Uniform integrability in periodic homogenization of fully nonlinear elliptic equations

This paper is devoted to the study of uniform W1,npn-p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{1,\frac{np}{n-p}}$$\end{document}- and W2,p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{2,p}$$\end{document}-estimates for periodic homogenization problems of fully nonlinear elliptic equations. We establish sharp, global, large-scale estimates under the Dirichlet boundary conditions. The main novelty of this paper can be found in the characterization of the size of the “effective” Hessian and gradient of viscosity solutions to homogenization problems. Moreover, the large-scale estimates work in a large class of non-convex problems. It should be stressed that our global estimates are new even for the standard problems without homogenization.


Introduction
This paper is devoted to the study of uniform integrability of the Hessian and gradient of viscosity solutions u ε ∈ C( ) to fully nonlinear periodic homogenization problems, of the type (1.0.1) In [7], the size of the "Hessian" of a continuous function at a point is characterized by the smallest opening of touching convex and concave paraboloids at that point.Here we extend this concept by allowing the touching to take place in a neighborhood of size ε around the reference point, and denote this quantity by H ε .Similarly, we characterize the size of the "gradient" by replacing paraboloids with cones, and denote it by G ε .See Definition 2.2 for more precise definitions for these quantities.Especially given u ∈ C( ), we designed H ε (u) (G ε (u)) in such a way that where 2 εe u(x) := (u(x + εe) + u(x − εe) − 2u(x))/ε 2 (resp., εe u(x) := (u(x + εe) − u(x))/ε) is the second (resp., first) ε-differential quotient in direction e ∈ ∂ B 1 ; see Remark 2.3.From the above inequalities, the L p -estimates of H ε (u) (G ε (u)) yield the same estimates for 2  εe u (resp.εe u) for all e ∈ ∂ B 1 .Thanks to this relation, denoting by u ε the solution to our homogenization problem (1.0.1), the L p -estimates of H ε (u ε ) (G ε (u ε )) can be understood as the so-called large-scale W 2, p -(resp., W 1, p -)estimates of u ε .
As another important remark, from our definition, H ε → H (G ε → G ) as ε → 0, where H (resp., G ) controls the standard Hessian (resp., gradient).The quantity H is the classical one introduced in [7].On the other hand, the quantity G for the gradient appeared here, and also independently in a very recent paper [25], for the first time.
The first main result of this paper is the uniform integrability of H ε (u ε ); see Definition 2.5 for domains of W 2, p -type.Theorem 1.1 (W 2, p -estimates) Let F ∈ C(S n × R n ) be a functional satisfying (2.0.1)-(2.0.4), ⊂ R n be a bounded domain, f ∈ C( ) ∩ L p ( ) for some finite p > p 0 , g ∈ C(∂ ) ∩ W 2, p ( ) and u ε ∈ C( ) be a viscosity solution to (1.0.1).Suppose either of the following: is a W 2,n -type domain, and p 0 < p < n; (ii) is a W 2,n+σ -type domain for some σ > 0, and p = n; (iii) is a W 2, p -type domain and p > n, all with size (δ, R).
Let us provide some motivation for the assumption (2.0.4).Roughly speaking, the assumption says that the effective problem F(D 2 v) = 0 has interior V M O-estimates for the Hessian of its (viscosity) solutions.Note that u ε converges to its effective profile ū, only, uniformly.This is too weak to ensure any closeness between their Hessian.Under the V M O-condition on D 2 ū, however, D 2 ū satisfies a small B M O-condition at an intermediate scale.We observe that P ± (D 2 (u ε − ū − ε 2 w( • ε ))) = o (1), at that scale, with w being an interior corrector.This is one of the key observations in Lemma 6.7, which is an approximation lemma for the interior W 2, p -estimates.
It should also be addressed that due to [16,Theorem 3.4], there is a large class of nonconvex functionals satisfying (2.0.4).More specifically, the result implies the following: if there exists a functional F * : S n × R n → R, which is convex in the first argument and satisfies (2.0.1)-(2.0.3), such that |(F − F * )(P, y) − (F − F * )(Q, y)| ≤ θ |P − Q| for all P, Q ∈ S n and all y ∈ R n , for some small constant θ , then the effective functional F satisfies (2.0.4).It is also noteworthy that unless the governing functional is continuously differentiable [15], (2.0.4) is a strictly relaxed assumption than assuming that F(D 2 ū) = 0 has interior C 2,α -estimates.For some further development in interior W 2, p -estimates for standard fully nonlinear problems, see e.g., [26].
We remark that throughout this paper, we do not assume continuous differentiability of F (or F).In fact, our result on the uniform L p -estimate for H ε (u ε ) only requires F to be continuous.Here we encounter another subtle issue that arises from the homogenization of L p -viscosity solutions.It is worth mentioning that homogenization of viscosity solutions has not yet been justified for equations with measurable ingredients.For the moment, the author is not sure whether the measurable ingredients would be homogenized either.We use the continuity of F (as well as the datum f ) to circumvent this issue.
The above estimates are sharp not only in terms of the regularity of the data, but also of the regularity of the boundary layer.The major challenge here arises from the fact that boundary flattening maps destroy the pattern of the rapid oscillation.For this reason, our analysis is quite different from, and in fact more complicated than, the argument for standard problems, c.f. [30].
As a matter of fact, the boundary estimates for the case p 0 < p < n are even new in the context of standard problems.The analysis is based on the following sharp W Then there exists a constant δ 0 ∈ (0, 1), depending only on n, λ, , R and p, such that if δ ≤ δ 0 , then G ε (u ε ) ∈ L np n− p ( ε ), and where C depends only on n, λ, , δ 0 , R and p.
To the best of author's knowledge, the closest result under the framework of standard problems is [11,Theorem 1.4], where interior gradient estimates are established in the Lorentz space.Their result shows that f ∈ L p,γ implies |Du| ∈ L np/(n− p),γ loc for any p ∈ ( p 0 , n) and any γ > 0. As for estimates for subcritical Sobolev exponents, the interior and boundary estimates (for C 2 -domains and C 1,α -data on boundaries) are obtained in [29] and respectively [30].
Our analysis is based on the decay estimate of the set of large "gradient", in the spirit of Caffarelli's approach in [7].We present a parallel study for the set of large gradient by replacing the touching paraboloids of the Hessian with cones.The proof relies on the general maximum principle as well as an elementary observation that the slope of supporting hyperplanes for convex envelopes to viscosity solutions in the Pucci class can be universally bounded from below.Recently, in the framework of standard problems (without homogenization), It is worthwhile to mention that the above estimates for the uniform integrability of the gradient are sharp in terms of the data, and that the domains are only required to be Reifenberg flat.Moreover, as a byproduct via the Sobolev embedding theorem, we obtain a uniform interior C 0,2−n/ p -estimate, which is rather well-understood in the setting of linear homogenization problems [2] and was proved in the setting of standard fully nonlinear problems [27].Nevertheless, our uniform C 0,2−n/ p -estimates are new in the framework of fully nonlinear homogenization problems.
Remark 1. 3 A uniform L p -estimate for the full Hessian (gradient), H (u ε ) (resp., G (u ε )), can be obtained under suitable hypotheses that ensure the regularity in small-scales.The passage from Theorem 1.1 (resp., 1.2) to the full estimates is by now standard, whence is omitted in this paper.
Let us briefly summarize the recent development of uniform estimates in the homogenization theory.Needless to say, the study has gained its interests due to a series of papers by Avellaneda and Lin.In particular, W 1, p -estimates are established in [3] for linear divergencetype equations, based on the study of Green functions.Later in [9], Caffarelli and Peral proved W 1, p -estimates for nonlinear divergence-type equations, via Calderón-Zygmund cube decomposition argument.In [24], Melcher and Schweizer proved the estimates via a more direct approach, based on the observation that ε-difference quotients solve the same class of equations.We would also like to mention [19], where uniform integrability estimates are established for nonlinear systems in divergence form.More recently, Byun and Jang proved, in [4], uniform W 1, p -estimates for linear divergence-type systems, under small B M O-condition on the periodically oscillating operators, up to Reifenberg flat domains.Some sharp "large-scale" estimates for linear divergence-type equations, without any regularity assumption on the governing operators can be found in [28].All the above results are concerned with periodic homogenization of solutions to either interior problems or Dirichlet problems.As for Neumann problems, some important sharp estimates can be found in [17].There is also a large amount of literature concerning uniform pointwise estimates for random homogenization, for which we would like to refer readers to a recent book [1] and the references therein.
Most of the aforementioned works are concerned with weak solutions to divergence-type problems.Uniform estimates for viscosity solutions to non-divergence type equations was done only recently in a collaboration [18] by Lee and the author, where pointwise C 1,α -and C 1,1 -estimates are proved for a class of non-convex functionals.The uniform integrability estimates established in this paper are new, even for linear equations in non-divergence form.
The paper is organized as follows.In the next section, we collect the notation, main assumptions and some preliminaries.Section 3 is devoted to several technical tools used in the subsequent analysis, yet of their own independent interests.In Sect.4, we study universal decay estimates for the set of large Hessian and gradient that will play an important role in the subsequent analysis.In Sect.5.1, we establish the uniform W 1, np n− p -estimates for both interior (Theorem 5.1) and boundaries (Theorem 6.1).Finally, in Sect.6.2, we prove the uniform W 2, p -estimates, whose proof is again divided into the case of interior (Theorem 5.4) and of boundaries (Theorem 6.5).

Preliminaries
We shall denote by B r (x) the n-dimensional ball centered at x with radius r , and by Q r (x) the n-dimensional cube centered at x with side-length r .By S δ (ν) we denote the slab centered at the origin with width δ in direction ν, i.e., S δ (ν) = {x ∈ R n : |x •ν| < δ}.By H t (ν) we denote the half-space in direction ν with the lowest level being t, i.e., H δ (ν) = {x ∈ R n : x • ν > t}.Also we shall write H 0 (ν) simply by H (ν).Moreover, by S n we denote the space of all symmetric (n × n)-matrices.
Throughout this paper, λ and will be fixed as some positive constants, with λ ≤ , and will also denote the lower and respectively upper ellipticity bound for the governing functional.By p 0 we shall denote Escauriaza's constant such that the generalized maximum principle holds for all p > p 0 ; note n 2 < p 0 < n and it depends only on n and the ellipticity bounds, λ and .For more details, we refer readers to [12] and [8].In addition, we denote by P − and P + the Pucci minimal and respectively maximal functional on S n , associated with ellipticity bounds λ and , such that where e i is the i-th eigenvalue of P.
Let ψ : (0, ∞) → (0, ∞) be a nondecreasing, strictly concave function such that ψ(0+) = 0, and be a domain in R n .We say where the supremum is taken over all balls B ⊂ .Given any g ∈ L 1 (R n ) with g ≥ 0, M(g) denotes the maximal function of g, i.e., where the supremum is taken over all balls B containing x.Given any g ∈ L 1 loc (R n ), I α (g) denotes the Riesz potential of g, i.e., with c α being a suitable normalization constant.
For definiteness, we shall assume that F ∈ C(S n × R n ) is a functional satisfying, for any P, Q ∈ S n and any y ∈ R n , (2.0.1) Under the first two assumptions, there exists a unique functional F : S n → R (the so-called effective functional), according to [13, Theorem 3.1], such that any limit ū ∈ C( ) of the sequence of viscosity solutions ) and ε > 0, under locally uniform convergence as ε → 0 is a viscosity solution to F(D 2 ū) = f in .We shall suppose that (2.0.4) where ψ : (0, ∞) → (0, ∞) is a concave, non-decreasing function satisfying ψ(0+) = 0 and κ > 0 is a fixed constant.More specifically, by (2.0.4), we indicate the following: given any ball B R ⊂ R n and any function Let us collect by now standard results regarding periodic homogenization for fully nonlinear problems in the following lemma.
and (2.0.2).Then there exists a unique functional F : S n → R, satisfying (2.0.1), such that for each P ∈ S n , F(P) is the unique constant for which there exists a viscosity solution to where c > 0 and α ∈ (0, 1) depend only on n, λ and .
Next, we introduce the set of large "Hessian", as well as the set of large "gradient", with room for errors of order ε 2 and respectively ε.These sets will play the main role throughout this paper, as our primary goal is to establish "large-scale" estimates.The set of large Hessian without any room for error has played a central role in the W 2. p -theory for standard fully nonlinear problems.Nevertheless, the set of large gradient seems to appear in this paper for the first time, in the literature.Needless to say, the sets with room for errors are entirely new, as far as the author is concerned.It should be stressed that one can generalize this concept to homogenization problems under various oscillating structures, such as quasi-periodic, almost-periodic or random environment.Definition 2.2 Let ⊂ R n be a bounded domain, ε ≥ 0, t > 0 and u ∈ C( ) be given.Let A ε t (u, ) (and L ε t (u, )) be defined as a subset of such that x 0 ∈ \A ε t (u, ) (resp., \L ε t (u, )) if and only if there exists a linear polynomial (resp., a constant a) for which and denote by H (u) (and G (u)) the function H 0 (u) (resp., G 0 (u)).
We remark that H ε (and G ε ) controls the second (resp.first) ε-differential quotients of u ε .Since the proof is essentially the same (and in fact shorter) as that of [6, Lemma 2.1], we shall only present the statement.
The following is the definition for the Reifenberg flat sets, which will appear in uniform boundary W 1, np n− p -estimates.Definition 2.4 Let be a domain, and U be a neighborhood of a point at ∂ .Set ∂ ∩ U is said to be (δ, R)-Reifenberg flat from exterior (or interior), if for any for some unit vector ν x 0 ,r ; here ν x 0 ,r may vary upon both x 0 and r .The set ∂ ∩ U is said to be (δ, R)-Reifenberg flat, if it is (δ, R)-Reifenberg flat from both exterior and interior.The domain is said to be Next, we define domains of W 2, p -type.Definition 2.5 Let p > 1 be a constant, be a domain, and U be a neighborhood of a point at ∂ .Set ∂ ∩U is said to be of W 2, p -type with size κ, if there exists a neighborhood We shall also need some covering lemmas.As for the analysis for interior estimates, we shall use the classical Calderón-Zygmund cube decomposition lemma, c.f. [14, Section 9.2] and [6, Lemma 4.1]: Lemma 2.6 (Calderón-Zygmund cube decomposition) Let A ⊂ Q 1 be a measurable set such that |A| ≤ η for some η ∈ (0, 1).Then there exists a finite collection F of cubes from the dyadic subdivision of Q 1 , such that |A ∩ Q| > η|Q| for all Q ∈ F , and |A ∩ Q| ≤ η| Q| for the predecessor of Q.
Let B be a measurable set such that A As for the boundary estimates, we shall utilize the Vitali-type covering lemma for Reifenberg flat domains [5,Theorem 2.8].We present its statement for the reader's convenience.

Some technical tools
Let us begin with an assertion that the difference between a viscosity solution and a viscosity sub-or super-solution belongs to the Pucci class in the viscosity sense.It is particularly important that one of two must be a solution.This assertion might be known for some experts.Still, we intend to present a proof because the assertion is not as simple as it sounds, apart from the fact that the author was not able to find a proof in the literature.It should be stressed that the assertion is yet to be known if we replace viscosity solution with viscosity super-or sub-solution (depending on what it is compared with).

Lemma 3.1 Let
⊂ R n be a bounded domain, and F ∈ C(S n × ) be a functional satisfying Proof Fix δ > 0, and denote δ = {x ∈ : dist (x, ∂ ) > δ}.Clearly, u, v ∈ C( δ ), and δ satisfies the uniform exterior sphere condition with radius at most δ −1 .Given any pair (τ, σ ) of real parameters such that 0 < σ < τ < δ, let v τ : → R be the sup-inf convolution of v over ; i.e., Such a regularization is by now considered standard.Among other important properties, we shall use the following, which can be found in [20, thm] that in , with c δ > 0 being a constant depending only on the sup-norm and the modulus of continuity of v over δ , and where F τ : S n × τ → R and g τ : τ → R are defined by F τ (P, x) = F(P, x − (τ − σ )Dv τ (x)) and g τ (x) = g(x − (τ − σ )Dv τ (x)).In particular, since τ |Dv τ | → 0 uniformly on , F τ → F locally uniformly in S n × δ and g τ → g uniformly on δ .Consider an auxiliary Dirichlet boundary value problem, in the L ∞ -viscosity sense, but then in the usual (C-)viscosity sense as f − g τ ∈ C( δ ); here, we refer to [8] the notion of L ∞ -viscosity solutions.Now letting τ → 0, and recalling that u τ → u, v τ → v and g τ → g uniformly on δ , we may conclude from the stability theory again that As δ > 0 was an arbitrary constant, the assertion of the lemma follows by sending δ → 0.
Let us close this section with a few results that are essentially due to [18], but extended so as to be adoptable in our subsequent analysis.We shall start with an interior L ∞ -approximation of viscosity solutions to periodic homogenization problems by those to the corresponding effective problems.The assertion is a slight generalization of [18,Lemma 3.1,4.2],which was established for bounded data.Here we shall extend the result to L p -integrable data.
Proof The assertion for f = 0 is a direct consequence of [18,Lemma 3.1,4.2].As for the general case, we consider an auxiliary boundary value problem, which admits a unique viscosity solution.By the general maximum principle, one may easily construct a barrier function to verify that for some c > 0 depending only on n, λ, and p.This combined with the assertion with f = 0 yields the conclusion.
With the above lemma at hand, we can extend the uniform pointwise C 1,α -estimates for fully nonlinear homogenization problems, established in [18], to a more general setting.[18,Theorem 4.1]) Let F ∈ C(S n × R n ) be a functional satisfying (2.0.1)-(2.0.3), ⊂ R n be a bounded domain, f ∈ C( ) ∩ L p ( ) for some p > p 0 , and u ε ∈ C( ) be a viscosity solution to
Proof This assertion is proved for the case f ∈ L ∞ in [18,Theorem 4.1].However, the same proof works equally well for any point x 0 ∈ featuring With the latter observation, the iteration argument in [18,Lemma 4.3] works, without any notable modification, once we invoke Lemma 3.2 as the approximation lemma, in place of [18,Lemma 4.2], in the proof there.Let us remark that the iteration technique for standard problems is by now understood as standard, c.f. [29, Remark 2.5].Therefore, we shall not repeat the detail here.

there exists a linear polynomial ε
x 0 such that for any x ∈ ∩ U , where C > 0 depends only on n, λ, , κ, α, p and diam (U ).

Remark 3.5
The reason that we state the above lemma for the case p > n is only because the other case, i.e., p 0 < p ≤ n, holds in a much general setting, namely for standard problems in the Pucci class, c.f. [21, Theorem 1.6]; of course, in the latter case, one needs to replace α p with α and f L p ( ∩U ) with ((

Universal decay estimates
This section is devoted to a global, universal decay estimate of the measure of the set for large "gradient" and "Hessian" of viscosity solutions to fully nonlinear equations, up to Reifenberg flat boundaries.As surprising as it may sound, our estimates would not see the boundary value of solutions, as long as the solutions are bounded up to the boundaries.Roughly speaking, this is because of the fact that the boundary layer, as a Reifenberg flat set, can be trapped in between a thin slab, which already has small measure and thus can be neglected.Of course, at a cost, the decay rate we establish here could be extremely slow, yet universal.Let us remark that such a global estimate is hinted in [30], which proves global universal decay estimate for Hessians, up to flat boundaries.

Set of large gradient
Let us begin with estimates for the set of large gradient.Throughout this section, given any u ∈ C( ) and t > 0, L t (u, ) is the subset of such that x 0 ∈ \ L t (u, ) if and only if there exists a constant a for which u , where C > 1, δ > 0 and μ > 0 depend only on n, λ, and p.
We shall split our analysis into two parts, each concerning interior and respectively boundary layer.Let us begin with the interior case first.As the analysis below will be of local character, we shall confine ourselves to the case = B 4 √ n and = Q 1 .The following lemma is the gradient-counterpart of [6,Lemma 7.5].
where m > 1, δ 0 > 0 and σ ∈ (0, 1) are constants depending at most on n, λ, and p. Proof As in the proof of [6, Lemma 7.5], we consider an auxiliary function Lemma 4.1].Due to the general maximum principle (which is available as f ∈ L p with p > p 0 ), one can argue as in the proof of [6,Lemma 4.5] (where the smallness of for some constant σ > 0, depending only on n, λ, and p, where w is the convex envelope of −w − in B 4 √ n .Our claim is, as again in the proof of [6,Lemma 7.5], that for some large constant m > 1, depending only on n, λ and .
The main observation here is that the gradient of the supporting hyperplanes for the convex envelope w at the contact set is universally bounded.This actually follows from a simple fact that by construction, for some m > 0 depending only on n, λ and .These inequalities follows from the specific choice of the barrier function ϕ in [6,Lemma 4 In what follows, we shall let m denote a generic positive constant depending only on n, λ and and allow it to vary at each occurrence.
Keeping in mind of these properties of ϕ, let x 0 ∈ {w = w } ∩ Q 1 be any.As w is the convex envelope of w in B 4 √ n , we can find a linear polynomial (as one of the supporting hyperplanes of for all x ∈ .Next, we observe that we have freedom to choose ϕ in such a way that , and the extension leaves the gradient free, we can find an extension such that sup by taking m larger if necessary.Finally, by the definition of w, we deduce from (2.0.1), (2.0.2) and the assumption that u Now we may argue as in [6,Lemma 7.7] to deduce a universal decay of the measure of the set with large gradient "from below".

Proof
The proof is almost the same with that of [6,Lemma 7.7].The involvement of the Riesz potential, which replaces the maximal function in the statement of the latter lemma, is due to the linear rescaling of the solutions. Fix Hence, due to the Calderón-Zygmund cube decomposition lemma, it suffices to prove that given any dyadic cube Suppose, by way of contradiction, that Q\(L k ∪ B k ) = ∅.Let x Q and s Q be the center and respectively the side-length of Let us consider the rescaled version of u and respectively f , with c > 1 being a constant to be determined solely by n and p, and let where the first inequality is ensured by choosing c > 1 large, depending only on n.Moreover, since provided that we choose c > c 0 .Furthermore, in the L p -viscosity sense.Thanks to (2.0.3)-(2.0.5), u Q and f Q fall under the setting of Lemma 4.2, from which we deduce that by choosing m > 1 larger from the beginning so that c −1 m becomes the constant appearing in the latter lemma.Rescaling back, we arrive at |L k+1 ∩ Q| ≤ σ |Q|, a contradiction to the choice of Q.Thus, the proof is finished.
As a corollary, we obtain a universal decay estimate in the interior.We shall only present the statement and omit the proof, as it is essentially the same with that of [6,Lemma 7.8].Lemma 4.4 Under the setting of Lemma 4.3, for any t > 0, where c > 1 and μ > 0 depend at most on n, λ, and p.
From now on, we shall study the estimates near boundaries.As mentioned earlier, the idea to combine the interior estimate with the small measure of the thin slab that contains the boundary layer is originally from [30, Lemma 2.9]; here we simply extend the argument to the framework of Reifenberg flat domains.

Reifenberg flat, and u ∈ C( ) be an L p -viscosity solution to
for any t > 0, where c > 0 and μ > 0 depend at most on n, λ, and p. 123 Due to (a properly rescaled form of) Lemma 4.4, we have, for any t > 1, The conclusion follows easily from the observation that |{x ∈ B 1 : |x • ν| < 2δ}| ≤ c n δ, and Next, we obtain universal decay estimates near boundary layers.
where m > 1 and δ 0 ∈ (0, 1 2 ) are constants depending at most on n, λ, and p. Proof Let m > 1 be a constant to be determined later, and set η = c 1 c 0 (δ where c 0 > 1 and μ > 0 are as in Lemma 4.5, and c 1 > 1 is a constant to be determined later, by n, λ, and p only.Fix any integer k ≥ 1.Then it follows from the latter lemma, as well as the relation (2.0.6) Assume for the moment that the claim is true.Then it follows from Lemma 2.7 (along with Then we first choose δ 0 sufficiently small such that 4c n c 0 δ 0 ≤ 1. Selecting m accordingly large such that 4c n c 0 δ −μ m −μ ≤ 1, we obtain that 2 , which finishes the proof.Henceforth, we shall prove the claim (2.0.6).Suppose by way of contradiction that ∩ B\(L k ∪ B k ) = ∅.Here it suffices to consider the case 2B \ ( ∩ B 2 ) = ∅, since the other case can be handled as in the interior analysis (see the proof of Lemma 4.3).
Set r B = rad B and choose and rescale u and f as follows, Arguing analogously as in the proof of Lemma 4.3, we may deduce from ≤ 1, provided that c > 1 is a large constant, depending at most on n and p.Moreover, in the L p -viscosity solution.Thanks to the scaling invariance of the Reifenberg flatness, ∂ B ∩ B 2 is (δ 0 , 2)-Reifenberg flat and contains the origin.Thus, we can employ Lemma 4.5 to deduce that for any t > 0. Rephrase the above inequality in terms of u, and deduce that where ω n is the volume of the n-dimensional unit ball.Evaluating this inequality at t = c −1 m, we reach contradiction against n c μ from the notation in the beginning of the proof).Now we have a boundary-analogue of Lemma 4.4.Let us skip the proof for the same reason as mentioned above the statement of the latter lemma.Lemma 4.7 Under the same hypothesis of Lemma 4.6, for any t > 0, where c > 1 and μ > 0 depend at most on n, λ, and p.
Finally, we are ready to prove the global universal decay asserted in the beginning of this section.
Proof of Proposition 4.1 With Lemmas 4.4 and 4.7, the assertion of this proposition follows easily via a standard covering argument.The exponent μ can be taken as the minimum between those in both lemmas.We omit the detail.

Set of large Hessian
Here we shall study universal decay estimates for the set of large Hessian.Note that an interior estimate is by now considered classical, and can be found in [6, Lemma 7.8], while an estimate near flat boundary is established rather recently in [30].Here we extend the result to Reifenberg flat boundaries.
Here given any u ∈ C( ) and t > 0, A t (u, ) is defined, as in [6, Section 7], as a subset of such that x 0 ∈ \A t (u, ) if and only if there exists a linear polynomial for which u(x) ≥ (x) − t 2 |x − x 0 | 2 for all x ∈ .Proposition 4.8 Let ⊂ R n be a bounded, (δ, R)-Reifenberg flat domain with R ∈ (0, 1], f ∈ L p ( ) be given, with some p > p 0 , and u ∈ C( ) ∩ L ∞ ( ) be an L p -viscosity solution to P − (D 2 u) ≤ f in .Then for any t > 0, μ , where C > 1, δ > 0 and μ > 0 depend at most on n, λ, and p. Proof Since the proof repeats most of the argument presented in the section above, we shall pinpoint the difference and skip the detail.First, observe that we can replace L t in Lemma 4.5 with A t , by simply applying [6, Lemma 7.5] (which holds equally well for L p -viscosity solutions with L p -integrable right-hand side, due to [8]) in place of Lemma 4.2 in the proof.Now the assertion of Lemma 4.6 holds true with L k now denoting the set A m k (u, ) ∩ B 1 , since the proof only uses Lemma 4.2.Finally, iterating the modified version of Lemma 4.6 would yield Lemma 4.7, again with L t (u, ) replaced by A t (u, ).Thus, a standard covering argument along with the L p -variant of [6, Lemma 7.5] and the modified version of Lemma 4.7 would yield the conclusion of this proposition.

W 1, np n−p -estimates
This section is devoted to uniform interior W 1, np n− p -estimates in fully nonlinear homogenization problems, for any p ∈ ( p 0 , n); note that np n− p is the (critical) Sobolev exponent of p.The estimate is optimal, and is even new in the context of standard fully nonlinear problems.
for some p ∈ ( p 0 , n), and u ε ∈ C( ). for some ε > 0, be a viscosity solution to ), and for any subdomain , where C > 0 depends only on n, λ, , q and p.
In what follows, we shall present our argument with = B 4 √ n and = Q 1 , as our analysis will be of local character.Also, unless stated otherwise, we shall always assume that F is a continuous functional on S n × R n satisfying (2.0.1)-(2.0.3), u ε is a viscosity solution to (2.0.1) with replaced by B 4 √ n , for some ε > 0, and Let us begin with an approximation lemma for the measure of the set with large "gradient".
Lemma 5.2 Let ⊂ R n be a bounded domain such that B 4 √ n ⊂ , and suppose that , where N > 1 and μ > 0 depend at most on n, λ, and p.

Proof Consider an auxiliary boundary value problem,
As √ n , for some c > 1 and ᾱ ∈ (0, 1) depending only on n, λ and .Thus, by taking c > 1 slightly larger if necessary.
On the other hand, due to Lemma 3.1, one can compute that w ε = δ −1 (u ε − h ε ) satisfies in the viscosity sense.Since we assume f L p (B 4 √ n ) ≤ δ, it follows from the general maximum principle that w ε for all t > 0. Rephrasing this inequality in terms of u ε , we deduce that for any s > 0. Thus, the conclusion follows from (2.0.2) and (2.0.3), as well as the assumption that The following lemma is an analogue of [6, Lemma 7.12].
Proof Fix an integer k ≥ 1.Let M > N be a large constant such that c 0 M −μ < 1, with c 0 , N > 1 and μ > 0 as in Lemma 3.2; note that M depends only on n, λ, and p.By Lemma 3.2 (along with The rest of our proof will resemble that of [6,Lemma 7.12].
, where Q is the predecessor of Q.Once this claim is justified, the conclusion is ensured by the Calderón-Zygmund cube decomposition lemma.
Suppose, by way of contradiction, that Q\(L ε k ∪ B k ) = ∅.Denote by x Q and s Q the center and respectively the side-length of √ n, it follows from the latter inequality that for any In addition, due to the assumption xQ / ∈ B k , as well as In what follows, we shall use c > 1 to denote a positive constant depending at most on n and p, and allow it to vary at each occurrence.
Let us consider the following rescaled versions of u ε and f , .
and thus in view of (2.0.1), u where ×R n ) and it satisfies (2.0.1)-(2.0.3).On the other hand, it follows immediately from (2.0.4) and (2.0.5) that with c > 1 sufficiently large, for any s > N with N > 1 as in the latter lemma.Thus, taking M > N larger if necessary such that c 0 c −μ M −μ < 1, and then rescaling back to u ε , we arrive at We are ready to prove the uniform "large-scale" interior W Proof of Theorem 5.1 After some suitable rescaling argument, it suffices to consider the case where where δ ∈ (0, 1) is to be determined by n, λ, and p only.
Set p = p+ p 0 2 ∈ ( p 0 , p), and apply Lemma 5.3, with p replaced by p .Observe that ≤ cδ for some c > 0 depending only on n and p.Hence, with η = (cδ) μ < 1 (by choosing δ smaller if necessary), Thus, according to the embedding theorem for the Riesz potential, (2.0.7) To this end, we choose δ ∈ (0, 1) as a sufficiently small constant such that clearly, it is the set of parameters n, λ, and p that determines how small δ should be.Then it follows from (2.0.6) and (2.0.7) that ≤ c, as desired.
With the additional assumption in the statement of the theorem, we can replace L ε s in Lemma 5.2 with L s .For we can now invoke the uniform interior C 1,α -estimate below ε-scale, [18, Theorem 4.1 (ii)], to deduce that the approximating solution

W 2,p -estimates
This section is devoted to interior W 2, p -estimates for viscosity solutions to a certain class of fully nonlinear homogenization problems.
Although Lemma 3.2 yields an error estimate between u ε and ū in L ∞ norm, we cannot expect ū is close to u ε in the viscosity sense (i.e., P ± (D 2 (u ε − ū)) = o(1) with P ± being the Pucci extremal operators), since D 2 u ε is supposed to be rapidly oscillating around D 2 ū in the small scales.
In the next lemma, we obtain the closeness between D 2 u ε and D 2 ū in the viscosity sense by incorporating interior correctors.To do so, we shall assume V M O-condition, or more exactly S. Kim B M O ψ -condition for some modulus of continuity ψ, for D 2 ū.This condition replaces the small oscillation condition for standard problems, F(D 2 u, x) = f , c.f. [6, Theorem 7.1].Let us remark that B M O ψ -regularity does neither imply nor follow from the boundedness of D 2 ū, and that it also allows D 2 ū to be discontinuous.
With suitable rescaling argument, it suffices to take care of the case where = B 4 In what follows, we shall always assume that F is a continuous functional satisfying (2.0.1)-( 2 ) is a viscosity solution to (2.0.1), unless stated otherwise.Moreover, we shall let c denote a positive generic constant depending at most on a set of fixed quantities, shown in the statement of each lemmas below, and we allow it to vary at each occurrence.
Proof Fix s > 1 and η > 0. Let δ > 0 be a constant to be determined later.Since the hypothesis of Lemma 3.2 is met, we can find a constant ε δ ∈ (0, 1), corresponding to δ, and a function ū ∈ C(B 3 Due to the John-Nirenberg inequality, we have (2.0.4) In addition, by [21, Lemma 2.5], we also have

for the definition of ), whence it follows from the relation
(2.0.5) Let ρ s ∈ (0, 1  4 ) be a constant to be determined later.Our idea is to subdivide Q 1 into two groups, say F and G, of dyadic cubes with side-length in between ρ s and 2ρ s such that here c > 1 is a constant to be determined by n, λ, , κ, ψ and q.Thanks to (2.0.4) and (2.0.5) (as well as an obvious fact that A ε s (g, E) ⊂ A s (g, E) for any g ∈ C(E)) yields that (2.0.6) Note that we can replace the domain B 2 √ n in the leftmost side to by utilizing the assumption that |u ε (x)| ≤ |x| 2 for all x ∈ \B 2 √ n .Thus, it will be enough to prove that with ε s,η > 0 sufficiently small and ε < ε s,η , for all Q ∈ F , where μ > 0 is a constant depending only on n, λ and .
To prove (2.0.7), let us fix a cube Q ∈ F .By definition, there is some where c n > 0 is a constant depending only on n.Consider an auxiliary function w : (2.0.9) in the viscosity sense.According to [13, Lemma 3.1], such a periodic viscosity solution exists (in C 0,α (R n ), with α ∈ (0, 1) universal) and unique, and due to (2.0.1) as well as(2.0.3), it satisfies for some constant c 0 > 0 depending only on n, λ and .Consider auxiliary functions φ ε , g : B → R defined by Suppose for the moment that the claim is true.With sufficiently small δ and ε s,η , whose smallness condition depending only on r B , κ, ψ(r B ), c 0 and c, we can deduce from (2.0.12) and (2.0.13) that φ ε L ∞ (B) ≤ 1 and respectively g L n (B) ≤ c n r B .Then we can apply (a rescaled form of) [6, Lemma 7.5]1 to φ ε and deduce that for any t > 0, where μ > 0 depends only on n, λ and ; here we also used that dist (Q, ∂ B) ≥ 1 2 r B , which is apparent from the choice of Q and B. Thus, setting t = Utilizing (2.0.10) (as well as a simple observation that ≤ δ with δ being small depending on r B , one can also replace B above with B 2 √ n .At this point, we select c ≥ 2c 0 , and ρ s = r B 2 √ n ∈ (0, 1  4 ) as a small constant such that κψ(r B ) ≤ η 1/μ , so that we arrive at (2.0.7), as desired.
Thus, we are only left with proving that (2.0.14) holds in the L p -viscosity sense.Let ϕ be a quadratic polynomial such that D 2 ϕ = P.For the moment, let us denote by W ε the function ε in B in the viscosity sense, so it follows from Lemma 3.1 that in B in the L p -viscosity sense.Similarly, we obtain P + (D 2 φ ε ) ≥ −|g| in B in the L pviscosity sense.This finishes the proof.
We are ready to use the by-now standard cube decomposition argument to obtain a geometric decay of |A ε s (u ε , ) ∩ Q 1 |.The idea is the same with [9] in the sense that we split the set A k = A ε m k (u ε , ) ∩ Q 1 into two parts, say D k and E k , where D k is the part of A k intersected by its Calderón-Zygmund cube covering whose side-length is at least ε ε η (these cubes are said to be of high frequency), whereas E k = A k \ D k , i.e., the part of A k intersected by the cubes of low frequency.As for D k , we can deduce a geometric decay via Lemma 5.5, and this part is almost the same with the argument for standard problem, e.g., [6,Lemma 7.13].
As for E k , the above lemma is no longer applicable, but at the same time we cannot argue as in [9] because we do not assume any structure condition on F so as to ensure sufficient regularity of u ε in small scales.Here we control E k directly from the fact that the set A k (or more exactly Q 1 \ A k ) allows error of order ε 2 for quadratic polynomials to touch u ε .
Proof As briefly mentioned in the discussion before the statement of this lemma, the set D k is the part of A k intersected by its Calderón-Zygmund covering whose side-length is no less than ε ε η .More exactly, we choose M > 1 sufficiently large such that |A 1 | ≤ η + cm −q < 1 due to Lemma 5.5.We shall fix ε η as the constant ε c −1 0 m,η from Lemma 5.5, with c 0 > 1 to be determined later.As whence there exists a Calderón-Zygmund covering, denoted by F ε k+1 , of A k+1 corresponding to the level η + cm −q .Define , where Q is the predecessor of Q.Once this is proved, from the fact that Q belongs to the Calderón-Zygmund covering of A k it follows immediately that The proof for the above claim mostly follows the argument for standard problems, e.g., [6, Lemma 7.12], except for the following two points: (i) we need to verify the hypothesis (2.0.4) for the effective functional at each iteration step, (ii) the set A k involves error of order ε 2 and may vary along with different scalings in the domain.
Denote by x Q and s Q the center and respectively the side-length of √ n via the above rescaling.Then due to the assumption for certain linear polynomial , satisfy that u n , provided that we choose c 0 > 1 to be large (depending only on n).Now as √ n in the viscosity sense, we have in the viscosity sense, where 2.0.1),(2.0.2) and (2.0.3).Noting that its effective functional : S n → R is given by FQ (P) = 1 c 0 m k F(c 0 m k P), for any P ∈ S n , H satisfies (2.0.4) (with the same modulus of continuity ψ and constant κ as F does).Moreover, with Rescaling back, we arrive at This finishes the proof for (2.0.18).
Next, we prove that from which (2.0.21) follows immediately.Suppose, by way of contradiction, that as desired, and the inclusion in (2.0.21) follows.
We are now ready to prove the large-scale interior W 2, p -estimates.
Proof of Theorem 5. 4 Fix p ∈ ( p 0 , ∞).With suitable rescaling of the problem, it is sufficient to prove the assertions for the case where , with ε η to be determined.Choose q = 2 p and p = p+ p 0 2 .Let M > 1 be as in Lemma 5.6 with p replaced by p .We can assume that cM p−q ≤ 1 4 by taking M larger, depending on the choice of p. Then we select η > 0 as a small constant satisfying M p η ≤ 1 4 .Then Now let ε η > 0 and k η > 1 be as in Lemma 5.6 corresponding to the specific choice of η.
Note that all the constants η, M, ĉ, ε η and k η involved in the statement of Lemma 5.6 now depend only on n, λ, , ψ, κ and p (as q depending solely on p, and γ solely on n, λ, and p).
With (2.0.17) at hand, we deduce that where A k and B k are as in the statement of Lemma 5.6.Since The proof can now be finished, as in [6, Proposition 7.2], and we omit the detail.
To prove the second assertion of Theorem 5.4, we shall modify some of the argument for the proof of Lemmas 5.5 and 5.6 in such a way that we estimate the measure of In what follows, we intend to highlight those changes, which are not so trivial, and then omit the detail for the argument that might repeat what is already written so far.

Estimates near boundary layers
(2.0.1) , where δ > 0 depends only on n, λ, and p, and C > 1 may depend further on R and diam (U ).
The above estimate is optimal as the power of integrability reaches the critical Sobolev exponent.This estimate is even new in the setting of the standard problems.To the best of the author's knowledge, the boundary estimate is proved up to the subcritical Sobolev exponent, i.e., W 1,q with q < np n− p , in [30].Following the spirit of [29], the proof in [30] relies heavily on pointwise C 1,α -approximation, and hence the estimate could not reach the critical exponent.
We shall set the starting point of our analysis, however, at a sub-optimal estimate below.We shall provide some motivations and remarks after the statement.
, and for any U U , , where δ > 0 depends only on n, λ, and p, and C > 1 may depend further on q, R and diam (U ).
We shall provide this estimate, mainly because of its independent interests.Of course, we will use this proposition in our subsequent analysis to ensure a better regularity for the approximating solutions.Still, this step could be conveniently replaced by the uniform boundary C 1,α -estimates [18], as the approximating solutions solve a "clean" version of the homogenization problems (i.e., f = 0 and g = a, with a a constant).
In view of the compact embedding of W 1, np n− p to C 0,2− n p , the above estimate seems to be the best one can expect with a C 0,2− n p -regular boundary data.The estimate is quite interesting in the sense that a Hölder regular function may not be (weakly) differentiable a.e., that is, one cannot expect C 0,2− n p -regular boundary data to be extended to a W 1, np n− p -regular function in a neighborhood of the boundary layer.In particular, one cannot deduce the above sub-optimal estimate from the former optimal estimate (Theorem 6.1).Hence, the above proposition shows certain regularizing effect arising from the presence of boundary layers.
Let us also remark that the sub-optimal estimate above improves the one in [30], in terms of the regularity of the boundary data, as the latter estimate requires C 1,α -regularity.
We shall first present a complete proof for Proposition 6.2 and then move onto that of Theorem 6.1.The proof for the former is based on a boundary C 0,2− n p -estimate for any viscosity solution belonging to the Pucci class, up to an L p -regular, with p > p 0 , right hand side.In particular, the functional may not oscillate under certain pattern in small scales, whence it has nothing to do with homogenization.
Proof of Proposition 6.2 Let us prove the first part of the assertion, and then mention the changes in the argument for the second part, as the latter is almost the same with the former.
After some standard rescaling procedure, one may prove the assertion for the case where 0 Applying the boundary Hölder regularity [22, Theorem 1.1] to u ε around each point for all x ∈ ∩ B 1 ; let us remark that although the statement of [22, Theorem 1.1] involves sup r >0 r α−2 f L n ( ∩B r (x 0 )) on the right-hand side, one can easily replace this norm with f L p ( ∩B 1 ) by taking α = 2 − n p , as the modification in the proof there is straightforward.
Let B ⊂ ∩ B 1/2 be a ball for which ∂(2B) ∩ ∂ ∩ B 1/2 = ∅.Let x B and ρ B denote the center and respectively the radius of B. Also let x B,0 be a point of intersection between ∂(2B) and ∂ ∩ B 1/2 .Consider the rescaling in the viscosity sense, where we wrote Thanks to (2.0.2), we also have ).The latter estimate can be translated in term of u ε as To this end, consider a covering F of ε ∩B 1/2 by balls B for which where k ε is a large positive integer of order − log 2 ε.Then where the last inequality is ensured from the fact that q < np n− p .This finishes the proof, for the first part of the assertion of the theorem.
As for the second part of the assertion, one may have already noticed that under the additional assumption, one can invoke Theorem 5.1 (ii) in place of (i) above, such that one can replace D ε u ε with Du ε first in (2.0.3), and then in (2.0.4), leaving everything else untouched.Thus, the conclusion follows.We leave out the detail to the reader.With Proposition 6.2 at hand, we may now proceed with the proof for Theorem 6.1.The idea is basically the same with the interior case (Lemma 5.2), but the presence of Reifenberg flat boundaries yields some additional technical difficulties.
As in the analysis in the interior case, we shall confine ourselves to the case U = B 2 and U = B 1 , as the analysis here is of local character around a boundary point.Unless specified otherwise, we shall always assume, from now on, that , is a bounded domain containing the origin, and u ε is a viscosity solution to (2.0.1) with U replaced by B 2 .
Proof Let â denote the integral average of g over B 2 .By the Poincaré inequality, we have ≤ c 0 η, where c 0 depends only on n and p.By the by taking c 0 larger.Denote by S the slab S 2δ (ν) for some unit vector ν such that ∂ ∩ B 2 ⊂ S; such a unit vector exists owing to the Reifenberg flatness of ∂ ∩ B 2 .Also let E be the half-space E, so that ∩ B 2 ⊂ E. Since This problem admits a unique viscosity solution h ε ∈ C(E ∩ B 3/2 ) because F and u ε − g are continuous, and ∂(E ∩ B 3/2 ) satisfies a uniform exterior sphere condition.Since we can deduce from Lemma 3.4, the Krylov theory [6, Corollary 5.7], as well as the fact that with both c 0 > 1 and α ∈ (0, 1) depending only on n, λ and .In particular, by taking c 0 larger if necessary, with In what follows, we shall denote by c 0 a constant which may depend at most on n, λ, and p, and we shall allow it to vary at each occurrence. As ≤ c 0 η that if δ ≤ δ p , with δ p depending only on n, λ, and p, then Combining this inequality with [12, Lemma 2], we deduce that With (2.0.7) at hand, we may now estimate the global Hölder norm of h ε .As ≤ c 0 , we may apply a variant of [6,Proposition 4.13] to derive that Consider another auxiliary function w ε = η −1 0 (u ε − â−h ε ), with η 0 to be determined later.This function is well-defined on ∩ B 3/2 , since ∩ B 3/2 ⊂ E ∩ B 3/2 .Due to Lemma 3.1, we may compute that . This together with (2.0.5) yields that with η 0 = c 0 max{δ α 2 , η} and c 0 > 1, With such a choice of η 0 , we also have f L p ( ∩B 3/2 ) ≤ η 0 , whence it follows from the general maximum principle that w ε L ∞ (B 3/2 ) ≤ c 0 .Now we can apply Proposition 4.1 to obtain that for any t > 0, (2.0.9) where μ > 0 depends only on n, λ, and p; this is another step that determines how small the Reifenberg flatness constant δ should be.Since the set L t (w ε , •) is invariant under vertical translation of given function w ε , the conclusion follows immediately from (2.0.6), (2.0.9), the choice of η 0 and the assumption that |u ε (x)| ≤ |x| for all x ∈ \ B 1 .
The following lemma is the boundary analogue of Lemma 5.3.

Proof
The proof resembles with that of Lemma 4.6.A key difference here is that now we need to be careful of the changes made in boundary data, when rescaling the problem; note that we did not encounter this issue in the proof of Lemma 4.6, since we did not need to see the boundary value at all.Henceforth, we shall proceed with the proof focusing on this issue, and try to skip any argument that only requires a minor modification of what is already shown so far.
Fix an integer k ≥ 1. Arguing as in the proof of Lemma 5.3, one may use Lemma 6.3, in place of Lemma 5.2, to deduce that , where c 0 > 1 and m > 1 are to be chosen later.
Let B ⊂ B 1 be any ball with center in ∩ B 2 and rad (B) ≤ 1. Suppose that |L ε k+1 ∩ B| > η 0 |B|.As in the proof of Lemma 4.6, we assert that Consider the following rescaled versions of u ε , f and g, where c > 1 is a constant to be determined later.In view of (2.0.1), we may compute that in the viscosity sense, where B = for any s > N , for some N > 1 depending only on n, λ and .To this end, we may follow the argument at the end of the proof of Lemma 5.3 to arrive at |L k+1 ∩ B| ≤ η 0 |B|, with suitable choice of η 0 , a contradiction.This finishes the proof.
The uniform boundary W 1, np n− p -estimates can now be proved as follows.
Proof of Theorem 6.1 One may argue exactly as in the proof of Theorem 5.1, by substituting Lemma 5.3 with Lemma 6.4.The additional term, namely the measure of C k in the notation of the latter lemma, is controlled by the W 1, np n− p -regularity of the boundary data g, as well as the strong (q, q)-type inequality, with q > 1, for the maximal function.We omit the detail to avoid repeating arguments.

W 2,p -estimates
Let us now move on to the uniform W 2, p -estimates around boundary points.As always, we use ε to denote the set {x ∈ : dist (x, ∂ ) > ε}. (2.0.1) provided that and for any U U , where C > 0 depends at most on n, λ, , ψ, κ, p, σ and diam (U ).
Let us begin with a sub-optimal estimate, namely a uniform boundary W 2,q -estimates, with q < p, provided that f ∈ L p ( ∩ U ) and g ∈ C The proposition below will be used later in our approximation lemma (Lemma 6.7 for the boundary estimate. for some p > p 0 , and u ε ∈ C( ∩ U ) be a viscosity solution to ) for all q ∈ ( p 0 , min{ p, n 2−α }), and for any subdomain U U , where C > 0 depends only on n, λ, , ψ, κ, q, α and diam (U ).

123
(ii) If p > n, assume that ∂ ∩ U is a C 1,α -hypersurface, and g ∈ C 1,α (∂ ∩ U ). Then H ε ∩U (u ε ) ∈ L q loc ( ε ∩ U ) for all q ∈ ( p 0 , min{ p, n 1−α }), and for any subdomain U U , Proof The proof is essentially the same with that of Proposition 6.2.After a suitable rescaling argument, it may suffice to prove the case where (2.0.3) for all x ∈ ∩ B 1 , where c > 0 depends only on n, λ, , κ and p n .Now for each ball B ⊂ ε ∩ B 1/2 with ∂(2B)∩∂ ∩ B 1/2 = ∅, we can make the following rescaling, , of u ε , where x B is the center of B, ρ B its radius and x B,0 the point of intersection between ∂(2B) and ∂ ∩ B 1/2 .Then we may repeat the proof of Proposition 6.2, utilizing Theorem 5.4 in place of Theorem 5.1, to deduce that Fix any q < p n .Then we can consider the same Besicovitch covering G, as in the proof of Proposition 6.2, of ε ∩ B 1/2 by balls B, such that the summation of the right-hand side of over all B ∈ G is bounded by a constant c.This finishes the proof.
As for the proof of Theorem 6.5, it suffices to consider the case where f ∈ L p and g = 0, since one may always substitute u ε with u ε − g and f with f + c|D 2 g|.
Since our analysis will be of local nature around a boundary point, and will be invariant under translation, we shall work from now on with domains with 0 ∈ ∂ , U = B 2 and U = B 1 .Unless specified otherwise, from now on, F ∈ C(S n × R n ) satisfies (2.0.1)-(2.0.4), ∂ ∩ B 2 is a C 1 -hypersurface containing the origin, and that there is a diffeomorphism for some p > p 0 , and u ε ∈ C( ∩ B 2 ) is a viscosity solution to (2.0.1) with U = B 2 and g = − , with a linear polynomial; we shall discuss later in detail the reason for the involvement of a linear polynomial in the boundary condition.
The difficulty of our analysis arises from the fact that homogenization problems are unfavorable towards boundary flattening argument, as one loses the oscillating pattern by the transformation.In one way or another, one will resort to the fact that the original problem in small scales is homogenized to a "nice" effective problem to improve the regularity, whence the level of difficulty remains the same.
For this reason, we shall study our problem (2.0.1) without flattening the boundary.This readily implies some notable changes in the approximation lemma below for the measure of the set of large "Hessian", compared to the interior case (Lemma 5.5) as well as those for standard problems in the setting of flat boundaries (e.g., [30,Lemma 2.14]).Lemma 6.7 Let ε, δ, α, ρ, p and q be constants with 0 < ε < 1, 0 < δ ≤ δ 0 , 0 < α < 1, ρ > 0 and p 0 < p < q < ∞ be given.Suppose that for some linear polynomial .Assume either of the following: L n (B 2 ) ≤ δρ and q > n.
dx, and let L T : R n → R n be the linear transformation induced by T ; i.e., L T (0) = 0 and DL T = T .In what follows, we shall denote by c 0 a constant depending at most on n and q, and we shall allow it to vary from one line to another.
By the Poincaré inequality, together with Let L T : R n → R n be the linear transformation such that DL T = T .Then by the Sobolev embedding theorem, one can infer that Now by the assumption that Let us now turn to the regularity of the linear polynomial , for which where the last inequality holds for any small δ, whose smallness condition depends only on n and q.Let h ε ∈ C( ∩ B 2 ) be a viscosity solution to The existence of such a viscosity solution is ensured by . By the maximum principle, we have (2.0.11) By the assumption on , ∂ ∩ B 2 is a C 1 -hypersurface whose Lipschitz norm is less than c 0 δ.Thus, by taking δ smaller if necessary, depending now on n, λ, and q, we can deduce from Proposition 6.6 (with α = 2 − n q ), (2.0.9) and (2.0.11) that .0.12) for some constant c > 0 depending only on n, λ, , κ, ψ and q.In particular, by the definition We can then replace ∩ B 2 above with by invoking the inequality |u ε (x)| ≤ |x| 2 for all x ∈ \B 1 .
As for this case, by the Sobolev embedding theorem, and In other words, ∂ ∩ B 2 is a C 1,1− n q -hypersurface whose C 1,α -norm is bounded by 2c 0 δρ.
For the rest of the proof, we shall denote by c α a positive constant depending at most on n and α, and it may vary at each occurrence.With (2.0.18) at hand, we claim that ∈ C Moreover, we may also compute, via (2.0.17) and |D | ≤ ρ −1 , that where D τ is the tangential gradient to ∂ ∩ B 2 .Combining the last two displays altogether with (2.0.7), we verify the claim (2.0.19).Now let h ε be the viscosity solution to (2.0.10), as in Case 1.The inequality in (2.0.11) continues to hold here.However, now with (2.0.18) and (2.0.19) at hand, Proposition 6.6 ensures that (2.0.12), hence (2.0.13) as well, holds for q ≤ n 1−α .The rest of the proof repeats that of Case 1 verbatim, so it is omitted.
In the above cases, the auxiliary Dirichlet problem for the approximating function h ε was imposed on the same domain ∩ B 2 .Thus, the integrability of H ε ∩B 2 (h ε ) cannot exceed the exponent determined by the regularity of the boundary layer.On contrast, this last case asks for q to go over the threshold.To achieve this goal, we shall consider another Dirichlet problem, whose boundary layer is much smoother (in fact, a hyperplane).However, the choice of such an auxiliary problem cannot be arbitrary, since the newly obtained approximating function should also be sufficiently close to the original solution on the boundary layer, so that their difference satisfies (2.0.15).To meet the latter requirement, we need to be C 0,αregular for some α > 0, not only on ∂ ∩ B 2 (as in (2.0.9)), but also over a slab S ∩ B 2 , with S = S c 0 δρ (ν), that contains ∂ ∩ B 2 .This is where a stronger assumption, |D | ≤ ρ α−1 , is used.
Let us explain this in more detail.In what follows, let E denote the half-space H −c 0 δρ (ν); note that ∩ B 2 ⊂ E. We shall keep writing by S the slab S c 0 δρ (ν); recall from (2.0.6) that ∂ ∩ B 2 ⊂ S.
Set v ε = u ε −φ ε cδ γ , with a possibly different c > 1 to that in the last display, yet depending on the same quantities.Then again by Lemma 3.1, we may compare (2.0.1) (with g = − ) with (2.0.22) and compute that ⎧ ⎪ ⎨ ⎪ ⎩ (2.0.26)As c > 1 and 0 < δ, γ < 1, it follows from the assumption that f L p ( ∩B 2 ) ≤ cδ γ .This implies, by the general maximum principle, that where the last inequality is ensured by γ ≤ α 4 , δ < 1 and ρ ≤ 1.This completes the proof.Our next step is to design a suitable iteration argument for the boundary estimate.Lemma 6.8 Let ε, δ, α, ρ, p and q be constants with 0 < ε < 1, 0 < δ ≤ δ 0 , 0 < α < 1, ρ > 0 and p 0 < p < q < ∞ be given.Suppose that |ξ t D (0)ξ − 1| ≤ δ for any ξ )) > δ p m kp }.Assume either of the following: kpq n }, and p < q < n; Fix any q > p such that q < n if p < n, q ≤ n + σ if p = n and q = 2 p if p > n.We may choose m larger in Lemma 6.8, but still depending on the quantities in the statement of the lemma, such that cm p−q ≤ 1 4 .Then we take δ sufficiently small such that cδ γ μ m μ ≤ 1  4 , which ensures that m p η ≤ (2.0.39) for some c > 0, depending only on n, λ, , κ, ψ, p and σ (only for the case p = n).Hence, it suffices to prove a uniform bound of the rightmost term in the above display.By the strong ( p p , p p )-type inequality for the maximal function and the assumption that f L p ( ∩B 2 ) ≤ δ, one can immediately prove that ∞ k=1 m kp β k ≤ c.

3 .
as in the case of Lemma 3.Proof of Lemma 3.4 With a similar modification shown in the proof of Lemma 3.2, we can extend f ∈ L ∞ ( ∩ U ), required in [18, Lemma 5.2 and 5.3], to L p ( ∩ U ).To extend the lemmas to ∂ ∩ U ∈ C 1,α (from C 1,1 ), we may combine the compactness argument in [23, Lemma 3.1] with [18, Lemma 3.1, 5.2].We skip the detail.
Now it follows from Lemma 3.3, along with the Kyrlov theory [6, Corollary 5.7], that for any x 0 ∈ B 3 √ n , there exists a linear polynomial ε

( 2 .
0.5) where c > 1 depends only on n and p.
To simplify the exposition, let us write by B the ball B 2ρ s √ n (x 0 ), by B the radius of B, i.e., r B = 2ρ s √ n, and by P the matrix (D 2 ū) B , i.e., P = 1 |B| B D 2 ū dx.Then by the choice of x 0 , |P| ≤ s c , and by (2.0.3) along with the John-Nirenberg inequality, Let us now turn to uniform W 1, np n− p -estimates near boundary layers.

1 2 .
With such a choice of m and δ, one can derive, by following computations in the proof of Theorem 5.1, that−c( p ,n) log m ε k=1 m kp α k ≤ c + c ∞ k=1 m kp (β k + γ k + δ k ).
as desired, by Lemma 2.7.Assume by way of contradiction that∩ B\(L ε k ∪ B k ∪ C k ) = ∅.For the same reason as in the proof of Lemma 5.2, we have 2r B > ε, by taking m > 1 large, depending only on n.Choose a point x B ∈ ∂ ∩ B 1 such that 2B ⊂ B 4r B (x B ). Select any xB ∈ ∩ B\(L ε k ∪ B k ∪ C k ).Since xB / ∈ ∩ B 2r B (x B )\L ε k and 2r B > ε, one can find some constant a for which |u B (x) − a| ≤ m k (4r B + |x − x B |), (2.0.10) for all x ∈ ∩ B 2 .Moreover, it follows from xB ∈ ∩ B 2r B (x B )\(B k ∪ C k ), one may deduce as in the proof of Lemma 5.3 that Note that F B ∈ C(S n × R n ) and it satisfies (2.0.1)-(2.0.3) for an obvious reason.Selecting a large c > 1, we observe from (2.0.10) and (2.0.11) that u B , f B and g B verify the hypothesis of Lemma 6.3.Due to the scaling invariance of the Reifenberg flatness, ∂ B ∩ B 2 is also (δ, 2)-Reifenberg flat.Hence, all the hypotheses of Lemma 6.3 are verified, from which it follows that ε ).