Λ) <∞ such that

(3.12) ∥∇uhom∥L∞(U1)+ 3βm [∇uhom]C0,β(U

1)≤ C3d(n−m)/2M ≤ C3nd%

2(d−σ).

To see this, apply the Holder estimate for ∇uhom in each ball of radius 3m+1 centeredat a point x ∈ U

1 to obtain, for a constant C(U0, d,Λ) <∞,

∥∇uhom∥L∞(B3m(x)) + 3βm [∇uhom]C0,β(B3m(x)) ≤ C ∥∇uhom∥L2(B3m+1(x))

≤ C ( ∣U ∣∣B3m+1(x)∣

)12

∥∇uhom∥L2(U)

≤ C3d(n−m)/2M ≤ C3nd%2(d−σ).

A covering of U 1 by such balls yields (3.12).


Pointwise bounds for the solution whom of the linearized equation follow nextfrom the Schauder estimates. These depend on the Holder seminorm of thelinearized coefficients. Combining Proposition 2.5 and (3.12) yields β(d,Λ, γ) > 0and C(data) <∞ such that

(3.13) 3βm [D2L (∇uhom)]C0,β(U

1)≤ C3nd%

2(d−σ).

The Schauder estimates yield, for C(M, U0,data) <∞ and β(d,Λ, γ) ∈ (0, 12], the

existence of α(d,Λ, γ) ∈ (0, 12] such that

∥∇whom∥L∞(U2)+ 3βm [∇whom]C0,β(U

2)(3.14)

≤ C (1 + (3βm [D2L (∇uhom)]C0,β(U

1))d2β ) sup

x∈U2

∥∇whom∥L2(B3m(x))

≤ C3n(%2/α)(d−σ) ∥∇f∥L2(U) .

Note that Schauder estimates with explicit dependence on the Holder seminormof the coefficients can be obtained from the usual statement (which can be foundfor instance in [19, Theorem 3.13]) and a straightforward scaling argument. Theexponents of 3 on the right side of each of the estimates (3.12), (3.13) and (3.14)above is of the form C%2(d − σ) for a constant C(d,Λ, γ) <∞. Eventually we willchoose the parameter % very small so that these factors grow as a very small power3n (and can thus be absorbed by other factors which are negative powers of 3n).

3.4. Approximation by a locally stationary equation. The first step in theproof of Theorem 1.1 is to show that w may be approximated by the solution wof a linear equation with locally stationary coefficients. In order to construct thisequation and obtain an estimate on the difference between w and w, we need torecall certain quantitative homogenization estimates from [5].

To give ourselves a little room, we put σ′ ∶= 12(σ + d) and σ′′ ∶= 1

2(σ′ + d). By [5,

Theorem 1.1], there exist α0(d,Λ) ∈ (0, 12], C(σ, δ,M, U0,data) <∞ and a randomvariable X satisfying

X ≤ O1(C)such that

(3.15)1

r∥u − uhom∥2L2(U) ≤ Cr−α0(d−σ) +X r−σ′′ .

We will compare u to solutions of the Dirichlet problem with affine boundary datain mesoscopic cubes. Recall that we denote by `ξ the affine function `ξ(x) ∶= ξ ⋅ x.For each ξ ∈ Rd and Lipschitz domain U ⊆ Rd, we denote by v(⋅, U, ξ) the minimizerin the definition of ν(U, ξ) which is the unique solution of the Dirichlet problem

(3.16) −∇ ⋅ (DpL(∇v(⋅, U, ξ), x)) = 0 in U,

v(⋅, U, ξ) = `ξ on ∂U.

We will use the following estimates for the solutions of (3.16): for every K ∈ [1,∞)and q ∈ [1,∞), there exist α1(d,Λ) > 0 and C(K, q, d,Λ) < ∞ and a random


variable X satisfying X ≤ O1(C) such that, for every M,N ∈ N with M ≤ N ,

(3.17) supξ∈B

K3Mq

(1 + ∣ξ∣)−23−d(N−M) ∑z∈3MZd∩◻N

3−2M ∥v(⋅, z +◻M , ξ) − `ξ∥2L2(z+◻M )

≤ C3−Mα1(d−σ) +X3−σ′′M

and

(3.18)

supξ∈B

K3Mq

(1 + ∣ξ∣)−23−d(N−M) ∑z∈3MZd∩◻N

∥∇v(⋅,◻N , ξ) −∇v(⋅, z +◻M , ξ)∥2L2(z+◻M )

≤ C3−Mα1(d−σ) +X3−σ′′N .

The first inequality (3.17) is a consequence of [5, Corollary 3.5]. The secondinequality (3.18) was essentially proved in [5], but was not stated in exactly thisform, so we give a proof of it in Appendix B. We will apply these inequalities withN ≥ n and with K and q chosen, depending only on (d,Λ), so that, by (3.12), thesupremums over ξ ∈ BK3qN can be replaced by supremums over ∣ξ∣ ≤ ∥∇uhom∥L∞(U).

To introduce the approximating equation, we define, for each ξ ∈ Rd, the linearcoefficient field

(3.19) aξ(x) ∶=D2pL (∇v (x, z′ +◻k, ξ) , x) , z′ ∈ 3kZd, x ∈ z′ +◻k.

Observe from the assumption (L2) that aξ is uniformly elliptic,

Id ≤ aξ ≤ ΛId in Rd.

It is clear that aξ is 3kZd–stationary and has a range of dependence of at most

2 (1 +√d ⋅ 3k). In particular, the theory of quantitative homogenization for linear

equations with finite range dependence (see [3]) applies to the linear equation withcoefficients aξ. We denote the homogenized coefficients corresponding to aξ by aξ.

Finally, we define the linear coefficient field a(⋅) by gluing together the aξ’s inmesoscopic boxes with the parameter ξ given by a local averaged slope of uhom:

(3.20) a(x) ∶= a(∇uhom)z+◻l(x), z ∈ 3lZd, x ∈ z +◻l.

This coefficient field a(⋅) is locally stationary in the sense that in each mesoscopiccube z +◻l with z ∈ 3lZd it is the restriction of the 3kZd–stationary field aξ withparameter ξ = (∇uhom)z+◻l .

We next prove that a(x) is close to D2pL(∇u(x), x) and deduce therefore, by an

argument similar to the proof of Lemma 2.4, that w is close to the solution w ∈H1(U)of the Dirichlet problem

(3.21) −∇ ⋅ (a∇w) = 0 in U,

w ∈ f +H10(U).

Lemma 3.1. There exist α(δ,data) ∈ (0, 12], C(σ, δ,M, U0,data) <∞ and a randomvariable X satisfying X ≤ O1(C) such that if % ≤ α, then

(3.22) ∥∇w −∇w∥L2(U) ≤ C ∥∇f∥L2+δ(U) (3−nα%2(d−σ)2 +X3−nσ) .


Proof. We have that a(x) =D2pL(F (x), x), where we define the vector field F by

F (x) ∶= ∇v (x, z′ +◻k, (∇uhom)z+◻l) , z ∈ 3lZd, z′ ∈ 3kZd, x ∈ z′ +◻k.We also define a vector field H similar to F but using a larger mesoscopic scale:

H(x) ∶= ∇v (⋅, z +◻l+1, (∇uhom)z+◻l) , z ∈ 3lZd, x ∈ z +◻l.Throughout the argument α, C and X will be as in the statement of the lemma,but may change in each occurrence.

Step 1. We show that, for fixed σ ∈ (0, d), there exist constants α(δ,data) > 0and C(σ,data) <∞, and a random variable X = O1(C) such that if % ≤ α, then

(3.23) ∥∇u −H∥2L2(U1)≤ C3−nα(d−σ) +X3−nσ

′.

By the Caccioppoli and triangle inequalities, for every z ∈ 3lZd ∩U 1 ,

∥∇u −∇v (⋅, z +◻l+1, (∇uhom)z+◻l)∥2

L2(z+◻l)

≤ C3−2l ∥u − v (⋅, z +◻l+1, (∇uhom)z+◻l)∥2

L2(z+◻l+1)

≤ C3−2l (∥u − uhom∥2L2(z+◻l+1) + ∥uhom − v (⋅, z +◻l+1, (∇uhom)z+◻l)∥2

L2(z+◻l+1)) .

Set σ′′ ∶= d+σ′2 . By (3.15) and (3.6), we find, for % sufficiently small,

∣◻l∣∣U

1 ∣∑

z∈3lZd∩U1

3−2l ∥u − uhom∥2L2(z+◻l+1) ≤ C3−2l ∥u − uhom∥2L2(U)

≤ C32(n−l) (3−α0(d−σ)n +X3−nσ′′) .

≤ C (3−α04

(d−σ)n +X3−nσ′) .

Next, for each z ∈ 3lZd ∩U 1 , define the affine function

`z(x) ∶= uhom(z) + (∇uhom)z+◻l ⋅ (x − z)

and compute, with the aid of (3.12),

∥`z − uhom∥L2(z+◻l+1) ≤ C3l(1+β) [∇uhom]C0,β(U1)≤ C3l−β(m−l)+d(n−m)/2M.

Thus by the triangle inequality,

∥uhom − v (⋅, z +◻l+1, (∇uhom)z+◻l)∥2

L2(z+◻l+1)

≤ 2 ∥v (⋅, z +◻l+1, (∇uhom)z+◻l) − `z∥2

L2(z+◻l+1)+ 2 ∥`z − uhom∥2L2(z+◻l+1)

≤ 2 ∥v (⋅, z +◻l+1, (∇uhom)z+◻l) − `z∥2

L2(z+◻l+1)+C32l−2β(m−l)+d(n−m).

Summing over z ∈ 3lZd ∩U 1 , and applying (3.17) and (3.6), yields

1

∣3lZd ∩U 1 ∣

∑z∈3lZd∩U

1

3−2l ∥uhom − v (⋅, z +◻l+1, (∇uhom)z+◻l)∥2

L2(z+◻l+1)

≤ C (3−α1(d−σ′′)l + 3−2β(m−l)+d(n−m)) +X3−lσ′′

≤ C (3−nα1(1−%)(d−σ′′) + 3−n%(2β−2d%)) +X3−nσ′′(1−2%).

Combining the above inequalities and taking % sufficiently small yields (3.23).


Step 2. We show that there exists α(δ, d,Λ) > 0 such that

(3.24)1

∣U ∣∥∇u −H∥2L2(U∖U

1)≤ C3−nα%

2(d−σ).

By (3.8) and (3.7),

1

∣U ∣∥∇u∥2L2(U∖U

1)≤ C (∣U ∖U

1 ∣∣U ∣

)δ

2+δ∥∇u∥2L2+δ(U) ≤ CM23−δ(n−m)/(2+δ).

Similarly, we have

1

∣U ∣∥∇uhom∥2L2(V ∖U

1)≤ CM23−δ(n−m)/(2+δ).

Likewise,

1

∣U ∣∥H∥2L2(U∖U

1)≤ ∣◻l∣

∣U ∣ ∑z∈3lZd∩U∖U

1

∥∇v (⋅, z +◻l+1, (∇uhom)z+◻l)∥2

L2(z+◻l)

≤ C ∣◻l∣∣U ∣ ∑

z∈3lZd∩U∖U1

∥∇v (⋅, z +◻l+1, (∇uhom)z+◻l)∥2

L2(z+◻l+1)

≤ C ∣◻l∣∣U ∣ ∑

z∈3lZd∩U∖U1

(1 + ∣(∇uhom)z+◻l ∣2)

≤ C 1

∣U ∣(∣V ∖U

1 ∣ + ∥∇uhom∥2L2(V ∖U1)) .

Putting these together gives (3.24).

Step 3. We argue that, for fixed σ ∈ (0, d), there exist constants α(δ, d,Λ, γ) > 0and C <∞, and a random variable X = O1(C) such that if % ≤ α, then

(3.25) ∥∇u − F ∥2L2(U1)≤ C3−nα%

2(d−σ) +X3−nσ′.

By the previous two steps, it suffices to show that

∥F −H∥2L2(U1 ) ≤ C3−kα(d−σ) +X3−nσ′ ≤ C3−nα%(d−σ)

2 +X3−nσ′.

This is an immediate consequence of (3.18) and (3.12).

Step 4. We show next that, there exist α(δ, d,Λ, γ) > 0, C <∞ and X such thatif % ≤ α, then, for every q ∈ [2,∞),

(3.26) ∥D2pL(∇u, ⋅) − a∥

Lq(U) ≤ C3−nα%2(d−σ)2/q +X3−σn.

For q = 2, we have, by (3.25),

∥D2pL(∇u, ⋅) −D2

pL(F, ⋅)∥L2(U1)≤ supx∈Rd

[D2pL(⋅, x)]C0,γ(Rd) ∥∇u − F ∥γ

L2(U1)

≤M0 (C3−nα%2(d−σ)2 +X3−nσ

′)γ,

and, since D2pL(p, ⋅) is bounded, we have, in view of (3.7),

∣U ∣− 12 ∥D2

pL(∇u, ⋅) −D2pL(F, ⋅)∥L2(U∖U

1)≤ C ∣U ∣−

12 ∣U ∖U

1 ∣12 ≤ C3−nα%

2(d−σ).

Putting these together, we get that

∥D2pL(∇u, ⋅) −D2

pL(F, ⋅)∥L2(U) ≤M0 (C3−nα%2(d−σ)2 +X3−nσ

′)γ.


Using again that D2pL(p, ⋅) is bounded, by interpolation we deduce

∥D2pL(∇u, ⋅) −D2

pL(F, ⋅)∥Lq(U) ≤ C (3−nα%2(d−σ)2 +X3−nσ

′)γ/q.

Using the inequality

(3.27) ∀ε ∈ (0,1], a, b ∈ (0,∞), (a + b)ε ≤ aε + εaε−1b,

we get

∥D2pL(∇u, ⋅) −D2

pL(F, ⋅)∥Lq(U) ≤ C3−nα%2(d−σ)2γ/q +X3−nσ.

Taking % sufficiently small, and recalling that a =D2pL(F, ⋅) gives the claim (3.26).

Step 5. The conclusion. By subtracting the equations for w and w, we find thatthe difference w − w satisfies

−∇ ⋅ (a∇ (w − w)) = ∇ ⋅ ((D2pL(∇u,x) − a)∇w) in U.

Testing this equation with w − w, which we notice belongs to H10(U), we obtain

∥∇ (w − w)∥L2(U) ≤ C ∥(D2pL(∇u,x) − a)∇w∥

L2(U) .

Finally, by the Holder inequality, the Meyers estimate and (3.26) with q = 4+2δδ ,

∥(D2pL(∇u,x) − a)∇w∥

L2(U) ≤ ∥D2pL(∇u,x) − a∥

L4+2δδ (U)

∥∇w∥L2+δ(U)(3.28)

≤ C ∥D2pL(∇u,x) − a∥

L4+2δδ (U)

∥∇f∥L2+δ(U)

≤ C ∥∇f∥L2+δ(U) (3−nα%2(d−σ)2 +X3−nσ) .

This completes the proof of the lemma.

3.5. Homogenization estimates for the locally stationary equation. Inview of Lemma 3.1, we are motivated to obtain homogenization estimates forthe locally stationary problem (3.21). This is accomplished in the next subsection,and here we prepare for the analysis by recording some estimates for the stationaryfields aξ(⋅).

We next check that the homogenized coefficients aξ corresponding to aξ are

close to what we expect, namely D2pL (ξ). This is a crucial point in the proof of

Theorem 1.1, for it is here that we really see that homogenization and linearizationmust commute.

For each η ∈ Rd (recall that `η(x) ∶= η ⋅ x), we have that

(3.29)1

2η ⋅ aξη = lim

j→∞E [ inf

w∈`η+H10(◻j)

⨏◻j1

2∇w ⋅ aξ∇w] .

Let us denote by wξ(⋅,◻j, η) the minimizer of the optimization problem inside theexpectation on the right side of the previous display.

Lemma 3.2. Let K,Q ∈ [1,∞). There exist an exponent α(data) ∈ (0, 12] andconstant C(Q,K,data) <∞ such that

(3.30) sup∣ξ∣≤K3qk

∣aξ −D2pL (ξ)∣

1 + ∣ξ∣≤ C3−kα.


Proof. Fix ∣ξ∣ ≤ K3Qk.

Step 1. For each j ∈ N with j ≥ k, we approximate aξ in ◻j by the coefficients

aξ,j(x) ∶=D2pL (∇v (x,◻j, ξ) , x) .

The claim is that, for q ∈ [2,∞),

(3.31) ∥aξ − aξ,j∥Lq(◻j) ≤ C(1 + ∣ξ∣) (3−kα1 +X3−n)2γq .

To prove (3.31), we first observe that, for every z ∈ 3kZd ∩◻j and x ∈ z +◻k,∣aξ(x) − aξ,j(x)∣ ≤ [D2L]

C0,γ(Rd) ∣∇v(x, z +◻k, ξ) −∇v(x,◻j, ξ)∣γ,

and therefore, by (3.18),

∥aξ − aξ,j∥L2(◻j) ≤ C3−d(j−k) ∑z∈3k∩◻j

∥∇v(⋅, z +◻k, ξ) −∇v(⋅,◻j, ξ)∥γL2(z+◻k)

≤ (1 + ∣ξ∣) (C3−kα1 +X3−k)γ2 .

Using also that

∥aξ − aξ,j∥L∞(◻j) ≤ ∥aξ∥L∞(◻j) + ∥aξ,j∥L∞(◻j) ≤ C,

we find that, for every q ∈ [2,∞),

∥aξ − aξ,j∥Lq(◻j) ≤ ∥aξ − aξ,j∥1− 2

q

L∞(◻j) ∥aξ − aξ,j∥2q

L2(◻j) ≤ (1 + ∣ξ∣) (C3−kα1 +X3−k)2γq .

This completes the proof of (3.31).

Step 2. For each j ∈ N with j ≥ k, let us denote by wξ,j(⋅, η) the minimizer of thevariational problem

infw∈`η+H1

0(◻j)⨏◻j

1

2∇w ⋅ aξ,j∇w.

In this step we show that, for some α(d,Λ) > 0,

(3.32) E [∥∇wξ(⋅,◻j, η) −∇wξ,j(⋅, η)∥2L2(◻j)] ≤ C ∣η∣2 3−kα.

Fix j ∈ N and η ∈ B1. The difference Wξ ∶= wξ(⋅,◻j, η)− wξ,j(⋅, η) belongs to H10(◻j)

and satisfies the equation

−∇ ⋅ aξ∇Wξ = ∇ ⋅ (aξ − aξ,j)∇wξ,j in ◻j.Testing this equation with Wξ yields

∥∇Wξ∥2L2(◻j) ≤ C ⨏◻j∇Wξ ⋅ aξ∇Wξ = C ⨏◻j

∇Wξ ⋅ (aξ,j − aξ)∇wξ,j.

From this, the Holder inequality, the Meyers estimate and (3.31), we obtain

∥∇Wξ∥L2(◻j) ≤ C ∥(aξ,j − aξ)∇wξ,j∥L2(◻j)≤ C ∥(aξ,j − aξ)∥

L4+2δδ (◻j)

∥∇wξ,j∥L2+δ(◻j)

≤ C (3−kα1 +X3−k)2δγ2+δ ∣η∣ .

Squaring and taking expectations yields (3.32) via (3.27).

Step 3. The conclusion. As was shown in the proof of Proposition 2.5, we have

wξ,j(⋅,◻j, η) = η ⋅Dξv (⋅,◻j, ξ)


and, moreover, by (2.22) and a testing of the equation for wξ,j by wξ,j −`η ∈H10(◻j),

⨏◻j1

2∇wξ,j(⋅,◻j, η) ⋅ aξ,j∇wξ,j(⋅,◻j, η) = 1

2η ⋅D2

pν (◻j, ξ) η.

By Lemma 2.6, we have that

E [∣D2L(ξ) −D2pν(◻j, ξ)∣] ≤ C3−jα(1 + ∣ξ∣2)Ð→ 0 as j →∞.

Combining (3.29) and (3.32), and using as well as the global Meyers estimate whichgives, for some δ(data) > 0, the deterministic bound

∥∇wξ(⋅,◻j, η)∥L2+δ(◻j) + ∥∇wξ,j(⋅,◻j, η)∥L2+δ(◻j)≤ C ∥∇wξ(⋅,◻j, η)∥L2(◻j) +C ∥∇wξ,j(⋅,◻j, η)∥L2(◻j) ≤ C ∣η∣,

we deduce that

lim supj→∞

∣12η ⋅ aξη −E [⨏◻j

1

2∇wξ,j(⋅,◻j, η) ⋅ aξ,j∇wξ,j(⋅,◻j, η)]∣ ≤ C ∣η∣2 3−kα.

Combining the previous displays now yields the lemma.

We next state some estimates for the first-order correctors for the (globallystationary) coefficients aξ. For each z ∈ 3l−1Zd and e ∈ B1, we take φe,z ∈H1(z +◻l)to be the solution of

(3.33) −∇ ⋅ a∇uhom(z) (e +∇φe,z) = 0 in z +◻l,φe,z = 0 on ∂(z +◻l).

As previously mentioned, the coefficient field aξ is 3kZd–stationary and has range ofdependence at most C3k. Therefore, after rescaling the equation by dilation factorof C3k and applying [3, Theorem 2.17], we obtain the following estimate: thereexist α(d,Λ) > 0 and C(σ, d,Λ) <∞ and, for every z ∈ 3l−1Zd and e ∈ B1, a randomvariable Xz ≤ O1(C) satisfying

(3.34) 3−l ∥∇φe,z∥H−1(z+◻l) + 3−l ∥a∇uhom(z) (e +∇φe,z) − a∇uhom(z)e∥H−1(z+◻l)

≤ C3−β(d−σ′′)(l−k) +Xz3−σ

′′(l−k) ≤ C3−nα(d−σ) +Xz3−nσ′.

Observe that here we also used the following consequence of (3.6):

l − k ≥ n − n(2% + %2)(d − σ)These also imply that

(3.35) 3−l ∥φe,z∥L2(z+◻l) ≤ C3−nα(d−σ) +Xz3−nσ′.

By testing (3.33) with φe,z itself, we also have the deterministic estimate

(3.36) ∥∇φe,z∥L2(z+◻l) ≤ C.

The Meyers estimate applied to x↦ e ⋅ x + φe,z(x) then yields

(3.37) ∥∇φe,z∥L2+δ(z+◻l) ≤ C ∥∇φe,z∥L2(z+◻l) ≤ C.

We also need to record some estimates for the dependence in ξ of aξ. The claimis that there exists β(d,Λ, γ) > 0 such that, for every q ∈ [2,∞), ξ, ξ′ ∈ Rd andz ∈ 3kZd,

(3.38) ∥aξ − aξ′∥Lq(z+◻k) ≤ C ∣ξ − ξ′∣β/q .


This estimate then implies that, for some β(d,Λ, γ) > 0,

(3.39) ∣aξ − aξ′ ∣ ≤ C ∣ξ − ξ′∣β .To prove (3.38), we first consider the case q = 2 and, recalling the definition (3.19)of aξ and using assumption (L1), we find

∥aξ − aξ′∥L2(z+◻k)

≤ ∥D2pL (∇v(⋅, z +◻k, ξ), ⋅) −D2

pL (∇v(⋅, z +◻k, ξ′), ⋅)∥L2(z+◻k)

≤ C supx∈Rd

[D2pL(⋅, x)]C0,β(Rd) ∥∇v(⋅, z +◻k, ξ) −∇v(⋅, z +◻k, ξ′)∥

β

L2(z+◻k)

≤ C ∣ξ − ξ′∣β .We then obtain (3.38) for general q ∈ [2,∞) from the case q = 2 and the fact that,by assumption (L2),

∥aξ − aξ′∥L∞(z+◻k) ≤ ∥aξ∥L∞(z+◻k) + ∥aξ′∥L∞(z+◻k) ≤ CΛ ≤ C.

3.6. Homogenization with locally stationary coefficients. By Lemma 3.1and the triangle inequality, to prove (3.3) it suffices to prove the estimate with win place of w. The advantage is that w satisfies an equation with locally stationarycoefficients which, by Lemma 3.2, have local homogenized coefficients that areclose to D2L(∇uhom). To complete the proof, we therefore need to establish aquantitative homogenization result for linear equations with coefficients whichare locally stationary. This is accomplished by a fairly straightforward (althoughtechnical) adaptation of the argument for (globally) stationary coefficients whichcan be found, for instance, in the proof of [3, Theorem 1.17].

The result is presented in the following proposition. Recall that the coefficientfield a(x) is defined above in (3.20) and w is the solution of (3.21).

Proposition 3.3. There exist α(δ,data) ∈ (0, 12] and C(σ, δ,M, U0,data) <∞ suchthat

(3.40)1

r∥∇w −∇whom∥H−1(U) +

1

r∥a∇w −D2L(∇uhom)∇whom∥

H−1(U)

≤ C ∥∇f∥L2+δ(U) (3−nα%2(d−σ) +X3−nσ

′) .

The idea of the proof of Proposition 3.3 is to compare the solution w of (3.21) toa two-scale expansion around whom. We construct, for each e ∈ Rd, a function φewhich will play the role of the first-order corrector (with slope e) in the two-scaleexpansion. Since the linear equation for w has a locally stationary structure, webuild φe by gluing (finite-volume) correctors φe,z for the stationary problems onthe mesoscopic scale, defined in (3.33). We fix a smooth function χ such that its3l−1Zd translates form a partition of unity. Precisely, we require

(3.41) 0 ≤ χ ≤ 1, χ ≡ 0 in Rd ∖◻l, ∥∇χ∥L∞(Rd) ≤ C3−l, ∑z∈3l−1Zd

χ(⋅ − z) ≡ 1.

We can construct such a function χ by mollifying the indicator function of ◻l−1, forinstance. We then define

φe(x) = ∑z∈3l−1Zd

χ(x − z)φe,z(x).


Finally, we define the competitor to w by

(3.42) T ∶= (1 − ζ)whom + ζ (whom ∗ ψ) + ζd

∑j=1

(∂xjwhom ∗ ψ)φej ,

where ζ ∈ C∞(Rd) is a cutoff function chosen to satisfy, for a constant C(U0, d) <∞,

(3.43) 0 ≤ ζ ≤ 1, ζ ≡ 1 on U 4 , ζ ≡ 0 in Rd ∖U

3 , ∥∇2ζ∥L∞(Rd) ≤ C3−2m,

and ψ is a mollifier on length scale 3l, that is, ψ ∈ C∞(Rd) satisfies

(3.44) ψ ≥ 0, ∫Rdψ(x)dx = 1, ψ ≡ 0 on Rd ∖B3l , ∥∇2ψ∥

L∞(Rd) ≤ C3−(d+2)l.

The purpose of the cutoff function ζ in the definition of T is the enforcement ofthe boundary condition: notice that T ∈ f +H1

0(U). It also cuts off the second termin the definition of T in the boundary layer where we do not have good estimateson φe and ∇whom. The reason for convolving ∇whom with the mollifier ψ is thatwe wish to differentiate T but only possess C0,β estimates on ∇whom (cf. (3.14)).Mollifying it on a mesoscopic scale does not change ∇whom very much, since itis a macroscopic function, but gives us some control on its gradient. Indeed,by (3.14), (3.44) and (3.6), we have, for small enough %,

3−l ∥whom ∗ ψ −whom∥L∞(U3)≤ C3n(%

2/α)(d−σ) ∥∇f∥L2(U) ,(3.45)

∥∇whom ∗ ψ −∇whom∥L∞(U3)≤ C3βl [∇whom]C0,β(U

2)(3.46)

≤ C3−β(m−l)3n(%2/α)(d−σ) ∥∇f∥L2(U)

≤ 3−nα%(d−σ) ∥∇f∥L2(U) ,

and, for j ∈ N,

∥∇j (∇whom ∗ ψ)∥L∞(U

3)≤ ∥∇whom∥L∞(U

2)∥∇jψ∥

L1(Rd)(3.47)

≤ Cj3−lj3(%2/α)(d−σ) ∥∇f∥L2(U) .

The proof of Proposition 3.3 now breaks into two basic steps, which are presentedin Lemmas 3.4 and 3.5: first we show that ∇T is close to ∇w in a strong norm; second,we show that the gradient and flux of T are close to the gradient and homogenizedflux of ∇whom in weak norms. Each of these steps relies on the homogenizationestimates for the functions φe,z from the previous section: see (3.34).

Lemma 3.4. There exist α(δ,data) ∈ (0, 12] and C(σ, δ,M, U0,data) <∞ such that

(3.48) ∥∇T −∇w∥L2(U) + ∥∇ ⋅ (a∇T )∥H−1(U)

≤ C ∥∇f∥L2+δ(U) (3−nα%2(d−σ) +X3−nσ

′) .

Lemma 3.5. There exist α(δ,data) ∈ (0, 12] and C(σ, δ,M, U0,data) <∞ such that

(3.49)1

r(∥∇T −∇whom∥H−1(U) + ∥a∇T −D2L(∇uhom)∇whom∥

H−1(U))

≤ C ∥∇f∥L2+δ(U) (r−α%2(d−σ) +X r−σ′) .


Before giving the proofs of these lemmas, we compute the gradient of T anddiscard some terms which (with good choices of the mesoscale parameters) aresmall.

Lemma 3.6. There exist α(δ,data) ∈ (0, 12] and C(σ, δ,M, U0,data) < ∞ and arandom variable X satisfying X = O1(C) such that if % ≤ α,

(3.50) ∥∇T − ζ ∑z∈3l−1Zd

d

∑j=1χ(⋅ − z)(∂xjwhom ∗ ψ) (ej +∇φej ,z)∥

L2(U)

≤ C ∥∇f∥L2+δ(U) (3−nα%2(d−σ) +X3−nσ

′) .

Proof. To shorten the notation, we write ∑z and ∑j in place of ∑z∈3l−1Zd and ∑dj=1,

respectively, and ∑z,j = ∑z∑j. We also denote χz ∶= χ(⋅ − z).A direct computation yields

∇T − ζ∑z,j

χz(∂xjwhom ∗ ψ) (ej +∇φej ,z)(3.51)

= (1 − ζ)∇whom + (whom ∗ ψ −whom)∇ζ+∑z,j

(∇ (ζχz) (∂xjwhom ∗ ψ) + ζχz∇ (∂xjwhom ∗ ψ))φej ,z.

To prove the lemma, we will show that each of the three terms on the right sideof (3.51) is bounded by the right side of (3.50). The first term is small becauseit is confined to a boundary layer: we use the Holder inequality, the Meyersestimate (3.9), (3.43), and (3.6) to get

∥(1 − ζ)∇whom∥L2(U) ≤ ∥1 − ζ∥L

4+2δδ (U)

∥∇whom∥L2+δ(U)

≤ C (∣U ∖U 4 ∣

∣U ∣)

δ4+2δ

∥∇f∥L2+δ(U)

≤ C3−(n−m)δ/(4+2δ) ∥∇f∥L2+δ(U) .

≤ C3−nα%2(d−σ) ∥∇f∥L2+δ(U) .

To estimate the second term, we use that ∇ζ is supported in U 3 ∖U

4 , and obtainby (3.43) and (3.45) that

∥(whom ∗ ψ −whom)∇ζ∥L2(U) ≤ C ∥∇ζ∥L∞(Rd) ∥whom ∗ ψ −whom∥L∞(U3)

≤ C3−(m−l)3n(%2/α)(d−σ) ∥∇f∥L2(U) ≤ C3−nα%(d−σ) ∥∇f∥L2(U) .

Combining (3.41), (3.43), and (3.47), we also get

supz,j

∥∇ (ζχz) (∂xjwhom ∗ ψ) + ζχz∇ (∂xjwhom ∗ ψ)∥L∞(Rd)

≤ C3−l3n(%2/α)(d−σ) ∥∇f∥L2(U) .


Using this and (3.35), we estimate the final term as

∥∑z,j

(∇ (ζχz) (∂xjwhom ∗ ψ) + ζχz∇ (∂xjwhom ∗ ψ))φej ,z∥L2(U)

≤ C3n(%2/α)(d−σ) ∥∇f∥L2(U)

∣◻l+1∣∣U ∣ ∑z,j

3−l ∥φej ,z∥L2(z+◻l+1)

≤ C3n(%2/α)(d−σ) ∥∇f∥L2(U) (3−nα(d−σ) +Xz3−nσ

′) .

Taking α smaller, if necessary, yields the lemma.

We now turn to the proof of Lemma 3.4.

Proof of Lemma 3.4. The rough idea is to plug T into the equation for w and checkthat the error we make is small: the claim is that

(3.52) ∥∇ ⋅ (a∇T )∥H−1(U) ≤ C ∥∇f∥L2+δ(U) (3−nα%2(d−σ) +X3−nσ

′) .

The proof of (3.52) occupies the first three steps below. In the fourth step weuse (3.52) to obtain the lemma. As above we write ∑z and ∑j in place of ∑z∈3l−1Zd

and ∑dj=1, respectively, to shorten the notation. We also write χz ∶= χ(⋅ − z).

Step 1. We organize the computation. According to (3.50),

∥∇ ⋅ a(∇T − ζ∑z,j

χz (∂xjwhom ∗ ψ) (ej +∇φej ,z))∥H−1(U)

(3.53)

≤ C ∥∇T − ζ∑z,j

χz(∂xjwhom ∗ ψ) (ej +∇φej ,z)∥L2(U)

≤ C ∥∇f∥L2+δ(U) (3−nα%2(d−σ) +X3−nσ

′) .

It therefore suffices to estimate the H−1 norm of

∇ ⋅ (∑z,j

ζχz(∂xjwhom ∗ ψ)a (ej +∇φej ,z))(3.54)

=∑z,j

(∂xjwhom ∗ ψ)ζχz∇ ⋅ az (ej +∇φej ,z)

+∑z,j

∇ ((∂xjwhom ∗ ψ)ζχz) ⋅ az (ej +∇φej ,z)

+∇ ⋅ (ζ∑z,j

χz(∂xjwhom ∗ ψ) (a − az) (ej +∇φej ,z)) ,

where, for each z ∈ 3l−1Zd, we abuse notation by defining

az ∶= a(∇uhom)z+◻l.

The first term on the right side of (3.54) is zero by the equation of φej ,z, and thusthe following steps are devoted to estimates of the other two terms.


Step 2. We show that

(3.55) ∥∑z,j

∇ ((∂xjwhom ∗ ψ)ζχz) ⋅ az (ej +∇φej ,z)∥H−1(U)

≤ C ∥∇f∥L2+δ(U) (3−nα%2(d−σ) +X3−nσ

′) .

We decompose each summand on the left as

∇ ((∂xjwhom ∗ ψ)ζχz) ⋅ a (ej +∇φej ,z)(3.56)

= ∇ ((∂xjwhom ∗ ψ)ζχz) ⋅ azej+∇ ((∂xjwhom ∗ ψ)ζχz) ⋅ (a (ej +∇φej ,z) − azej) .

It is in the estimate of the first term that we use the equation for whom. To see this,write

∑z,j

∇ ((∂xjwhom ∗ ψ)ζχz) =∑z

∇ ⋅ (azχz(∇whom ∗ ψ)ζ) ,

and decompose the term on the right as

∑z

∇ ⋅ (azχz(∇whom ∗ ψ)ζ) = ∇ ⋅ (ζD2pL (∇uhom)∇whom)(3.57)

+∇ ⋅ (ζ∑z

χzaz (∇whom ∗ ψ −∇whom))

−∇ ⋅ (ζ∑z

χz (D2pL (∇uhom) − az)∇whom) .

By the equation for whom, we have

∇ ⋅ (ζD2pL (∇uhom)∇whom) = ∇ ⋅ ((ζ − 1)D2

pL (∇uhom)∇whom) ,

and thus, by the Holder inequality, (3.9) and (3.43),

∥∇ ⋅ (ζD2pL (∇uhom)∇whom)∥

H−1(U) ≤ ∥(1 − ζ)D2pL (∇uhom)∇whom∥

L2(U)

≤ ∥1 − ζ∥L

4+2δδ (U)


≤ C3−nα%2(d−σ) ∥∇f∥L2+δ(U) .

We estimate the H−1(U) norm of the other two terms on the right side of (3.57)by showing that what is under the divergence sign is small in L2(U). We have,by (3.46),

∥∇ ⋅ (ζ∑z

χzaz (∇whom ∗ ψ −∇whom))∥H−1(U)

≤ ∥ζ∑z

χzaz (∇whom ∗ ψ −∇whom)∥L2(U)

≤ C ∥∇whom ∗ ψ −∇whom∥L2(U3)

≤ C3−nα%(d−σ) ∥∇f∥L2(U)


and, similarly, by the Holder inequality, (3.9), (3.12) and (3.39),

∥∇ ⋅ (ζ∑z

χz (D2pL (∇uhom) − az)∇whom)∥

2

H−1(U)

≤ ∥ζ∑z

χz (D2pL (∇uhom) − az)∇whom∥

L2(U)

≤ supz∈3l−1Zd∩U

3

∥D2pL (∇uhom) − az∥L∞(z+◻l+1) ∥∇whom∥L2(U) .

To estimate the first factor on the right, we use Lemma 3.2 and (3.12), noticingalso that for every z ∈ 3l−1Zd ∩U

3 we have z +◻l+1 ⊆ U 1 , to obtain

∥D2pL (∇uhom) − az∥L∞(z+◻l+1)

≤ ∥D2pL ((∇uhom)z+◻l) − a(∇uhom)z+◻l

∥L∞(z+◻l+1)

+ ∥D2pL (∇uhom) −D2

pL ((∇uhom)z+◻l)∥L∞(z+◻l+1)

≤ C(1 + ∥∇uhom∥L∞(U1))3−αk +C (3βl [∇uhom]C0,β(U

1))β

≤ C(3−αk + 3−β(m−l))3nd%2(d−σ).

Therefore, collecting the above estimates, we obtain by (3.57) and (3.6) that

(3.58) ∥∑z

∇ ⋅ (azχz(∇whom ∗ ψ)ζ)∥H−1(U)

≤ C ∥∇f∥L2(U) 3−nα%2(d−σ).

To estimate the second term on the right side of (3.56), we use (3.34) and (2.5) toobtain, for each z ∈ 3l−1Zd,

∥∇ ((∂xjwhom ∗ ψ)ζχz) ⋅ (az (ej +∇φej ,z) − azej)∥H−1(z+◻l)

≤ C (∥∇ ((∂xjwhom ∗ ψ)ζχz)∥L∞(z+◻l) + 3l ∥∇2 ((∂xjwhom ∗ ψ)ζχz)∥L∞(z+◻l))

× 3l (3−nα(d−σ) +Xz3−nσ′) .

Applying (3.47), (3.41), and (3.43) we see that

∥∇ ((∂xjwhom ∗ ψ)ζχz)∥L∞(z+◻l) + 3l ∥∇2 ((∂xjwhom ∗ ψ)ζχz)∥L∞(z+◻l)

≤ C3−l+n(%β2/α)(d−σ) ∥∇f∥L2(U) .

We now obtain (3.55) by combining the two previous displays with (3.56), (3.58)and (2.3), the latter in view of the fact that

∣◻l∣∣U ∣ ∑

z∈3l−1Zd∩U3

Xz = O1(C),

and then taking the parameter % > 0 sufficiently small.


Step 3. We estimate the H−1(U) norm of the divergence of the third term onthe right side of (3.54). The claim is that

(3.59) ∥∇ ⋅ (ζ∑z,j

χz(∂xjwhom ∗ ψ) (a − az) (ej +∇φej ,z))∥H−1(U)

≤ C ∥∇f∥L2(U) 3−nα%2(d−σ).

For this it is enough to bound the L2 norm of the term inside of the parenthesis.By the Holder inequality, (3.14) and (3.37), we have

∥ζ∑z,j

χz(∂xjwhom ∗ ψ) (a − az) (ej +∇φej ,z)∥L2(U)

≤ ∥ζ∑z,j

χz(∂xjwhom ∗ ψ) (a − az) (ej +∇φej ,z)∥L2(U)

≤ C ∥∇whom∥L∞(U2)∑

z

∥χz(a − az)∥L

4+2δδ (U)

(1 +∑j

∥∇φej ,z∥L2+δ(z+◻l))

≤ C ∥∇f∥L2(U) 3n(%β2/α)(d−σ)∑z

∥χz(a − az)∥L

4+2δδ (U)

.

Furthermore, to estimate the last sum, using (3.12), (3.38), and letting [z] denotethe nearest point of 3lZd to z ∈ 3l−1Zd, we have, for q ∈ [2,∞),

∥a −∑z

χzaz∥Lq(U

1)≤ ∑z∈3l−1Zd∩U

1

∥a − az∥Lq(z+◻l)

≤ ∑z∈3l−1Zd∩U

1

∥a(∇uhom)[z]+◻l− a(∇uhom)z+◻l

∥Lq(z+◻l)

≤ C ∑z∈3l−1Zd∩U

1

∣(∇uhom)[z]+◻l − (∇uhom)z+◻l ∣β/q

≤ C (3βl [∇uhom]C0,β(U1))β/q

≤ C (3−β(m−l)3d(n−m)/2)β/q .

Putting the last two displays together, using (3.6), and taking α smaller, if necessary,we obtain (3.59).

Step 4. The conclusion. We combine (3.53) and (3.54) with the estimates (3.55)and (3.59), recalling that the first term on the right in (3.54) is zero, to deduce (3.52).Since T − w ∈H1

0(U), the equation for w gives

⨏U∇ (T − w) ⋅ a∇w dx = 0.

The estimate (3.52) yields

∣⨏U∇ (T − w) ⋅ a∇T dx∣ ≤ C ∥T − w∥H1(U) ∥∇ ⋅ a∇T ∥H−1(U)

≤ C ∥∇T −∇w∥L2(U) ∥∇ ⋅ a∇T ∥H−1(U) .


Combining the previous two displays yields

∥∇T −∇w∥2L2(U) ≤ ⨏U∇ (T − w) ⋅ a∇ (T − w) dx

≤ C ∥∇T −∇w∥L2(U) ∥∇ ⋅ a∇T ∥H−1(U)

and thus∥∇T −∇w∥L2(U) ≤ C ∥∇ ⋅ a∇T ∥H−1(U) .

Combining this with (3.52) completes the proof of (3.48) and the lemma.

Proof of Lemma 3.5. We estimate the terms on the left side of (3.49) by using (3.50)to reduce the desired inequalities to bounds on the functions φe,z which are thenconsequences of quantitative homogenization estimates for linear equations. Forconvenience we denote

E ′ ∶= ∥∇f∥L2+δ(U) (3−nα%2(d−σ) +X3−nσ

′) .


(3.60)1

r∥∇T −∇whom∥H−1(U) ≤ CE ′.

Using (3.34), (3.47) and (3.50), we see that

1

r∥∇T − ζ (∇whom ∗ ψ)∥H−1(U)

≤ 1

r∥ζ

d

∑j=1∂xj (whom ∗ ψ)∇φej∥

H−1(U)

+CE ′

≤d

∑j=1

∥∇ (ζ (∇whom ∗ ψ))∥L∞(U) ∥∇φej∥H−1(U) +CE′ ≤ CE ′.

Next, we have, by (3.46),

1

r∥ζ (∇whom ∗ ψ −∇whom)∥H−1(U) ≤ C ∥ζ (∇whom ∗ ψ −∇whom)∥L2(U)

≤ C ∥∇whom ∗ ψ −∇whom∥L∞(U3)≤ CE ′.

Finally,

1

r∥(1 − ζ)∇whom∥H−1(U) ≤ ∥(1 − ζ)∇whom∥L2(U)

≤ ∥1 − ζ∥L

4+2δδ (U)


≤ C3−nα%2(d−σ) ∥∇f∥L2+δ(U) ≤ CE ′.

The triangle inequality and the previous three displays yield (3.60).

Step 2. To prepare for the estimate for the fluxes, we show that

(3.61) ∥D2L(∇uhom) − ∑z∈3l−1Zd

χzaz∥L∞(U

3)

≤ C3−nα%(d−σ),

where we abuse notation by defining, for each z ∈ 3l−1Zd,az ∶= a(∇uhom)z+◻l

.


By (2.19), Lemma 3.2 and (3.12), we have that, for every z ∈ 3l−1Zd ∩ U 2 and

x ∈ z +◻l,∣D2L(∇uhom(x)) − az ∣ ≤ C3−nα%(d−σ) + ∣D2L(∇uhom(x)) −D2L(∇uhom(z))∣

≤ C3−nα%(d−σ) + [D2L]C0,β(Rd) ∣∇uhom(x) −∇uhom(z)∣β

≤ C3−nα%(d−σ) +C [D2L]C0,β(Rd) (3βl [∇uhom])β

≤ C3−nα%(d−σ) +C (3−β(m−l)3d(n−m)/2)β .

After redefining α, by (3.6) this becomes

supz∈3l−1Zd∩U

2

supx∈z+◻l

∣D2L(∇uhom(x)) − az ∣ ≤ C3−nα%(d−σ) +C3−nα%(d−σ).

This yields (3.61) after summing over z by the triangle inequality, recalling alsothat % ≤ %.


(3.62)1

r∥a∇T −D2L (∇uhom)∇whom∥

H−1(U) ≤ CE′.

Let a ∶= ∑z∈3l−1Zd χzaz. By (3.34), (3.47) and (3.50),

1

r∥a∇T − ζa (∇whom ∗ ψ)∥H−1(U)

≤ 1

r∥ζ

d

∑j=1∂xj (whom ∗ ψ) (a (ej +∇φej) − aej)∥

H−1(U)

+CE ′

≤d

∑j=1

∥∇ (ζ (∇whom ∗ ψ))∥L∞(U) ∥a (ej +∇φej) − aej∥H−1(U) +CE′ ≤ CE ′.

Next we use (3.14) and (3.61) to see that

1

r∥ζ (D2L(∇uhom) − a) (∇whom ∗ ψ)∥

H−1(U)

≤ C ∥ζ (D2L(∇uhom) − a) (∇whom ∗ ψ)∥L2(U)

≤ C ∥D2L(∇uhom) − a∥L∞(U

3)∥∇whom∥L2(U

2)

≤ C ∥∇f∥L2(U) 3−nα%(d−σ)(1−%/α2) ≤ CE ′,

for sufficiently small %. Finally, as in Step 1, we have

1

r∥ζD2L(∇uhom) (∇whom ∗ ψ −∇whom)∥

H−1(U) ≤1

r∥ζ (∇whom ∗ ψ −∇whom)∥H−1(U)

≤ CE ′

and1

r∥(1 − ζ)D2L(∇uhom)∇whom∥

H−1(U) ≤1

r∥(1 − ζ)∇whom∥H−1(U) ≤ CE ′.

Combining the three previous displays yields (3.62).

Proof of Proposition 3.3. The statement is an immediate consequence of the triangleinequality and Lemmas 3.4 and 3.5.


3.7. The conclusion. We conclude the proof of Theorem 1.1 by summarizing howthe previous lemmas fit together to give the theorem.

Proof of Theorem 1.1. As discussed above, it suffices to prove (3.3). For the esti-mate of the gradient term, we have

1

r∥∇w −∇whom∥H−1(U) ≤ ∥∇w −∇w∥L2(U) +

1

r∥∇w −∇whom∥H−1(U)

≤ C ∥∇f∥L2+δ(U) (3−nα%2(d−σ)2 +X3−nσ) .

Here we used the triangle inequality, (2.4), Lemma 3.1 and Proposition 3.3. Forthe fluxes, the triangle inequality and (2.4) yield

1

r∥D2

pL (∇u, ⋅)∇w −D2pL (∇uhom)∇whom∥

H−1(U)

≤ ∥(D2pL(∇u,x) − a)∇w∥

L2(U) + ∥a∥L∞(U) ∥∇w −∇w∥L2(U)

+ 1

r∥a∇w −D2L(∇uhom)∇whom∥

H−1(U) .

The three terms on the right are controlled by (3.28), Lemma 3.1 and Proposition 3.3,respectively. We have shown that the left side of (3.3) is bounded by

C ∥∇f∥L2+δ(U) (3−nα%2(d−σ)2 +X3−nσ)

for % specified in the statements of (3.28), Lemma 3.1 and Proposition 3.3. Recallingthat r ≥ 3n−1, we obtain (3.3) and hence the theorem.

3.8. A corollary. In view of its application in the next section, we finish thissection by restating the result of Theorem 1.1 in “minimal scale form” in the casethat the domain U is a ball.

Corollary 3.7. Let σ ∈ (0, d), δ ∈ (0, 12] and M ∈ [1,∞). There exist α(δ,data) ∈(0, 12], C(σ, δ,M,data) <∞ and a random variable Xσ, satisfying the bound

(3.63) Xσ = Oσ (C)such that the following statement holds. For every r ∈ [Xσ,∞) and pair u,u ∈W 1,2+δ(Br) satisfying

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

−∇ ⋅ (DpL (∇u,x)) = 0 in Br,

−∇ ⋅ (DpL (∇u)) = 0 in Br,

u − u ∈H10(Br),

∥∇u∥L2+δ(Br) + ∥∇u∥L2+δ(Br) ≤M,

function f ∈W 1,2+δ(Br) and pair w,w ∈H1(Br) satisfying

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

−∇ ⋅ (D2pL (∇u,x)∇w) = 0 in Br,

−∇ ⋅ (D2pL (∇u)∇w) = 0 in Br,

wε,w ∈ f +H10(Br),


(3.64)1

r∥w −w∥L2(Br) ≤ C ∥∇f∥L2+δ(Br) r

−α(d−σ).


4. The large-scale C0,1–type estimate for differences

In this section we prove our second main result, Theorem 1.2, on the large-scaleC0,1 estimate for differences. The argument follows the one introduced in [5] forproving the large-scale C0,1 estimate for solutions. Since differences of solutions ofthe homogenized equation satisfy a C1,β estimate (this is by the Schauder estimates,since L ∈ C2,β), we should expect to be able to transfer this higher regularity todifferences of solutions of the heterogeneous equation by a excess decay iteration(as in [5, Lemma 5.1]). What is needed to apply this idea is an approximationresult, essentially a quantitative homogenization estimate for differences, whichstates that differences of solutions of the heterogeneous equation are close to thoseof the homogenized equation. This is accomplished by interpolating between thehomogenization error estimates for the linearized equation (Theorem 1.1) and thenonlinear equation (Theorem 2.1).

Proposition 4.1 (Error estimate for differences). Fix M ∈ [1,∞) and σ ∈ (0, d).There exist α(data) ∈ (0, 12], C(σ,M,data) <∞ and a random variable X satisfying

X ≤ Oσ(C)

such that the following statement holds. For every R ≥ X and pair u, v ∈ H1(BR)of solutions of the equations

−∇ ⋅DpL(∇u,x) = 0 and −∇ ⋅DpL(∇v, x) = 0 in BR

which satisfy

(4.1)1

R∥u − (u)BR∥L2(BR) ≤M and

1

R∥v − (v)BR∥L2(BR) ≤M,

the solutions u, v ∈H1(BR/2) of the Dirichlet problems for the homogenized equation

(4.2) −∇ ⋅DpL(∇u) = 0 = ∇ ⋅DpL(∇v) in BR/2,

u = u, v = v on ∂BR/2,

satisfy the estimate

(4.3) ∥u − v − (u − v)∥L2(BR/2) ≤ CR−α(d−σ) ∥u − v − (u − v)BR∥L2(BR) .

Proof. Let σ ∈ (0, d) and choose Xσ to be the maximum of the random variablesin the statements of Corollaries 2.2 and 3.7. Fix R ≥ Xσ and u, v ∈ H1(BR)satisfying (4.2). We split the argument in two cases: (i) the oscillation of u − v ismuch smaller compared those of u and v, in which case we apply the homogenizationresult for the linearized equation, or (ii) it is not, and we apply the homogenizationresult for the original nonlinear equation and conclude by the triangle inequality.

We fix a small parameter ε0 ∈ (0,1] which will be chosen in Step 3 of the proof.

Step 1. We consider the case that

(4.4) ∥u − v − (u − v)BR∥L2(BR) ≥ ε0 (∥u − (u)BR∥L2(BR) + ∥v − (v)BR∥L2(BR))

and prove that there exist C(σ,data) <∞ and α1(d,Λ) ∈ (0,1) such that

(4.5) ∥u − v − (u − v)∥L2(BR/2) ≤ Cε−10 R

−α1(d−σ) ∥u − v − (u − v)BR∥L2(BR) .


We take u, v ∈H1(BR/2) to be the solutions of

−∇ ⋅ (DpL (∇u)) = 0 in BR/2,

u = u on ∂BR/2,and

−∇ ⋅ (DpL (∇v)) = 0 in BR/2,

v = v on ∂BR/2.

As R ≥ Xσ, Corollary 2.2 and the Meyers estimate implies that there exist α1(d,Λ) ∈(0,1) and C(σ,data) <∞ such that

∥u − u∥L2(BR/2) + ∥v − v∥L2(BR/2)

≤ CR−α1(d−σ) (∥u − (u)BR∥L2(BR) + ∥v − (v)BR∥L2(BR)) .

Therefore, the triangle inequality and the assumption (4.4) together yield (4.5).

Step 2. We consider the alternative case to the one in Step 1, namely that

(4.6) ∥u − v − (u − v)BR∥L2(BR) ≤ ε0 (∥u − (u)BR∥L2(BR) + ∥v − (v)BR∥L2(BR)) .

We show that there exist α(data), β(data) ∈ (0, 12] and C(σ,M,data) <∞ such that

∥u − v − (u − v)∥L2(BR/2)(4.7)

≤ C ((ε0M)β +MR−α(d−σ)) ∥u − v − (u − v)BR∥L2(BR) .

Using the bound (4.1), the assumption (4.6) implies

∥u − v − (u − v)BR∥L2(BR) ≤ ε0RM.

The difference u − v is a solution of the equation

−∇ ⋅ (a(x)∇(u − v)) = 0 in BR.

where

a(x) ∶= ∫1

0D2pL (t∇u(x) + (1 − t)∇v(x), x) dt.

Note that Id ≤ a ≤ ΛId. By the Meyers estimate and the Caccioppoli inequality,there exists δ(d,Λ) ∈ (0, 12] such that

(4.8) ∥∇u −∇v∥L2+δ(BR/2) ≤ CR−1 ∥u − v − (u − v)BR∥L2(BR) ≤ Cε0M.

Analogously, since u − v solves −∇ ⋅ (a(x)∇(u − v)) = 0 in BR/2 with

a(x) ∶= ∫1

0D2L (t∇u(x) + (1 − t)∇v(x), x) dt,

we get by the global Meyers estimate that

∥∇u −∇v∥L2+δ(BR/2)(4.9)

≤ C ∥∇u −∇v∥L2+δ(BR/2) ≤ CR−1 ∥u − v − (u − v)BR∥L2(BR) .

Let w,w ∈H1(BR/2) be the solutions of the Dirichlet problems

−∇ ⋅ (D2

pL(∇u,x)∇w) = 0 in BR/2,

w = u − v on ∂BR/2.

and

−∇ ⋅ (D2

pL(∇u)∇w) = 0 in BR/2,

w = u − v on ∂BR/2.


In view of (4.8), the interior and Global Meyers estimates as well as the factthat u−v−(u−v) ∈H1

0(BR/2), we may apply Corollary 3.7 to obtain α(data) ∈ (0, 1)and C(σ,M,data) <∞ such that

∥w −w∥L2(BR/2) ≤ CR1−α(d−σ) ∥∇u −∇v∥L2+δ(BR/2)(4.10)

≤ CR−α(d−σ) ∥u − v − (u − v)BR∥L2(BR) .

On the other hand, applying Lemma 2.4 we find constants β(δ,data) > 0 andC(δ,data) <∞ such that

∥∇u −∇v −∇w∥L2(BR/2) ≤ C ∥∇u −∇v∥1+βL2+δ(BR/2)

≤ C (ε0M)βR−1 ∥u − v − (u − v)BR∥L2(BR) .

By Poincare’s inequality we then obtain

(4.11) ∥u − v −w∥L2(BR/2) ≤ C (ε0M)β ∥u − v − (u − v)BR∥L2(BR) .

A similar argument, applying the proof of Lemma 2.4 to L instead of L, togetherwith Proposition 2.5 and (4.9), yields

(4.12) ∥u − v −w∥L2(BR/2) ≤ C (ε0M)β ∥u − v − (u − v)BR∥L2(BR) .

Combining (4.10), (4.11) and (4.12), we obtain (4.7).

Step 3. The conclusion. Combining Steps 1 and 2 and defining ε0 ∶= R−α1(d−σ)/2

completes the proof.

We are now ready to complete the proof of Theorem 1.2. What remains is adeterministic argument, along the lines of [5, Lemma 5.1], which uses a Campanato-type iteration and Proposition 4.1 to transfer the higher regularity enjoyed bydifferences of solutions of the homogenized equation to differences of solutions of theheterogeneous equation. We prove a slightly more general version of Theorem 1.2by including a mesoscopic C1,α estimate.

Proposition 4.2. Fix σ ∈ (0, d) and M ∈ [1,∞). There exist constants α(data) ∈(0, 12], C(σ,M,data) <∞ and a random variable X satisfying

X ≤ Oσ (C)

such that the following holds. For every R ≥ 2X and u, v ∈H1(BR) satisfying

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

−∇ ⋅ (DpL(∇u,x)) = 0 in BR,

−∇ ⋅ (DpL(∇v, x)) = 0 in BR,

∥∇u∥L2(BR) , ∥∇v∥L2(BR) ≤M,

and every r ∈ [X , 12R], we have the estimate

∥∇(u−v)∥L2(Br) + ( rR+ 1

rd−σ)−α 1

rinf`∈P1

∥u−v−`∥L2(Br) ≤C

R∥u−v−(u−v)BR∥L2(BR) .

Proof. Fix σ ∈ (0, d), we let X be the maximum of the random variables in thestatements of Corollary 2.2, Theorem 2.3 and Proposition 4.1. Without loss ofgenerality, we may assume that (u − v)BR = 0 and that X ≥H for a large constant


H(σ,M,data) to be fixed in the course of the proof. Indeed, if X ≤H ≤ R, we have,for s ∈ [X ,H], that

∥∇(u − v)∥L2(Bs) ≤ (Hs)d2

∥∇(u − v)∥L2(BH) ≤C

R∥u − v∥L2(BR) ,

and similarly for the other term. We begin by applying Theorem 2.3 to obtain

(4.13) supr∈[X ,R]

(1

r∥u − (u)Br∥L2(Br) +

1

r∥v − (v)Br∥L2(Br)) ≤ C.

We can consequently apply (4.3) to get, for r ∈ [X ,R],

(4.14) ∥u − v − (ur − vr)∥L2(Br) ≤ Cr−α(d−σ) ∥u − v − (u − v)B2r∥L2(B2r) ,

where ur ∈ u +H10(Br) and vr ∈ v +H1

0(Br) solve

(4.15) −∇ ⋅ (DpL (∇ur)) = −∇ ⋅ (DpL (∇vr)) = 0.

Next, by Corollary 2.2, there exist constants C(σ,data) <∞ and α1(d,Λ) ∈ (0,1)such that

(4.16) ∥u − ur∥L2(Br) + ∥v − vr∥L2(Br)

≤ Cr−α1(d−σ) (∥u − (u)B2r∥L2(B2r) + ∥v − (v)B2r∥L2(B2r)) ≤ Cr1−α1(d−σ).

Combining (4.13) and the previous inequality we get

supr∈[X ,R]

(1

r∥ur − (ur)Br∥L2(Br) +

1

r∥vr − (vr)Br∥L2(Br)) ≤ C.

Applying the second estimate of Theorem 2.3, we obtain, for every r ∈ [X ,R],

(4.17)1

rinf`∈P1

∥u − `∥L2(Br) +1

rinf`∈P1

∥v − `∥L2(Br) ≤ C ( rR+ 1

rd−σ)α1

,

and together with (4.16) this leads to

(4.18)1

rinf`∈P1

∥ur − `∥L2(Br) +1

rinf`∈P1

∥ur − `∥L2(Br) ≤ C ( rR+ 1

rd−σ)α1

.

We choose ε0 small enough and H large enough, both depending on (δ, θ,M,data),so that r ≥H implies

C (εβ0 + r−α(d−σ)) ≤ 1.

Therefore, by (A.8) and the previous two displays, we get, for θ ∈ (0,1] andr ∈ [X , ε0R],

inf`∈P1

∥ur − vr − `∥L2(Bθr)

(4.19)

≤ Cθ2 inf`∈P1

∥ur − vr − `∥L2(Br) +Cθ− d

2 ( rR+ 1

rd−σ)α1β

∥ur − vr − (ur − vr)Br∥L2(Br)


and, by applying (4.14) once more, we also get

inf`∈P1

∥u − v − `∥L2(Bθr)

(4.20)

≤ Cθ2 inf`∈P1

∥u − v − `∥L2(Br) +Cθ− d

2 ( rR+ 1

rd−σ)α1β

∥u − v − (u − v)Br∥L2(Br) .

Taking α ∶= α1β2d and setting

E1(r) ∶= ( rR+ 1

rd−σ)−α 1

rinf`∈P1

∥u − v − `∥L2(Br)

and

D1(r) ∶= sups∈[r,ε1R]

1

s∥u − v − (u − v)Bs∥L2(Bs) ,

the previous inequality implies that, for every r ≤ s ≤ ε0R,

E1(θs) ≤ Cθ1/2E1(s) +Cθ−2−d2 ( sR+ 1

sd−σ)α

D1(r).

Choosing θ ∶= (2C)−2, ε1 ∈ (0, ε0] and k ∈ N such that θk+1ε1R < r ≤ θkε1R, aniteration argument yields

(4.21)k

∑j=0E1(θ−jr) ≤ E1(ε1R) +C (ε1 +H−(d−σ))αD1(r).

Letting `j be the affine function realizing the infimum in E1(θ−jr), we see that

∣∇`0 −∇`k∣ ≤ Ck

∑j=0E1(θ−jr) ≤ CE1(ε1R) +C (ε1 +H−(d−σ))αD1(r).

Therefore,

D1(r) ≤ E1(r) +C ∣∇`0∣ ≤ CE1(ε1R) +C ∣∇`k∣ +C (ε1 +H−(d−σ))αD1(r).By choosing ε1 small enough and H large enough, so that

C (ε1 +H−(d−σ))α ≤ 1

2,

we deduce, after reabsorption, that

sups∈(r,ε1R]

1

s∥u − v − (u − v)Bs∥L2(Bs) ≤ CE1(ε1R) +C ∣∇`k∣ .

We finally obtain that

E1(ε1R) + ∣∇`k∣ ≤C

R∥u − v − (u − v)BR∥L2(BR) ,

and thus

(4.22) supr∈[X ,R]

1

r∥u − v − (u − v)Bs∥L2(Br) ≤

C

R∥u − v − (u − v)BR∥L2(BR) .

This completes the proof of the first part of the statement, in view of the Caccioppoliinequality. The second statement is equivalent to

supr∈[X ,R]

E1(r) ≤C

R∥u − v − (u − v)BR∥L2(BR) ,


which also follows easily from (4.21) and (4.22).

We conclude this section by recording a second large-scale regularity estimate,this one for solutions of the linearized equation. The proof is omitted since it isessentially the same as the one of Theorem 1.2, except that it is simpler because inplace of Proposition 4.1 we just use a rescaling of Theorem 1.1.

Proposition 4.3 (Large-scale C0,1 estimate for linearized equations). Fix σ ∈ (0, d)and M ∈ [1,∞). There exist constants α(data) ∈ (0, 12] and C(σ,M,data) <∞ anda random variable X satisfying

X ≤ Oσ (C)

such that the following holds. For every R ≥ 2X and u,w ∈H1(BR) satisfying

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

−∇ ⋅ (DpL(∇u,x)) = 0 in BR,

−∇ ⋅ (D2pL(∇u,x)∇w) = 0 in BR,

∥∇u∥L2(BR) ≤M,

we have the estimate, for every r ∈ [X , 12R],

( rR+ 1

rd−σ)−α

inf`∈P1

1

r∥w − `∥L2(Br) + ∥∇w∥L2(Br) ≤

C

R∥w − (w)BR∥L2(BR) .

The estimate of the first term in the inequality above represents a “C1,α estimatedown to mesoscopic scales.” The analogous estimate for differences was statedin Proposition 4.2.

5. Large-scale C1,1-type regularity: proof of Theorem 1.3

We turn to the proof of Theorem 1.3. In the first subsection, we prove statements(i) and (ii) of the theorem, which together are the Liouville-type results characterizingthe set L1 of solutions exhibiting at most linear growth. In Subsection 5.2, wedevelop a large-scale regularity theory for a linearized equation around an elementof L1 and show quantitatively that the first-order linearized correctors describe thetangent spaces of the “manifold” L1. Finally, in Subsection 5.3 we complete theproof of Theorem 1.3 by obtaining the large-scale C1,1–type excess decay estimate.

5.1. The Liouville-type results. In this subsection we present the proof ofTheorem 1.3 (i) and (ii), namely the Liouville-type classification of L1. After theproof (i) and (ii), we will prove a similar result for difference of two solutions in L1.

Proof of Theorem 1.3(i)–(ii). Fix σ ∈ (0, d) and M ∈ [1,∞). We set r0 ∶= X ,where X is the maximum of minimal scales in Theorem 1.2, Corollary 2.2 andTheorem 2.3, and a constant H(σ,M,data) <∞ to be fixed.

Step 1. The proof of statement (ii) of Theorem 1.3. Fix p ∈ BM and let `(x) ∶= p ⋅x.Let rm ∶= 2mr0. Let vm be the solution of the Dirichlet problem

−∇ ⋅ (DpL(∇vm, x)) = 0 in Brm ,

vm = ` on ∂Brm .


Set wm ∶= vm+1 − vm. As ∣∇`∣ ≤ M, we may apply Corollary 2.2 to obtain, forevery m ∈ N,

(5.1) ∥vm − `∥L2(Brm) ≤ Cr1−βm

with β = α(d − σ). The triangle inequality thus gives us, for all m ∈ N,

(5.2) ∥wm∥L2(Brm) ≤ Cr1−βm .

Then, by the Lipschitz regularity for differences of solutions in Theorem 1.2, wehave that, for j ≤m,

(5.3) ∥∇wm∥L2(Brj )≤ C

rm∥wm∥L2(Brm) ≤ Cr−βm .

We also get by the above inequality and Poincare’s inequality that, for j ≤m,

(5.4) ∣(wm)Brj − (wm)Br0 ∣ ≤ Cj

∑i=1

∥wm − (wm)Bri∥L2(Bri)≤ Crjr−βm .

Using the above inequality we obtain, again by Poincare’s inequality, that for allk ∈ N and j ∈ 0, . . . , k,

(5.5) ∥wk − (wk)Br0∥L2(Brj )≤ Crjr−βk .

Set nowun ∶= vn − (vn)Br0 ,

so that, noticing that

un − vm =n−1∑j=m

(wj − (wj)Br0) +m−1∑j=0

(wj)Br0 ,

we have by the triangle inequality that

∥un − `∥L2(Brm) ≤ ∥vm − `∥L2(Brm) +n−1∑j=m

∥wj − (wj)Br0∥L2(Brm) +m−1∑j=0

∣(wj)Br0 ∣ .

The first two terms on the right can be estimated using (5.1) and (5.5), respectively.For the last term we instead use (5.2) and (5.4) to obtain

m−1∑j=0

∣(wj)Br0 ∣ ≤m−1∑j=0

∥wj∥L2(Brj )+

m

∑j=0

∣(wj)Br0 − (wj)Brj ∣ ≤ Cr1−βm .

Therefore, we have shown that, for n ∈ N and m ∈ 0, . . . , n,

∥un − `∥L2(Brm) ≤ Cr1−βm .

Having arrived at this estimate, we now observe that, for every j,m,n ∈ N withj ≤m < n, we have that ∇un −∇um = ∑n−1

i=m∇wi, and hence by (5.3) we get

(5.6) ∥∇un −∇um∥L2(Brj )≤ C2−β(m−j)r−βj .

Thus un∞n=j is a Cauchy sequence in H1(Brj) and, consequently, we deduce by adiagonal argument that there exists u ∈ L1 such that, for all j ∈ N,

(5.7) ∥u − `∥L2(Brj )≤ Cr1−βj .

This yields (1.3).


Step 2. The proof of statement (i) of Theorem 1.3. Fix u ∈ L1 satisfying

(5.8) lim supr→∞

1

r∥u − (u)Br∥L2(Br) ≤M.

By the Lipschitz estimate and (5.8) we get

(5.9) supr∈[X ,R]

1

r∥u − (u)Br∥L2(Br) ≤ C lim sup

r→∞

1

r∥u − (u)Br∥L2(Br) ≤ C.

Denoting

E(r) ∶= 1

rinf`∈P1

∥u − `∥L2(Br),

we get by applying Theorem 2.3 that, for r ∈ [X ,R],

(5.10) E(θr) ≤ CθαE(r) +Cθ− d+22 r−β,

where β = α(d − σ). Denote by `r be the affine function minimizing E(r). Noticethat, by (5.9), ∣∇`r∣ ≤ C for all r ≥ X . Taking θ ∈ (0,1) so small that Cθα ≤ 1

2 andsetting rm ∶= θ−mr0, we obtain, for all m ∈ N,

(5.11) E(rm−1) ≤1

2E(rm) +Cr−βm .

It follows, after reabsorption, that

n−1∑j=m

E(rj) ≤ E(rn) +Cn

∑j=m+1

r−βj ≤ E(rn) +Cr−βm .

The term on the right is uniformly bounded in n, because (5.9) implies that

(5.12) lim supn→∞

E(rn) ≤ C.

The above two displays give that E(rn)→ 0 as n→∞ and, thus

(5.13)∞∑j=m

E(rj) ≤ Cr−βm .

Furthermore, since

∣∇`rj −∇`rj+1 ∣ ≤ C (E(rj) +E(rj+1)) ,

we see by telescoping that, for all m,n ∈ N such that n >m,

∣∇`rm −∇`rn ∣ ≤ Cr−βm ,

and the right-hand side converges to zero as m→∞. Therefore ∇`rnn is a Cauchysequence and consequently there is an affine function ` such that

limn→∞

∇`rn = ∇`.

We hence obtain that

∣∇`rm −∇`∣ ≤ Cr−βm .

Plugging this into (5.13) proves, after easy manipulations, for (x) ∶= ∇` ⋅ x andr ∈ [X ,∞),

(5.14) ∥u − (u)Br − ∥L2(Br)

≤ Cr1−β.


Furthermore, by Step 1 there is u ∈ L1 corresponding such that, for r ≥ X ,

(5.15) ∥u − ∥L2(Br)

≤ Cr1−β.

Together with (5.14) this yields

(5.16) ∥u − u − (u − u)Brj ∥L2(Brj )≤ Cr1−δj .

Thus, applying the Lipschitz estimate for differences we see that, for all r ≥ X ,

1

r∥u − u − (u − u)Br∥L2(Br) ≤ C lim sup

r→∞

1

r∥u − u − (u − u)Br∥L2(Br) = 0.

Therefore u− u is a constant, and the statement (1.3) follows from (5.15) by setting

` = u − u + . The proof is complete.

We conclude this subsection with a quantitative Liouville-type result for differ-ences of solutions.

Lemma 5.1. Fix σ ∈ (0, d) and M ∈ [1,∞). There exists constants C(σ,M,data) <∞ and α(data) ∈ (0,1), and a random variable X satisfying

(5.17) X ≤ Oσ(C).

such that the following holds. Suppose that u, v ∈ L1 satisfy


1

r(∥u − (u)Br∥L2(Br) ∨ ∥v − (v)Br∥L2(Br)) ≤M.

Then there is an affine function ` such that, for r ≥ X ,

(5.19) ∥u − v − `∥L2(Br) ≤ Cr−α(d−σ) ∥`∥L2(Br)

and

(5.20) supr∈[X ,∞)

∥∇u −∇v∥L2(Br) ≤ C infr∈[X ,∞)

∥∇u −∇v∥L2(Br) .

Proof. Let X be the maximum of the minimal scales in Theorem 1.3(i)–(ii), Corol-lary 2.2, Proposition 4.1, and a constant H(σ,M,data) <∞ to be selected below.Then (5.17) holds. Set z ∶= u − v and denote

D(r) ∶= 1

rinf`∈P1

∥u − `∥L2(Br) ∨ inf`∈P1

∥v − `∥L2(Br)

and

E(r) ∶= 1

rinf`∈P1

∥z − `∥L2(Br) .

Let ur, vr, and zr = ur − vr stand for the homogenized solutions at the scale rprovided by Proposition 4.1 applied at the scale R = r. In particular, we have that

(5.21) ∥z − zr∥L2(Br) ≤ Cr−β ∥z − (z)Br∥L2(Br) .

Next, Theorem 1.3(i) implies, by taking H so large that CH− 12α(d−s) ≤ 1, that there

exists β ∶= 12α(d − s) such that

(5.22) D(r) ≤ 1

2r−β.


Corollary 2.2 then yields, by taking H larger and β smaller, if necessary, that

D(r) ∶= 1

rinf`∈P1

∥ur − `∥L2(Br) ∨ inf`∈P1

∥vr − `∥L2(Br) ≤ r−β.

The basic decay estimate for the differences of homogenized solutions is given inProposition A.1: it states that

inf`∈P1

∥zr − `∥L2(Bθr)

≤ Cθ2 inf`∈P1

∥zr − `∥L2(Br/2) +C (D(r) ∨ 1)η θ− d2D(r)γ ∥zr − (zr)Br∥L2(Br/2) .

Using (5.21) for the differences together with the triangle inequality, yields, againfor smaller β,

E(θr) ≤ C (θ + θ− d+22 H−β)E(r) +Cθ− d+22 r−β ∣∇`r∣.

Comparing this to (5.10) allows us to reason exactly as in the proof of Theorem 1.3(i)to obtain an affine function ` such that

(5.23)1

r∥z − (z)Br − `∥L2(Br) ≤ Cr

1−β ∣∇`∣ ,

giving (5.19). The above display also yields, for a positive constant A(d), that, forr ≥ X ≥H,

∣1r∥z − (z)Br∥L2(Br) −A ∣∇`∣∣ ≤ CH−β ∣∇`∣.

This gives (5.20) by Caccioppoli and Poincare’s inequalities provided that we takea lower bound H for X such that 2CH−β ≤ A. The proof is complete.

5.2. Linearized equations around first-order correctors. In this subsection,we establish some estimates for the linearized equations around first-order correctors,namely the full slate of large-scale Ck,1 estimates as well as a quadratic estimatefor linearization errors. The former is essentially a verbatim consequence of thearguments of [3, Chapter 3] combined with homogenization results already obtainedin this paper. The only work to be done is to slightly post-process the statementsin [3] to be compatible with the statement of the result we need.

To state the next theorem, we introduce the linearized correctors. Let φ ∈ L1. ByTheorem 1.3(i), we have that there is a unique affine function `φ such that

(5.24) limr→∞

1

r∥φ − `φ∥L2(Br) = 0.

For an open set U ⊆ Rd, we denote

A[φ](U) ∶= w ∈H1loc(U) ∶ −∇ ⋅ (D2

pL(∇φ, ⋅)∇w) = 0 in U .We next define, for each k ∈ N,

Ak[φ] ∶= w ∈ Ak[φ](Rd) ∶ limr→∞

r−k−1 ∥w∥L2(Br) = 0

and denote the D2pL (∇`φ)–harmonic polynomials of degree k by

Ak[φ] ∶= q ∈ Pk ∶ −∇ ⋅ (D2pL(∇`φ)∇q) = 0 .

The next theorem, which can be compared to [3, Theorem 3.8], asserts that thevector space Ak[φ] essentially homogenizes to Ak[φ] and, in particular, has the


same dimension. It also gives a large-scale Ck,1 estimate which states that solutionsof the linearized equation (around φ) can be approximated by elements of Ak[φ] inthe same way that a harmonic function can be approximated by a polynomial ofdegree k (or an analytic function by its kth order Taylor series).

Theorem 5.2 (Ck,1 regularity for linearized equation around an element of L1).Fix σ ∈ (0, d) and M ∈ [1,∞). There exist δ(σ,data) ∈ (0, 12], C(data) <∞ and arandom variable X satisfying the estimate

(5.25) X ≤ Oσ (C(σ,M,data))such that, for every φ ∈ L1 satisfying lim supr→∞

1r ∥φ − (φ)Br∥L2(Br) ≤M and k ∈ N,

the following statements are valid:

(i)k There exists C(k,M,data) <∞ such that, for every w ∈ Ak[φ], there exists

q ∈ Ak[φ] such that, for every r ≥ X ,

(5.26)1

r∥w − q∥L2(Br) ≤ Cr

−δ ∥q∥L2(Br) .

(ii)k For every p ∈ Ak[φ], there exists w ∈ Ak[φ] satisfying (5.26) for every r ≥ X .

(iii)k There exists C(k,M,data) < ∞ such that, for every R ≥ X and w ∈A[φ](BR), there exists ξ ∈ Ak[φ] such that, for every r ∈ [X ,R], we havethe estimate

(5.27) ∥w − ξ∥L2(Br) ≤ C ( rR

)k+1

∥w∥L2(BR) .

Proof. Fix σ ∈ (0, d) and φ ∈ L1. Let X be the maximum of minimal scales appearingin Theorem 2.3, Corollary 3.7 and Theorem 1.3(i)–(ii), which we already proved.Then Xσ ≤ Oσ(C). Fix r ≥ 2X . By Theorem 1.3 (i) we have that there is an affinefunction `φ such that

∥φ − `φ∥L2(Br) ≤ Cr1−α(d−σ).

Let w ∈ A[φ](Br). Consider the following system of homogenized solutions:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

−∇ ⋅ (DpL(∇ur)) = 0 in Br,

−∇ ⋅ (D2pL(∇ur)∇wr) = 0 in B4r/5,

−∇ ⋅ (D2pL(∇`φ)∇w) = 0 in B3r/4,

ur = φ on ∂Br,

wr = w on ∂B4r/5,

w = wr on ∂B3r/4.

We claim that

(5.28) ∥w −w∥L2(Br/2) ≤ Cr−α(d−σ) ∥w − (w)Br∥L2(Br) .

This estimate, which asserts that w can be well-approximated by a D2pL(∇`φ)–

harmonic function, is analogous to [3, Lemma 3.4]. Indeed, using this estimate asa replacement for [3, Lemma 3.4], we may repeat, essentially verbatim, the proofof [3, Theorem 3.8] to obtain the theorem.


To show (5.28), we first obtain by Corollary 3.7 and Meyers estimate that

(5.29) ∥w −wr∥L2(B4r/5) ≤ Cr−α(d−σ) ∥w − (w)Br∥L2(Br) .

Second, by Theorem 2.3 we have that

∥φ − ur∥L2(Br) ≤ Cr1−α(d−σ).

Thus, in view of (5.2), we get by the triangle inequality that

∥ur − `φ∥L2(Br) ≤ Cr1−α(d−σ).

Then Caccioppoli inequality yields that

∥∇ur −∇`φ∥L2(B3r/4)≤ Cr

∥ur − `φ∥L2(Br) ≤ Cr−α(d−σ)

and, by Proposition A.1 we arrive at

∥∇ur −∇`φ∥L∞(Br/2)≤ Cr−α(d−σ).

We deduce, by the equations,

−∇ ⋅ (D2pL(∇`φ)∇(wr −w)) = ∇ ⋅ ((D2

pL(∇ur) −D2pL(∇`φ))∇wr) ,

and hence there is possibly smaller α(data) ∈ (0, 12] such that

∥∇wr −∇w∥L2(B3r/4) ≤ Cr−α(d−σ)−1 ∥wr − (wr)∥L2(B4r/5) .

We now obtain (5.28) by the triangle inequality, Poincare’s inequality, Caccioppoliestimate and (5.29), concluding the proof.

We next present a result concerning the superlinear response to the linearapproximation of L1 by linearized correctors in A1. This states roughly that A1[φ]is the tangent space of L1 at φ and that the “manifold” L1 has a structure thatis at least C1,δ for some small δ > 0. It is important for our purposes that theseestimates hold “all the way down to the microscopic scale,” quantified by a minimalscale X with optimal stochastic integrability. This result is of independent interest.

Lemma 5.3 (Superlinear response to linearization for L1). Fix σ ∈ (0, d) andM ∈ [1,∞). There exist constants δ(d,Λ) ∈ (0, 12] and C(σ,M,data) < ∞, and arandom variable X satisfying X = Oσ(C) such that the following two statements arevalid:

(i) For every u, v ∈ L1 satisfying


1

r(∥u − (u)Br∥L2(Br) ∨ ∥v − (v)Br∥L2(Br)) ≤M,

there exists w ∈ A1[u] such that, for every r ≥ X ,

(5.31) ∥∇v −∇u −∇w∥L2(Br) ≤ C (∥∇u −∇v∥L2(Br))1+δ

.

(ii) For every u ∈ L1 and w ∈ A1[u] satisfying

(5.32) lim supR→∞

1

r∥u − (u)BR∥L2(BR) ≤M,

there exists v ∈ L1 such that, for all r ≥ X ,

(5.33) ∥∇v −∇u −∇w∥L2(Br) ≤ C (∥∇w∥L2(Br))1+δ

.


Proof. Let X be the maximum of minimal scales in Theorem 1.3(i)–(ii), Theorem 5.2and Lemma 5.1, as well as a constant H(σ,M,data) <∞ to be fixed. Clearly wehave X ≤ Oσ(C).

Step 1. The proof of statement (i). Fix u, v ∈ L1 satisfying (5.30). Denotez = v − u. Throughout the proof, for given r ∈ [X ,∞) we denote by ψr a functionrealizing the infimum below:

∥z − ψr∥L2(Br) = infψ∈A1[u]

∥z − ψ∥L2(Br) .

We first collect some consequences of our preliminary results. By (5.20),

(5.34) supt∈[X ,∞)

∥∇z∥L2(Bt) ≤ C inft∈[X ,∞)

∥∇z∥L2(Bt) .

Moreover, by Theorem 5.2 and Lemma 5.1, we find ψ0 ∈ A1[u] such that, for r ≥ X ,

∥z − ψ0∥L2(Br) ≤ Cr−2β ∥ψ0∥L2(Br) ,

where we denote β ∶= 12α(d − σ). Taking H so large that CH−β ≤ 1

2 , we also obtainby the previous two displays that, for all r ≥ X ,

(5.35)1

r∥z − ψ0∥L2(Br) ≤ Cr

−2β inft∈[X ,∞)

∥∇z∥L2(Bt) .

Define

r ∶= inft∈[X ,∞)

(∥∇z∥L2(Bt))− δ0β ,

where δ0 is as in (5.39) below, coming from the Meyers estimate for z. On the onehand, for r ≥ X ∨ r we have by (5.35) that

(5.36) infψ∈A1[u]

1

r∥z − ψ∥L2(Br) ≤

1

r∥z − ψ0∥L2(Br) ≤ Cr

−β inft∈[X ,∞)

(∥∇z∥L2(Bt))1+δ0

.

On the other hand, if r > X , then for r ∈ [X , r) we get

(5.37) inft∈[X ,∞)

(∥∇z∥L2(Bt))δ0 = r −β ≤ r−β.

We next focus on the case r > X in more detail. Observe that z solves

(5.38) −∇ ⋅ (D2pL(∇u, ⋅)∇z) = ∇ ⋅ (∫

1

0(D2

pL(∇u+t∇z, ⋅)−D2pL(∇u, ⋅)) dt∇z) .

Letting wr solve, for r ∈ [X , r),

−∇ ⋅ (D2

pL(∇u, ⋅)∇wr) = 0 in Br/2,

wr = z − ψr on ∂Br/2,

we have that

∥∇z−∇ψr−∇wr∥L2(Br/2) ≤ C ∥∫1

0(D2

pL(∇u+t∇z, ⋅)−D2pL(∇u, ⋅)) dt∇z∥

L2(Br/2).

It follows by the Meyers estimate and regularity of p↦D2pL(p, ⋅) that there exist

constants C(data) <∞ and δ0(data) ∈ (0, 12] such that, for all δ ∈ [0,2δ0],

(5.39) ∥∫1

0(D2

pL(∇u+t∇z, ⋅)−D2pL(∇u, ⋅)) dt∇z∥

L2(Br/2)≤ C (∥∇z∥L2(Br))

1+δ.


Since we assume that r ∈ [X , r), we obtain by (5.37), (5.34), and Poincare’sinequality that

∥z − ψr −wr∥L2(Br/2) ≤ Cr1−β inf

t∈[X ,∞)(∥∇z∥L2(Bt))

1+δ0.

Furthermore, by Theorem 5.2, there is ψ ∈ A1[u] such that

∥ψ −wr∥L2(Bθr)≤ Cθ2 ∥wr∥L2(Br) .

We then obtain, by the triangle inequality, that

∥z − ψ − ψr∥L2(Bθr)≤ Cθ2 ∥z − ψr∥L2(Br) +Cθ

− d2 ∥z − ψr −wr∥L2(Br/2) ,

so that for r ≥ θ−1X , by choosing θC = 12 , we arrive at

1

θr∥z − ψθr∥L2(Bθr) ≤

1

2

1

r∥z − ψr∥L2(Br) +Cr


(∥∇z∥L2(Bt))1+δ0

.

An iteration yields, for every r ∈ [X , r], that

1

r∥z − ψr∥L2(Br) ≤ C

1

r∥z − ψr∥L2(Br) +Cr


(∥∇z∥L2(Bt))1+δ0

,

and the first term can be bounded by (5.36). In conclusion, we have proved that,for all r ≥ X ,

1

r∥z − ψr∥L2(Br) ≤ Cr


(∥∇z∥L2(Bt))1+δ0

.

Furthermore, we get by the above display and the fact that ψ2jX , ψ2j+1X ∈ A1[u]that, for all j ∈ N and r ∈ [X ,2jX ],

1

r∥ψ2jX − ψ2j+1X ∥L2(Br) ≤ C

1

2jX∥ψ2jX − ψ2j+1X ∥L2(B

2jX )

≤ C2−jβ inft∈[X ,∞)

(∥∇z∥L2(Bt))1+δ0

,

which leads easily to

sups∈[X ,∞)

1

r∥ψX − ψs∥L2(Br) ≤ C inf

t∈[X ,∞)(∥∇z∥L2(Bt))

1+δ0.

Therefore, we obtain, with w = ψX ∈ A1[u], that, for all r ≥ X ,

1

r∥z −w∥L2(Br) ≤ C inf

t∈[X ,∞)(∥∇z∥L2(Bt))

1+δ0.

Finally, by the equations of z and w, the Meyers and Caccioppoli estimatesand (5.34), we obtain, for every r ≥ X ,

∥∇z −∇w∥L2(Br) ≤C

r∥z −w∥L2(B2r) +C inf

t∈[X ,∞)(∥∇z∥L2(Bt))

1+δ0.

Combining previous two displays yields (5.31).

Step 2. The proof of statement (ii). Fix u ∈ L1 satisfying (5.32) and w ∈ A1[u].Observe first that by Theorem 5.2 we have that

(5.40) supt∈[X ,∞)

∥∇w∥L2(Bt) ≤ C inft∈[X ,∞)

∥∇w∥L2(Bt) .


Given small η ∈ (0,1) to be fixed shortly, we may assume that

(5.41) supt∈[X ,∞)

∥∇w∥L2(Bt) ≤ η.

Indeed, otherwise

inft∈[X ,∞)

∥∇w∥L2(Bt) ≥η

C,

and we may simply take v = u in this case to obtain (5.33) trivially. We henceforthassume (5.41).

By Theorem 1.3(i) and Theorem 5.2, we find that there exist affine functions `wand ù such that, for every r ≥ X ,

∥w − `w∥L2(Br) ≤ Cr−α(d−σ) ∥w∥L2(Br) and ∥u − ù∥L2(Br) ≤ Cr

1−α(d−σ).

Observe that we have

∣∇`w∣ ≤C

r∥`w − (`w)∥L2(Br) ≤ Cr

−α(d−σ)−1 ∥`w∥L2(Br) +C ∥∇w∥L2(Br) ,

so thus by sending r →∞ yields by (5.41) that

∣∇`w∣ ≤ Cη.Theorem 1.3(ii) then yields v ∈ L1 such that

∥v − ù − `w∥L2(Br) ≤ Cr1−α(d−σ).

By the Caccioppoli inequality,

supt∈[X ,∞)

∥∇u −∇v∥L2(Bt)

≤ lim supt→∞

C

t∥(u − ù) − (v − ù − `w)∥L2(B2t) + lim sup

t→∞

C

t∥`w∥L2(Bt) ≤ Cη.

Step 1 yields the existence of w ∈ A1[u] such that

supt∈[X ,∞)

∥∇v −∇u −∇w∥L2(Bt) ≤ C ( inft∈[X ,∞)

∥∇u −∇v∥L2(Bt))1+δ

.

By Caccioppoli inequality, similarly as in Step 1, we get, for r, t ∈ [X ,∞),

∥∇v −∇u −∇w∥L2(Bt) ≤ Ct−α(d−σ) +C (∥∇u −∇v∥L2(Br))

1+δ.

It follows by the Lipschitz estimate for w − w ∈ A1[u] that

∥∇w −∇w∥L2(Br) ≤ C lim supt→∞

∥∇w −∇w∥L2(Bt)

≤ C supt∈[X ,∞)

∥∇v −∇u −∇w∥L2(Bt) +C (∥∇u −∇v∥L2(Br))1+δ

≤ C (∥∇u −∇v∥L2(Br))1+δ

.

Therefore, we have that

∥∇v −∇u −∇w∥L2(Br) ≤ C (∥∇u −∇v∥L2(Br))1+δ

.

Since ∥∇u −∇v∥L2(Br) ≤ Cη, we may choose (Cη)δ = 12 , and conclude that

∥∇u −∇v∥L2(Br) ≤ 2 ∥∇w∥L2(Br) ,

giving (5.33) by the last three displays, and concluding the proof.


5.3. The large-scale C1,1-type excess decay estimate. We have now assembledthe ingredients necessary to complete the proof of Theorem 1.3.

Proof of Theorem 1.3 (iii). We fix σ ∈ (0, d) and M ∈ [1,∞). For these parameters,let X be the maximum of random minimal scales appearing in Theorem 1.2,Theorem 2.3, Theorem 5.2 and Lemma 5.3. Let R ≥ X . Suppose that u ∈ L(BR)satisfies 1

R ∥u − (u)BR∥L2(BR) ≤M.

The proof proceeds in several steps. In the first four steps we show an inter-mediate result, that is, we show that for all α ∈ (0,1) there exists a constantC(σ,α,M,data) <∞ and φ ∈ L1 such that

(5.42) ∥∇u −∇φ∥L2(Br) ≤ C ( rR

)α 1

R∥u − (u)BR∥L2(BR) .

In the last two steps we demonstrate how this can be improved to (1.11).

We denote, in short,

(5.43) MR ∶= infφ∈L1

1

R∥u − φ − (u − φ)BR∥L2(BR) ,

which obviously satisfies MR ≤ C by the normalization

(5.44)1

R∥u − (u)BR∥L2(BR) ≤M.

Step 1. We first show that, for all η ∈ (0, 1], there exist constants γ(η,M, σ,data) ∈(0, 12] and H(η,M, σ,data) <∞ such that if γR ≥ X ∨H, then, for r ∈ [X ∨H,γR],(5.45) inf

φ∈L1

∥∇u −∇φ∥L2(Br) ≤ η.

The parameter η will be fixed in the end of Step 3. Set r0 ∶= γR and assume thatr0 ≥ X ∨H. To prove (5.45), we first observe by Theorem 2.3 and (5.44) that thereexists ` ∈ P1 such that ∣∇`∣ ≤ C and

∥u − `∥L2(B2r0)≤ Cr0 (γα + γ−1−

d2H−β(d−σ)) .

By statement (ii) of Theorem 1.3, there exists φ ∈ L1 such that

∥φ − `∥L2(Br0)≤ Cr1−β(d−σ)0 .

Combining above estimates with the Caccioppoli estimate implies that there existsφ ∈ L1 such that

∥∇u −∇φ∥L2(Br0)≤ C (γα + γ−1− d2H−β(d−σ)) .

By the Lipschitz estimate in Theorem 1.2 we then obtain that, for all r ∈ [X , r0],

∥∇u −∇φ∥L2(Br) ≤ C (γα + γ−1− d2H−β(d−σ)) .

We then choose γ(η,C,α) ∈ (0, 12] small and then H(γ, η,C, σ, β) <∞ large so that

C (γα + γ−1− d2H−β(d−σ)) ≤ η.

We therefore have that (5.45) is valid.

Step 2. Statement of the induction assumption. Fix α ∈ (0,1). Let r0 = γRand assume that r0 ≥ X ∧H, where γ and H are as in Step 1. Let θ ∈ (0, 12] tobe a constant to be fixed and set, for j ∈ N, rj ∶= θjr0. Take n ∈ N0 be such that


X ∨H ∈ (rn+1, rn]. We assume inductively that for some m ∈ 0, . . . , n there existssuch that, for all j ∈ 0, . . . ,m,

(5.46) infφ∈L1

∥∇u −∇φ∥L2(Brj )≤ θαj inf

φ∈L1

∥∇u −∇φ∥L2(Br0).

This is trivially true for m = 0, which serves as our initial step for induction. Wedenote by φj a member of L1 realizing the infimum in infφ∈L1 ∥∇u −∇φ∥L2(Brj )

.

Step 3. We show the induction step, that is, for α ∈ (0, 1), there exists θ(α,data) ∈(0,1) and φm+1 ∈ L1 such that

(5.47) ∥∇u −∇φm+1∥L2(Brm) ≤ θα ∥∇u −∇φm∥L2(Brm+1),

which obviously proves the induction step. Let h solve

−∇ ⋅ (D2

pL(∇φm, ⋅)∇h) = 0 in Brm/2,

h = u − φm on ∂Brm/2,

Similarly to the proof of Lemma 5.3, we have that

∥∇u −∇φm +∇h∥L2(Brm/2) ≤ C (∥∇u −∇φm∥L2(Brm))1+δ

.

Now (5.45) implies that

(5.48) ∥∇u −∇φm +∇h∥L2(Brm/2) ≤ Cηδ ∥∇u −∇φm∥L2(Brm) .

Furthermore, by Theorem 5.2, Caccioppoli inequality and the triangle inequality,we find ψ ∈ A1 [φm] such that

(5.49) ∥∇h −∇ψ∥L2(Brm+1)≤ Cθ ∥∇h∥L2(Brm/2) .

By (5.48) and the triangle inequality, we get

∥∇h∥L2(Brm/2) ≤ (1 +Cηδ) ∥∇u −∇φm∥L2(Brm) ,(5.50)

and hence

(5.51) ∥∇h −∇ψ∥L2(Brm+1)≤ Cθ ∥∇u −∇φm∥L2(Brm) .

Now, applying Lemma 5.3, we find φm+1 ∈ L1 such that

∥∇φm −∇φm+1 −∇ψ∥L2(Brm+1)≤ C (∥∇ψ∥L2(Brm+1)

)1+δ

.

By the Lipschitz estimate, (5.49) and (5.50), we have that

∥∇ψ∥L2(Brm+1)≤ C ∥∇ψ∥L2(Brm/2) ≤ C ∥∇u −∇φm∥L2(Brm) .

Thus we get by (5.45) that

∥∇φm −∇φm+1 −∇ψ∥L2(Brm+1)≤ Cηδ ∥∇u −∇φm∥L2(Brm) .

By the triangle inequality we have that

∥∇u −∇φm+1∥L2(Brm+1)≤ ∥∇h −∇ψ∥L2(Brm+1)

+ θ− d2 ∥∇u −∇φm +∇h∥L2(Brm/2)

+ ∥∇φm −∇φm+1 −∇ψ∥L2(Brm+1).

The terms on the right can be estimated using (5.51), (5.48) and (5.3), respectively,and we arrive at

∥∇u −∇φm+1∥L2(Brm+1)≤ C (θ + θ− d2 ηδ) ∥∇u −∇φm∥L2(Brm) .


We then choose first θ so that Cθ1−α = 12 , and then η so that Cθ−α−

d2 ηδ = 1

2 . Withthese choices, we get (5.47) from the above display.

Step 4. Proof of C1,1−. We show that for all α ∈ (0,1) there exists a constantC(α,σ,M,data) <∞ and φ ∈ L1 such that

(5.52) ∥∇u −∇φ∥L2(Br) ≤ C ( rR

)α

MR,

where MR is defined in (5.43). In the last two steps we demonstrate how this canbe improved to (1.11).

We show that the corrector can be chosen uniformly in m, that is, φm can bereplaced with φ ∈ L1 for all m. By (5.20), we have that, for r ∈ [X ,∞),

∥∇φm+1 −∇φm∥L2(Br) ≤ C ∥∇φm+1 −∇φm∥L2(Brm+1).

The triangle inequality implies that

∥∇φm+1 −∇φm∥L2(Brm+1)≤ ∥∇u −∇φm+1∥L2(Brm+1)

+ θ− d2 ∥∇u −∇φm∥L2(Brm) .

It follows, by the triangle inequality and (5.46), that for r ∈ [X ,∞) we get

∥∇φm+1 −∇φm∥L2(Br) ≤ C (rmr0

)α

infψ∈L1

∥∇u −∇ψ∥L2(Br0).

Summation then yields, for r ∈ [rm+1, rm],

∥∇φm −∇φn+1∥L2(Br) ≤ C ( rr0

)α

infψ∈L1

∥∇u −∇ψ∥L2(Br0).

Therefore, by the triangle inequality, we get

(5.53) ∥∇u −∇φn+1∥L2(Br) ≤ C ( rr0

)α

infψ∈L1

∥∇u −∇ψ∥L2(Br0),

and hence we may take φ = φn+1 to obtain (5.52) by applying Caccioppoli inequalityand giving up a volume factor.

Step 5. In this step we show that there exist constants β(data) > 0, C(σ,M,data) <∞ and ψ ∈ A2[φ] such that

(5.54) ∥∇u −∇φ −∇ψ∥L2(Br) ≤ C ( rR

)1+β

MR.

We immediately choose α = 2+δ2+2δ in (5.52) and β = (1 + δ)α − 1 = δ

1+δ , where δ is asin Lemma 5.3. We will show that (5.54) is valid with this β. Denote z = u − φ andlet ψm ∈ A2[φ] be such that

∥∇z −∇ψm∥L2(Brm) = infψ∈A2[φ]

∥∇z −∇ψ∥L2(Brm) .

By (5.52),

(5.55) ∥∇ψm∥L2(Brm) ≤ C (rmR

)α

MR.

Let h ∈H1(Brm/2) solve

−∇ ⋅ (D2

pL(∇φ, ⋅)∇h) = 0 in Brm/2,

h = z − ψm on ∂Brm/2,


so that, as in the proof of Lemma 5.3,

∥∇z −∇ψm −∇h∥L2(Brm/2) ≤ C (∥∇z∥L2(Brm))1+δ

.

Using (5.52) and (5.45), recalling that MR ≤ C, we get

∥∇z −∇ψm −∇h∥L2(Brm+1)≤ C (rm+1

R)1+β

MR.

On the other hand, by Theorem 5.2, there exists ψm ∈ A2[φ] such that

∥∇h −∇ψm+1∥L2(Brm+1)≤ Cθ2 ∥∇h∥L2(Brm/2) .

Therefore, by the triangle inequality and the previous two displays, by choosing θsmall appropriately, we get

infψ∈A2[φ]

∥∇z −∇ψ∥L2(Brm+1)

≤ ∥∇z −∇(ψm + ψm+1)∥L2(Brm+1)

≤ ∥∇h −∇ψm+1∥L2(Brm+1)+ ∥∇z −∇ψm −∇h∥L2(Brm+1)

≤ Cθ2 ∥∇h∥L2(Brm/2) + 2θ−d2 ∥∇z −∇ψm −∇h∥L2(Brm/2)

≤ θα(1+δ) ∥∇z −∇ψm∥L2(Brm) +C (rm+1R

)1+β

MR.

An iteration then gives

∥∇z −∇ψm∥L2(Brm) ≤ C (rmR

)1+β

MR.

By the triangle inequality, we obtain

∥∇ψm+1 −∇ψm∥L2(Brm) ≤ C (rmR

)1+β

MR.

Since ψm+1 − ψm ∈ A2[φ], we have that, for r ∈ [rm,R],

∥∇ψm+1 −∇ψm∥L2(Br) ≤ C ( r

rm) ∥∇ψm+1 −∇ψm∥L2(Brm) .

Combining, for r ∈ [rm,R],

∥∇ψm+1 −∇ψm∥L2(Br) ≤ C (rmr

)β

( rR

)1+β

MR.

Thus, by summing over the scales, we obtain

∥∇ψm −∇ψn∥L2(Brm) ≤ C (rmR

)1+β

MR.

The triangle inequality thus yields

∥∇z −∇ψn∥L2(Brm) ≤ C (rmR

)1+β

MR,

from which (5.54) follows easily by taking ψ = ψn.


Step 6. Conclusion. We show that there exist a constant C(σ,M,data) < ∞and φ ∈ L1 such that

(5.56) ∥∇u −∇φ∥L2(Br)

≤ C ( rR

)MR.

Let ψ be as in Step 5. By Theorem 5.2, we find p ∈ A2[φ] and p ∶= 12 (∇2p)x⊗2,

correspondingly, such that

∥ψ − p∥L2(Br) ≤ Cr−β ∥p∥L2(Br) and ∥ψ − p∥

L2(Br)≤ Cr−β ∥p∥L2(Br) .

Moreover, we have that

∥p∥L2(BR) ≤ C ∥p∥L2(BR) ≤ CR ∥∇ψ∥L2(BR) ≤ CRMR.

We now obtain, for r ∈ [X ,R], that

∥∇ψ∥L2(Br)

≤ C ( rR

) ∥∇ψ∥L2(BR) ≤ C ( r

R)MR.(5.57)

Furthermore, clearly ψ − ψ ∈ A1[φ] and, by the above inequality, (5.54) and (5.52),

∥∇(ψ − ψ)∥L2(Br)

≤ ∥∇z −∇ψ∥L2(Br) + ∥∇z∥L2(Br) + ∥∇ψ∥L2(Br)

≤ C ( rR

)α

MR.

Lemma 5.3 then yields that there is φ ∈ L1 such that

∥∇φ −∇φ +∇(ψ − ψ)∥L2(Br)

≤ C (∥∇(ψ − ψ)∥L2(Br)

)1+δ

,

and thus, since MR ≤ C,

(5.58) ∥∇φ −∇φ +∇(ψ − ψ)∥L2(Br)

≤ C ( rR

)1+β

MR.

By the triangle inequality we deduce that

∥∇u −∇φ∥L2(Br)

≤ ∥∇(u − φ − ψ)∥L2(Br) + ∥∇(φ − (φ + ψ − ψ))∥L2(Br)

+ ∥∇ψ∥L2(Br)

.

The terms on the right can be estimated by (5.54), (5.58) and (5.57), respectively.We thus obtain (5.56) and the proof is now complete.

Remark 5.4. In the course of proving (1.11), we also obtain an estimate whichcan be interpreted as large-scale C2,β-type estimate. Indeed, we showed that thereexist β(data) and C(σ,M,data) < ∞ such that the following holds. Let R ≥ Xand suppose that u ∈ L(BR) satisfying 1

R ∥u − (u)BR∥L2(BR) ≤M. Then there exists

φ ∈ L1 and ψ ∈ A2[φ] such that, for r ∈ [X ,R],

(5.59) ∥∇u −∇φ −∇ψ∥L2(Br) ≤ C ( rR

)1+β

infξ∈L1

1

R∥u − ξ − (u − ξ)BR∥L2(BR) .

6. Improved regularity of the homogenized Lagrangian

In this section, we prove Theorem 1.4. In addition to the hypotheses stated inSection 1.2, we suppose that there exists an exponent γ ∈ (0,1] such that

(6.1) P⎡⎢⎢⎢⎢⎣supp∈Rd

⎛⎝[DpL(p, ⋅)

1 + ∣p∣]C0,γ(Rd)

+ [D2pL(p, ⋅)]C0,γ(Rd)

⎞⎠≤M0

⎤⎥⎥⎥⎥⎦= 1.


In particular, unlike the rest of the paper, here we assume L(p, ⋅) has some regularityon the unit scale. A condition like this is needed to control the very small scalesand allow us to convert the large-scale C0,1 estimate for differences into pointwiseestimates, which seems to be necessary in order to improve the scaling of thelinearization error.

Proposition 6.1 (Pointwise gradient estimate for solutions). Assume (6.1) holdsfor some γ ∈ (0,1]. Fix σ ∈ (0, d) and M ∈ [1,∞). Let X be the random variable inTheorem 2.3. Then there exists β(γ,data) ∈ (0, 12] such that, for every R ≥ 4 andu ∈H1(BR) satisfying

−∇ ⋅ (DpL(∇u,x)) = 0 in BR,

∥∇u∥L2(BR) ≤M,


(6.2) ∥∇u∥C0,β(B1) ≤ CXd2R−1 ∥u − (u)BR∥L2(BR) .

Proof. By the Proposition A.3, there exists β(γ,data) ∈ (0, 12] such that

∥∇u∥C0,β∧γ(B1) ≤ C ∥∇u∥L2(B2) .

Giving up a volume factor and applying Theorem 2.3, we have

∥∇u∥L2(B2) ≤ C (R ∧X )d2 ∥∇u∥L2(BX∧R) ≤ CX

d2R−1 ∥u − (u)BR∥L2(BR) .

Proposition 6.2 (Pointwise gradient estimate for differences). Assume that (6.1)holds for some γ ∈ (0,1]. Let M ∈ [1,∞). Then there exist α(γ,data), δ(γ,data) ∈(0, 12], C(γ,M,data) <∞ and a random variable X satisfying

X = Oδ (C)

such that the following holds. For every R ≥ 4 and u, v ∈H1(BR) satisfying

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

−∇ ⋅ (DpL(∇u,x)) = 0 in BR,

−∇ ⋅ (DpL(∇v, x)) = 0 in BR,

∥∇u∥L2(BR) , ∥∇v∥L2(BR) ≤M,


∥∇u −∇v∥C0,α(B1/2) ≤XR

∥u − v∥L2(BR) .

Proof. Pick any σ ∈ (0, d) and let X be the maximum of the random variables inTheorems 1.2 and 2.3. Let u, v ∈H1(BR) be as in the statement of the propositionand observe that the difference w ∶= u − v satisfies the equation

∇ ⋅ (a∇w) = 0 in BR

with

a(x) ∶= ∫1

0D2pL(t∇u(x) + (1 − t)∇v(x), x)dt


which satisfy, by (6.1) and the previous lemma,

[a]C0,γ(β∧γ′)(B1) ≤Mγ0 +CM0 ([∇u]C0,β∧γ′(B1) + [∇v]C0,β∧γ′(B1))

γ

≤Mγ0 +CM0 (X

d2M)

γ

≤ CM0 (Xd2M)

γ.

The Schauder estimates imply that

∥∇w∥L∞(B1/2) + [∇w]C0,γ(β∧γ′)(B1/2) ≤ C [a]d

2γ(β∧γ′)C0,γ(β∧γ′)(B1)

∥∇w∥L2(B1) .

By Theorem 1.2,

∥∇w∥L2(B1) ≤ CXd2 ∥∇w∥L2(BX ) ≤ CX

d2R−1 ∥w∥L2(BR) .

Putting these together, we find that

∥∇w∥L∞(B1/2) + [∇w]C0,γ(β∧γ′)(B1/2) ≤ C (M1γ

0 Xd2M)

d2(β∧γ′)

X d2R−1 ∥w∥L2(BR) .

Allowing the constant C to depend on (γ, γ′,M0,M,data), we obtain, for a largeexponent q(γ′,data) ∈ (1,∞),

∥∇w∥L∞(B1/2) + [∇w]C0,γ(β∧γ′)(B1/2) ≤ CXqR−1 ∥w∥L2(BR) .

This completes the proof.

We now give the proof of Theorem 1.4.

Proof of Theorem 1.4. Let φξ denote the first-order corrector for the nonlinearequation with slope ξ, and ψξ,η the first-order corrector for the linearized equationaround x↦ x + φξ(x) with slope η. In other words, for every ξ, η ∈ Rd, the gradientfields ∇φξ and ∇ψξ,η are Zd–stationary, have mean zero and satisfy

(6.3) −∇ ⋅ (DpL(ξ +∇φξ(x), x)) = 0 in Rd

and

(6.4) −∇ ⋅ (D2pL(ξ +∇φξ(x), x) (η +∇ψξ,η(x))) = 0 in Rd.

One can see from the proof of Proposition 2.5 or alternatively, by differentiatingthe equation for φξ in ξ, that

∂ξi∇φξ = ∇ψξ,ei .

We also have the formula

D2L(ξ)η = E [∫[0,1]d

D2pL (ξ +∇φξ(x), x) (η +∇ψξ,η(x)) dx] ,

together with the estimates

(6.5) E [∥∇φξ −∇φξ′∥2L2([0,1]d)] ≤ C ∣ξ − ξ′∣2 and E [∥∇ψξ,η∥2L2([0,1]d)] ≤ C ∣η∣2.


Thus

∣(D2L(ξ) −D2L(ξ′)) η∣

≤ E [∫[0,1]d

∣D2pL (ξ +∇φξ(x), x) −D2

pL (ξ′ +∇φξ′(x), x)∣ ∣η +∇ψξ,η(x)∣ dx]

+E [∫[0,1]d

∣D2pL (ξ′ +∇φξ′(x), x)∣ ∣∇ψξ,η(x) −∇ψξ′,η(x)∣ dx] .

To estimate the first term on the right side, we observe that, by (6.1), Holder’sinequality and (6.5),

E [∫[0,1]d

∣D2pL (ξ +∇φξ(x), x) −D2

pL (ξ′ +∇φξ′(x), x)∣ ∣η +∇ψξ,η(x)∣ dx]

≤ CE [∫[0,1]d

∣(ξ +∇φξ(x)) − (ξ′ +∇φξ′(x))∣γ ∣η +∇ψξ,η(x)∣ dx]

≤ CE [∫[0,1]d

(∣ξ − ξ′∣ + ∣∇φξ′(x) −∇φξ(x)∣)2 dx]γ2

×E [∫[0,1]d

∣η +∇ψξ,η(x)∣2 dx]12

≤ C ∣ξ − ξ′∣γ ∣η∣ .To estimate the other term, we have that

E [∫[0,1]d

∣D2pL (ξ′ +∇φξ′(x), x)∣ ∣∇ψξ,η(x) −∇ψξ′,η(x)∣ dx]

≤ CE [∫[0,1]d

∣∇ψξ,η(x) −∇ψξ′,η(x)∣2 dx]12

.

Therefore to complete the proof, it suffices to show that

(6.6) E [∫[0,1]d

∣∇ψξ,η(x) −∇ψξ′,η(x)∣2 dx] ≤ C ∣η∣2 ∣ξ − ξ′∣2γ .

To prove (6.6), we argue similarly as in Lemma 2.4. Let ζ ∶= ψξ,η − ψξ′,η so that ∇ζis a Zd–stationary gradient field satisfying

−∇ ⋅ (D2pL(ξ +∇φξ(x), x)∇ζ) = ∇ ⋅ f , in Rd

where f is defined by

f = (D2pL(ξ +∇φξ(x), x) −D2

pL(ξ′ +∇φξ′(x), x))∇ψξ′,η.We have that

E [∫[0,1]d

∣∇ζ(x)∣2 dx]

≤ E [∫[0,1]d

∣f(x)∣2 dx]

≤ E [∫[0,1]d

∣(ξ +∇φξ(x)) − (ξ′ +∇φξ′(x))∣2γ ∣∇ψξ′,η(x)∣2 dx]

≤ E [∫[0,1]d

∣(ξ +∇φξ(x)) − (ξ′ +∇φξ′(x))∣2γ(2+δ)

δ ]δ

2+δE [∫

[0,1]d∣∇ψξ′,η ∣2+δ dx]

22+δ.


By Proposition 6.2 and the argument for (1.9),

E [∫[0,1]d

∣(ξ +∇φξ(x)) − (ξ′ +∇φξ′(x))∣2γ(2+δ)

δ ]δ

2+δ≤ C ∣ξ − ξ′∣2γ .

By the Meyers estimate and Proposition 4.3, noticing that the latter impliesbounds on ∇ψξ′,η analogous to (1.9) by the same argument, we get, for X as inProposition 4.3,

E [∫[0,1]d

∣∇ψξ′,η ∣2+δ dx]2

2+δ≤ E

⎡⎢⎢⎢⎣(∫

[−1,2]d∣∇ψξ′,η ∣2 dx)

2+δ2⎤⎥⎥⎥⎦

22+δ

≤ C ∣η∣2E [X d2(2+δ)] ≤ C ∣η∣2 .

Combining the previous three displays yields (6.6) and completes the proof.

Appendix A. Regularity for constant Lagrangians

In this appendix we recall some classical regularity estimates for solutions ofconstant-coefficient equations, summarized in the following proposition. Versions ofthe results stated here can be found in books such as [14], but for our purposes inthis paper we require somewhat sharper, more quantitative statements which wecould not find in the literature. For this reason we give complete proofs here.

Proposition A.1 (C1,β regularity for differences). Fix γ ∈ (0,1), R ∈ (0,∞), andK,M ∈ (0,∞). Let F ∶ Rd → R be a Lagrangian satisfying [F ]C2,γ ≤ K and, for everyp ∈ Rd,

(A.1) Id ≤D2pF (p) ≤ ΛId.

Suppose that u and v are local F -minimizers in BR. Then u, v ∈ C2,γ(BR/2) andthe following statements hold.

There exist θ(d,Λ) ∈ (0, 1) and C(d,Λ) <∞ such that, for every x, y ∈ BR/2,

(A.2) ∣∇u(x) −∇u(y)∣ ≤ C (∣x − y∣R

)θ

1

Rinf`∈P1

∥u − `∥L2(BR) .

There exists a constant C(K, γ, d,Λ) <∞ and η(d,Λ) <∞ such that

(A.3)1

Rinf`∈P1

∥u − `∥L2(BR) ≤M

implies

(A.4) R ∥∇2u∥L∞(BR/2)

≤ C (M ∨ 1)ηM

and, for every x, y ∈ BR/4,

(A.5) R ∣∇2u(x) −∇2u(y)∣

≤ C (∣x − y∣R

)γ

( infq∈P2

∥∇u −∇q∥L2(BR/2) + (M ∨ 1)ηM1+γ) .


Suppose that

(A.6)1

Rinf`∈P1

∥u − `∥L2(BR) +1

Rinf`∈P1

∥v − `∥L2(BR) ≤M.

Then there exist constants C(K, γ, d,Λ) <∞ and η(d,Λ) <∞ such that, forany s ∈ (0, R2 ], we have both

(A.7) ∥∇u −∇v∥L∞(BR/2) ≤ C (M ∨ 1)η 1

R∥u − v − (u − v)BR∥L2(BR)

and, for r ∈ (0, s],

(A.8) inf`∈P1

∥u − v − `∥L2(Br) ≤ C (rs)2

inf`∈P1

∥u − v − `∥L2(Bs)

+C (M ∨ 1)η (sr)d2

( sR

)1+γ

Mγ ∥u − v − (u − v)BR∥L2(BR) .

Remark A.2. For the estimate (A.2) it is enough to assume (A.1). In fact, issuffices to assume that F ∈ C1 and, for every p1, p2 ∈ Rd,

∣p1 − p2∣2 ≤ (DpF (p1) −DpF (p2)) ⋅ (p1 − p2) ≤ Λ∣p1 − p2∣2.

Proof of Proposition A.1. We divide the proof into five steps. In the first step wewill prove (A.2), and in the second both (A.6). In Step 3 we will prove (A.4)using (A.7), and in Step 4 we will show (A.8). Finally, in the last step, we willprove (A.5) using (A.7) and (A.8).

Step 1. We first prove (A.2). Due to the first variation and smoothness of F , usatisfies the equation

∇ ⋅ (D2pF (∇u)∇2u) = 0.

By (A.1), we may apply the classical De Giorgi-Nash-Moser theory to obtain that∇u ∈ C0,θ and, in particular,

supx,y∈Br

∣∇u(x) −∇u(y)∣ ≤ C ( rR

)θ

∥∇u − (∇u)BR/2∥L2(BR/2).

Furthermore, for all ` ∈ P1 we have

∇ ⋅ (DpF (∇u) −DpF (∇`)) = 0,

and again (A.1) provides us a Caccioppoli inequality

(A.9) ∥∇u − (∇u)BR/2∥L2(BR/2)= inf`∈P1

∥∇u −∇`∥L2(BR/2) ≤C

Rinf`∈P1

∥u − `∥L2(BR) .

Combining the estimates proves (A.2).

Step 2. We next prove (A.7). The difference u − v satisfies the equation

∇ ⋅ (b(x)∇(u − v)) = 0,

where

b(x) ∶= ∫1

0D2pF (t∇u(x) + (1 − t)∇v(x)) dt.

Observe that, by (A.1), Id ≤ b(x) ≤ ΛId. By freezing the coefficients at the originwe get that

∇ ⋅ (b(0)∇(u − v)) = ∇ ⋅ ((b(0) − b(x))∇(u − v)) .


By the smoothness of F and (A.2) (applied for both u and v), we get, for allx ∈ BR/2,

∣b(0) − b(x)∣ ≤ CK(∣x∣R

)θγ

Mγ

Fix s ∈ (0, R2 ) and let ws ∈ u − v +H10(Bs) solve ∇ ⋅ (b(0)∇ws) = 0 in Bs. Then

∥∇(u − v) −∇ws∥L2(Bs) ≤ C ( sR

)θγ

Mγ ∥∇(u − v)∥L2(Bs) .

Since ws satisfies the equation with constant coefficients, we find a constantC(K, d,Λ) such that, for any r ∈ (0,1),

inf`∈P1

∥ws − `∥L2(Br) ≤ C (rs)2

inf`∈P1

∥ws − `∥L2(Bs) .

By Poincare’s inequality, the Caccioppoli inequality for u − v, and the triangleinequality, the above two displays imply

inf`∈P1

∥u − v − `∥L2(Br) ≤ C (rs)2

inf`∈P1

∥u − v − `∥L2(Bs)

+C (sr)d2

( rR

)θγ

Mγ ∥u − v − (u − v)Bs∥L2(Bs) .

Setting

E1(s) ∶= (Rs)1+θγ

inf`∈P1

∥u − v − `∥L2(Bs) , E0(s) ∶=R

s∥u − v − (u − v)Bs∥L2(Bs) ,

we find a small constant ε(K, d,Λ) ∈ (0,1) such that

E1(εs) ≤1

2E1(s) +CMγE0(s)

This implies, after taking supremum and reabsorbing, that

(A.10) sups∈(0,R)

E1(s) ≤ CE1(R) +CMγ sups∈(0,R)

E0(s).

Notice that the term on the right is finite since both u and v are Lipschitz continuous.Letting `s be the affine function realizing the infimum in the definition of E1(s),we obtain by the triangle inequality that

∣∇`s −∇`2s∣ ≤ C1

R( sR

)θγ

(E1(s) +E1(2s))

≤ C 1

R( sR

)θγ

(E1(R) +Mγ sups∈(0,R)

E0(s)) .

After telescoping, we deduce, for s ∈ (0,R), the estimate

supr∈(0,s)

∣∇`r∣ ≤ ∣∇`s∣ +C1

R( sR

)θγ

(E1(R) +Mγ sups∈(0,R)

E0(s)) .


Using the trivial estimate ∣∇`s∣ ≤ CR−1E0(s), we also get, for any s ∈ (0,R],

supr∈(0,s]

E0(r) ≤ R supr∈(0,s]

∣∇`r∣ + ( sR

)θγ

supr∈(0,s]

E1(s)

≤ CE0(s) +C ( sR

)θγ

(E1(R) +Mγ sups∈(0,R]

E0(s))

Choosing, in particular, s = s∗, where

s∗ ∶= R [2CMγ ∨ 1]−1θγ ,

and using E1(R) ≤ E0(R) and

supr∈(s∗,R]

E0(r) ≤ (Rs∗

)1+ d

2

E0(R) = (2CMγ ∨ 1)d+22θγ E0(R),

we can reabsorb supr∈(0,s∗)E0(r) from the right to get

sups∈(0,R)

E0(s) ≤ C (M ∨ 1)d+22θ E0(R).

Thus, (A.7) follows.

Step 3. We prove (A.4). For fixed h ∈ Rd with ∣h∣ ≪ R, we set v(x) ∶= u(x + h),which is still F -minimizer in BR−∣h∣. Moreover, (A.3) gives, for small enough ∣h∣,

supy∈BR/2−∣h∣

( 2

Rinf`∈P1

∥u − `∥L2(BR/2(y)) +2

Rinf`∈P1

∥v − `∥L2(BR/2(y))) ≤ 2d2+1M.

We can thus apply (A.7) for u − v in BR/2−∣h∣(y). Dividing the resulting inequalitywith ∣h∣ and recalling that u ∈ H2

loc(BR), we may send ∣h∣ → 0 and obtain, fory ∈ BR/2,

1

s∥∇u − (∇u)Bs(y)∥L2(Bs(y))

≤ C (M ∨ 1)d+22θ

1

R∥∇u − (∇u)BR/4(y)∥L2(BR/4(y))

.

Now (A.4) follows by the previous inequality and an application of the Caccioppoliestimate, for the differentiated equation ∇ ⋅ (D2

pF (∇u)∇2u) = 0, giving

∥∇2u∥L2(Bs/2(y))

≤ Cs

∥∇u − (∇u)Bs∥L2(Bs(y))

≤ C (M ∨ 1)d+22θ

1

R∥∇u − (∇u)BR/4(y)∥L2(BR/4(y))

,

after letting s→ 0 and applying the Caccioppoli estimate (A.9).

Step 4. We then prove (A.8). We return to Step 2, and observe that by using (A.4)we actually get an improved estimate

∣b(0) − b(x)∣ ≤ C (M ∨ 1)γη (∣x∣R

)γ

Mγ,


and using this as in the Step 2, we also get

inf`∈P1

∥u − v − `∥L2(Br) ≤ C (rs)2

inf`∈P1

∥u − v − `∥L2(Bs)

+C (M ∨ 1)γη (sr)d2

( rR

)γ

Mγ ∥u − v − (u − v)Bs∥L2(Bs) .

Setting this time

E1(s) ∶= (Rs)1+γ

inf`∈P1

∥u − v − `∥L2(Bs) ,

we find a small constant ε(K, d,Λ) ∈ (0,1) such that

E1(εs) ≤1

2E1(s) +C (M ∨ 1)γηMγE0(s)

By (A.7), E0(s) ≤ C (M ∨ 1)ηE0(R), and thus, after reabsorption, we deduce (A.8).

Step 5. We finally prove (A.5). As in Step 3, (A.8) and (A.9) yield that, forr ≤ s ≤ R

2 ,

inf`∈P1

∥∇u − `∥L2(Br) ≤ C (rs)2

inf`∈P1

∥∇u − `∥L2(Bs) +C (M ∨ 1)γη (sr)d2

( sR

)1+γ

M1+γ.

Setting

E2(s) ∶= (Rs)1+γ

inf`∈P1

∥∇u − `∥L2(Bs) ,

we find ε(K, γ, d,Λ) ∈ (0,1) such that

E2(εs) ≤1

2E2(s) +C (M ∨ 1)γηM1+γ.

Thus, we get after reabsorption that

sups∈(0,R

2)E2(s) ≤ C (E2 (

R

2) + (M ∨ 1)γηM1+γ) .

Letting `s be the affine function realizing the infimum in E2(s), we have analogouslyto Step 3 that

∣∇`2s −∇`s∣ ≤C

R( sR

)γ

(E2 (R

2) + (M ∨ 1)γηM1+γ) .

Thus ∇`ss is a Cauchy sequence and there exists an affine function ` such that

∣∇`s −∇`∣ ≤C

R( sR

)γ

(E2 (R

2) + (M ∨ 1)γηM1+γ) .

A similar argument shows also that

∣`s(0) − `(0)∣ ≤ C ( sR

)1+γ

(E2 (R

2) + (M ∨ 1)γηM1+γ) .

As a consequence we obtain that that for this ` we have, for s ∈ (0, R2 ), that

∥∇u − `∥L2(Bs)

≤ C ( sR

)1+γ

( inf`∈P1

∥∇u − `∥L2(BR/2) + (M ∨ 1)γηM1+γ) .


It is straightforward to show that translating this estimate implies that u ∈ C2,γ

and that `(x) = ∇q(x), where

q(x) ∶= u(0) +∇u(0) ⋅ x + 1

2x ⋅ ∇2u(0)x,

and the previous inequality reads as

(A.11) ∥∇u −∇q∥L2(Bs) ≤ C ( sR

)1+γ

( inf`∈P1

∥∇u − `∥L2(BR/2) + (M ∨ 1)γηM1+γ) .

Furthermore, we have the equation

−∇ ⋅ (D2pF (∇u)∇ (∇u −∇q)) = −∇ ⋅ ((D2

pF (∇u) −D2pF (∇u(0)))∇2u(0)) .

The term on the right can be controlled as

∣(D2pF (∇u(x)) −D2

pF (∇u(0)))∇2u(0)∣ ≤ C ∣∇u(x) −∇u(0)∣γ ∣∇2u(0)∣ ,

and hence, by (A.4),

∣(D2pF (∇u(x)) −D2

pF (∇u(0)))∇2u(0)∣ ≤ CR

(∣x∣R

)γ

(M ∨ 1)η(1+γ)M1+γ.

We thus get a Caccioppoli estimate, using also (A.11),

∥∇2u −∇2u(0)∥L2(Bs/2)

≤ Cs

∥∇u −∇q∥L2(Bs) +C

R( sR

)γ

(M ∨ 1)η(1+γ)M1+γ

≤ CR

( sR

)γ

( inf`∈P1

∥∇u − `∥L2(BR/2) + (M ∨ 1)η(1+γ)M1+γ) .

By translating the previous estimate and applying the triangle inequality yields,for any x, y ∈ BR/2 with ∣x − y∣ ≤ R

32 ,

∣∇2u(x) −∇2u(y)∣≤ 2d ∥∇2u −∇2u(x)∥

L2(B4∣x−y∣(x))+ 2d ∥∇2u −∇2u(y)∥

L2(B4∣x−y∣(y))

≤ CR

(∣x − y∣R

)γ

( inf`∈P1

∥∇u − `∥L2(BR/2) + (M ∨ 1)η(1+γ)M1+γ) ,

which is (A.5) in the case ∣x − y∣ ≤ R32 after noticing that we trivially have

inf`∈P1

∥∇u − `∥L2(BR/2) ≤ infq∈P2

∥∇u −∇q∥L2(BR/2) .

If, on the other hand, ∣x − y∣ > R32 , the estimate follows easily by setting xj = j

32x

and yj = j32y, j ∈ 0,1, . . . ,32, and applying the previous estimate with x = xj and

y = xj+1, and similarly for yj and yj+1, and we thus deduce (A.5) also in this case.The proof is complete.

Proposition A.3 (C1,β Schauder estimate). Fix γ′ ∈ (0, 1] and K ∈ (0,∞). Supposethat F ∶ Rd ×B1 → R is a Lagrangian satisfying, for every p ∈ Rd,

(A.12) [DpF (p, ⋅)

1 + ∣p∣]C0,γ′(B1)

≤ K

and

(A.13) Id ≤D2pF (p, ⋅) ≤ ΛId.


There exist constants β(K, γ′, d,Λ) ∈ (0, γ′] and C(K, γ′, d,Λ) such that, for everylocal F -minimizers u ∈H1(B1),

(A.14) ∥∇u∥L∞(B1/2) + [∇u]C0,β(B1/2) ≤ C (1 + ∥∇u∥L2(B1)) .

Proof of Proposition A.3. Let ur be the minimizer of

minv∈u+H1

0(Br)⨏BrF (∇v(x),0)dx,

so that, by the first variation,

−∇ ⋅ (DpF (∇u,0) −DpF (∇ur,0)) = ∇ ⋅ (DpF (∇u,x) −DpF (∇u,0))

in Br. From this, together with (A.12), we obtain

∥∇u −∇ur∥L2(Br) ≤ Crγ′ (1 + ∥∇u∥L2(Br))

Then, for θ ∈ (0,1), we get by (A.2) (recalling Remark A.2) and the triangleinequality that

(A.15) ∥∇u − (∇u)Bθr∥L2(Bθr)

≤ Cθβ ∥∇u − (∇u)Br∥L2(Br)+Cθ− d2 rγ′ (1 + ∥∇u∥L2(Br)) .

Similarly to Step 2 of the proof of Proposition A.1 this yields, for β′ ∶= 12 (β ∧ γ′),

that

supr∈(0,1)

(r−β′∥∇u − (∇u)Br∥L2(Br) + ∥∇u∥L2(Br)) ≤ C (1 + ∥∇u∥L2(B1)) .

By translating this estimate, we conclude by an iteration argument that

∥∇u∥L∞(B1/2) + [∇u]C0,β′ ≤ C (1 + ∥∇u∥L2(B1)) .

Going back to (A.15), we see that

(A.16) ∥∇u − (∇u)Bθr∥L2(Bθr)

≤ Cθβ ∥∇u − (∇u)Br∥L2(Br)+Cθ− d2 rγ′ (1 + ∥∇u∥L2(B1)) ,

and may repeat the argument to obtain Holder exponent β′ ∈ (0, β) ∩ [0, γ′]. Thuswe conclude with (A.14) after relabelling β′.

Appendix B. Homogenization estimates

In this appendix we derive the estimate (3.18) from estimates in [5]. In fact, (3.18)is an immediate consequence of the triangle inequality, a basic energy estimate andthe following stronger form of (3.17): there exist α(d,Λ) > 0 and C(K, q, d,Λ) <∞and a random variable X satisfying X ≤ O1(C) such that, for every M,N ∈ N withM ≤ N ,

(B.1) supξ∈B

K3Mq

(1 + ∣ξ∣)−23−d(N−M) ∑z∈3MZd∩◻N

3−2M ∥v(⋅, z +◻M , ξ) − `ξ∥2L2(z+◻M )

≤ C3−Mα(d−σ) +X3−σ′N .


What [5, Corollary 3.5] gives us is the following bound, valid for every z ∈ 3MZd:

(B.2) supξ∈B

K3Mq

(1 + ∣ξ∣)−23−2M ∥v(⋅, z +◻M , ξ) − `ξ∥2L2(z+◻M )

≤ C3−Mα(d−σ) +Xz3−σ′M ,

where the random variables Xzz∈3MZd are identically distributed, satisfy thebound Xz ≤ O1(C), and each Xz is measurable with respect to F(z +◻M), that is,it depends on the restriction of the Lagrangian to z +◻M . In particular, for everyz, z′ ∈ 3MZd, the random variables Xz and Xz′ are independent provided that thecubes z +◻M and z′ +◻M are not adjacent. To obtain (B.1), it suffices to prove

3−d(N−M) ∑z∈3MZd∩◻N

Xz ≤ 2E [X0] +O1 (3−d(N−M)) .

This is a very crude large deviation-type estimate that is relatively straightforwardto derive. It can be obtained for instance from [5, Lemma 2.14] or its proof.

Acknowledgments. SA was partially supported by the National Science Foun-dation through grant DMS-1700329. SF was partially supported by NSF grantsDMS-1700329 and DMS-1311833. T.K. was supported by the Academy of Finlandand the European Research Council (ERC) under the European Union’s Horizon2020 research and innovation programme (grant agreement No 818437).

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(S. Armstrong) Courant Institute of Mathematical Sciences, New York University,251 Mercer St., New York, NY 10012

E-mail address: [email protected]

(S. J. Ferguson) Courant Institute of Mathematical Sciences, New York University,251 Mercer St., New York, NY 10012

E-mail address: [email protected]

(T. Kuusi) Department of Mathematics and Statistics, University of HelsinkiE-mail address: [email protected]

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