Ars Inveniendi Analytica (2023), Paper No. 2, 41 pp.

DOI 10.15781/52x3-ja93

ISSN: 2769-8505

00footnotetext: © William M. Feldman00footnotetext:  Licensed under a Creative
Commons Attribution License (CC-BY).

Large scale regularity of almost minimizers of the one-phase problem in
periodic media

William M Feldman

University of Utah

Communicated by Scott Armstrong

Abstract. We prove that minimizers and almost minimizers of one-phase free
boundary energy functionals in periodic media satisfy large scale (1)
Lipschitz estimates (2) free boundary flat implies Lipschitz estimates. The
proofs are based on techniques introduced by De Silva and Savin [17] for
almost minimizers in homogeneous media.

Keywords. Homogenization, free boundary, large scale regularity, almost
minimizers.

##  1\. Introduction

We consider minimizers of the one phase free boundary energy functional in a
domain $U\subset\varmathbb{R}^{d}$:

(1.1) $J(u,U)=\int_{U}\nabla u\cdot a(x)\nabla u+Q(x)^{2}{\bf 1}_{\\{u>0\\}}\
dx.$

Here the heterogeneous media $Q:\varmathbb{R}^{d}\to(0,\infty)$ and
$a:\varmathbb{R}^{d}\to M_{d\times d}^{sym}(\varmathbb{R})$ are
$\varmathbb{Z}^{d}$-periodic measurable functions, sufficiently regular (to be
specified), satisfying the ellipticity assumptions, for some $\Lambda>1$,

(1.2) $\Lambda^{-1}I\leqslant a(x)\leqslant\Lambda I\ \hbox{ and }\
\Lambda^{-1}\leqslant Q(x)\leqslant\Lambda.$

Consider the domain $U$ to be very large compared to the unit period scale
associated with the coefficients.

The goal of this paper is to prove that almost minimizers of the energy
functional $J$ satisfy a Lipschitz estimate and a large scale “flat implies
Lipschitz" estimate for the free boundary $\partial\\{u>0\\}$. The results are
also new for minimizers, but the generality of almost minimizers is useful for
applications. In a forthcoming paper [19] we will apply these regularity
results to obtain quantitative estimates for homogenization rate of a shape
optimization problem.

This follows the now classical Avellaneda and Lin [7] idea of inheriting the
$C^{1,\alpha}$ regularity iteration from a homogenized problem. The proof
follows the recent quantitative strategy for large scale regularity introduced
by Armstrong and Smart [6]. In terms of free boundary regularity theory we are
using the very powerful ideas introduced by De Silva and Savin in [17].

Local minimizers of (1.1) satisfy the Euler-Lagrange equation

(1.3) $\begin{cases}-\nabla\cdot(a(x)\nabla u)=0&\hbox{in }\\{u>0\\}\cap U\\\
n\cdot a(x)n|\nabla u|^{2}=Q(x)^{2}&\hbox{on }\partial\\{u>0\\}\cap
U\end{cases}$

which is called a one-phase Bernoulli-type free boundary problem. We will
study global (almost) energy minimizers which satisfy the additional property
that

$J(u,B_{r})\leqslant(1+(r/R_{0})^{\beta})J(v,B_{r})$

for any $B_{r}\subset U$ and $v\in u+H^{1}_{0}(B_{r})$. Here $R_{0}\gg 1$
introduces an additional length scale to the problem associated with the
almost minimality property, it is a slow scale compared to the unit scale
oscillations of the coefficients. This includes the case of minimizers when
$R_{0}=+\infty$.

In the large scale limit there is $\Gamma$-convergence to an effective energy
functional

$J_{0}(u,U)=\int_{U}\nabla u\cdot\bar{a}\nabla u+\langle Q^{2}\rangle{\bf
1}_{\\{u>0\\}}\ dx.$

Here $\langle\cdot\rangle$ is the average over a period cell, and $\bar{a}$ is
the effective matrix from classical divergence form elliptic homogenization
theory. The Euler-Lagrange equation associated with critical points of this
functional is the classical Bernoulli-type free boundary problem

(1.4) $\begin{cases}-\nabla\cdot(\bar{a}\nabla u)=0&\hbox{in }\\{u>0\\}\cap
U\\\ n\cdot\bar{a}n|\nabla u|^{2}=\langle Q^{2}\rangle&\hbox{on
}\partial\\{u>0\\}\cap U.\end{cases}$

The free boundary problem (1.4) has a well developed regularity theory for
minimizers and more general solutions of the Euler-Lagrange equation. In
higher dimensions the free boundary may have singular points, but if the free
boundary is sufficiently flat in some ball $B_{r}$ then it will be smooth in
$B_{r/2}$. The theory of almost minimizers of $J_{0}$ has been developed more
recently [17, 14, 15]. We will show that this type of flat implies smooth
result for almost minimizers of the effective energy $J_{0}$ is inherited by
almost minimizers of the heterogeneous energy $J$.

The structure of the paper follows the Armstrong and Smart [6] quantitative
approach to large scale regularity. First we prove a suboptimal quantitative
homogenization result, then we employ the quantitative homogenization result
within a $C^{1,\alpha}$ iteration to achieve a Lipschitz estimate of the free
boundary. In order to establish the suboptimal quantitative homogenization
result we need to prove some initial regularity of the free boundary.

In this direction, our first main result is a large scale Lipschitz estimate
for almost minimizers of $J$. There are many proofs of the Lipschitz estimate
for Bernoulli-type problems, see the book of Velichkov [29]. Despite some
effort the only proof which we were able to adapt to homogenization setting,
even for minimizers, is the recent proof of De Silva and Savin [17]. Our
argument closely follows theirs with some inputs from homogenization theory.

######  Theorem 1.1 (Large scale Lipschitz estimate).

Suppose that $a$ and $Q$ satisfy (1.2), $a$ is $\varmathbb{Z}^{d}$-periodic,
$1\leqslant R\leqslant c(d,\Lambda,\beta)R_{0}$, and $u\in H^{1}(B_{R}(0))$
satisfies, for all $0\leqslant r\leqslant R$,

$J(u,B_{r}(0))\leqslant(1+(r/R_{0})^{\beta})J(v,B_{r}(0))$

where $v\in u+H^{1}_{0}(B_{r})$ is the $a$-harmonic replacement. Then for all
$1\leqslant r\leqslant R$

$\|\nabla u\|_{{\underline{L}}^{2}(B_{r})}\leqslant C(d,\Lambda)(1+\|\nabla
u\|_{\underline{L}^{2}(B_{R})}).$

If, additionally, $a\in C^{0,\gamma}$ then

$|\nabla u(0)|\leqslant C(d,\Lambda,\gamma,[a]_{C^{0,\gamma}})(1+\|\nabla
u\|_{\underline{L}^{2}(B_{R})}).$

Note that $Q$ does not need to be periodic for this result, the upper and
lower bounds are enough. Periodicity of $a$ is of course important, interior
Lipschitz estimates do not hold for general bounded elliptic coefficient
fields.

The other basic regularity estimates of free boundary theory require the
Lipschitz estimate and then follow by natural adaptations of established
arguments: non-degeneracy (Lemma 4.3), free boundary perimeter (Lemma 4.5) and
Hausdorff dimension estimates (Lemma 4.6). We also give a proof of inner and
outer density estimates in Section A.4, this is not directly needed for our
main result but we include it for usefulness in applications.

With these important intermediate results in hand, we establish our
(suboptimal) quantitative homogenization result in Section 5. We show that a
regularization $\overline{u}$ of $u$ in the domain $\\{u>0\\}\cap B_{r}$ is
close to minimizing $J_{0}$ with the following type of minimality condition

$J_{0}(\overline{u},B_{r})\leqslant
J_{0}(v,B_{r})+C\left[(r/R_{0})^{\beta}+r^{-\alpha}\right]|B_{r}|\ \hbox{ for
all }\ v\in\overline{u}+H^{1}_{0}(B_{r}).$

Note that there are two error terms in this energy estimate, the first from
the almost minimality property of $u$ which is good for $r\ll R_{0}$, and the
second from the homogenization which is good for $r\gg 1$ (the periodicity
scale). Importantly both terms are summable over a geometric sequence of
scales between $1$ and $R_{0}$. This allows us to apply De Silva and Savin’s
[17] $C^{1,\alpha}$ iteration for flat almost minimizers to obtain a flat
implies Lipschitz result:

######  Theorem 1.2 (Flat free boundaries are (large scale) Lipschitz graphs).

Assume that $a$ and $Q$ satisfy (1.2), both are $\varmathbb{Z}^{d}$-periodic,
and $a$ is Lipschitz continuous. Suppose that $u$ is an $(R_{0},\beta)$-almost
minimizer in $B_{R}$, $1\leqslant R\leqslant c(d,\Lambda,\beta)R_{0}$, and
$\|\nabla u\|_{L^{\infty}(B_{R})}\leqslant L$. There is a constant
$\delta_{0}(d,\Lambda,L,\|\nabla a\|_{\infty})$ so that if

$\inf_{\nu\in S^{d-1}}\sup_{x\in B_{R}}\frac{1}{R}\left|u(x)-\tfrac{\langle
Q^{2}\rangle^{1/2}}{(\nu\cdot\bar{a}\nu)^{1/2}}(x\cdot\nu)_{+}\right|\leqslant\delta_{0}$

then, for all $y\in\partial\\{u>0\\}\cap B_{R/2}$ and for all $1\leqslant
r\leqslant R$

$\inf_{\nu\in S^{d-1}}\sup_{x\in B_{r}(y)}\frac{1}{r}\left|u(x)-\tfrac{\langle
Q^{2}\rangle^{1/2}}{(\nu\cdot\bar{a}\nu)^{1/2}}((x-y)\cdot\nu)_{+}\right|\leqslant
C(d,\Lambda,L,\|\nabla
a\|_{\infty})\left[(r/R)^{\alpha}\delta_{0}+r^{-\omega}\right].$

And, as a consequence, $\partial\\{u>0\\}\cap B_{r/2}$ is within distance
$C(d,\Lambda,L,\|\nabla a\|_{\infty})$ of a $1$-Lipschitz graph.

######  Remark 1.3.

A large scale flatness assumption is needed because even minimizers of $J_{0}$
can have singular free boundary points in higher dimensions [22].

######  Remark 1.4.

Note that both results are new even for minimizers $R_{0}=+\infty$. We
consider the more general case of almost minimizers because it adds only minor
additional difficulties and it is useful for applications.

######  Remark 1.5.

We must point out that it is extremely important that we are considering
energy minimizers or almost minimizers and not general local minimizers or
critical points (1.3). It is not true in general that solutions of (1.3), or
even local energy minimizers of $J$, converge to solutions of (1.4) in the
large scale limit [9, 18, 20].

######  Remark 1.6.

The matching of the periodicity lattices of $a$ and $Q$ is actually not
important at all. In fact the arguments easily extend to non-periodic $Q$ with
a quantitative estimate on the convergence of the spatial averages

$\sup_{x}\sup_{r\geqslant
R}|\frac{1}{r^{d}}\int_{x+[-\frac{r}{2},\frac{r}{2})^{d}}Q(y)^{2}\ dy-\langle
Q^{2}\rangle|\leqslant CR^{-\alpha}$

where the important feature of the Hölder rate is that it is summable over any
geometric sequence of scales. Sufficiently mixing stationary ergodic random
fields $Q$ where the previous estimate holds without the supremum and with a
random variable $C(x)$ satisfying some tail bounds should also be easy to fit
into the arguments.

We also expect that analogous results hold for finite range dependence random
fields $a(x)$ a la [6]. The proofs are set up to be adaptable to this case but
this is a more difficult extension and there could be unforeseen challenges.

######  Remark 1.7.

The result Theorem 1.2 implies a Liouville result for large scale flat
minimizers on the whole space. This is independently interesting. The
Liouville result and related discussion can be found below in Section 6.1. In
particular, related to the difficulty of obtaining $C^{1,\alpha}$ estimates of
the free boundary all the way down to the unit scale, see Remark 6.8.

######  Remark 1.8.

The large scale flatness assumption would typically be justified in an
application by proving an initial sub-optimal convergence rate to a regular
limiting object. The result Theorem 1.2 then takes this large scale flatness
and iterates it down to all scales above the unit scale. There is
$C^{1,\alpha}$ improvement of flatness at intermediate scales which could be
useful in some cases, we did not include the statement here in the
introduction but it can be found below in Section 6.

######  Remark 1.9.

Generally it is preferable to remove microscopic regularity hypotheses on the
coefficient field, which are philosophically unrelated to large scale
regularity. There are some impediments to completely removing these
assumptions in Theorem 1.2. In Theorem 1.1 we use $a\in C^{0,\gamma}$ to
obtain pointwise gradient bounds on $u$. The pointwise gradient bounds are
convenient, but could probably be replaced by the large scale gradient bound
in Theorem 1.1 for most of the elements in the proof of Theorem 1.2.

The key place where we do seem to need local regularity of $u(x)$, and
therefore of $a(x)$, is in the strong non-degeneracy estimate Lemma 4.3.
Strong non-degeneracy can be proved from weak non-degeneracy using a now
standard Harnack chain type argument due to Caffarelli [11] which needs to
start from weak non-degeneracy in a, possibly, very small ball near the free
boundary, see also Remark 4.4 below.

Also in the course of proving strong non-degeneracy we use a standard Hopf
Lemma barrier construction in Lemma 4.2. This is the only place that Lipschitz
as opposed to $C^{0,\gamma}$ is used in order to write the equation in non-
divergence form. Hopf Lemma does hold for Hölder continuous coefficients, see
[3], so the Lipschitz assumption could possibly be relaxed to just
$C^{0,\gamma}$ with more work.

###  1.1. Literature

The idea that elliptic problems in heterogeneous media could inherit
regularity, up to a certain threshold, from their large scale effective limit
was first introduced by Avellaneda and Lin in a series of papers [7]. Often
this kind of regularity theory can be used to obtain optimal quantitative
convergence estimates for the homogenization limit. This idea was re-
emphasized by Armstrong and Smart [6] who were able to adapt the Avellaneda
and Lin idea for the first time in random media. The idea of [6] is to first
obtain a suboptimal quantitative homogenization result, which can then be used
within the $C^{1,\alpha}$ iteration as long as the quantitative rate is
summable over geometric sequences of scales. It is exactly this idea which we
are applying in the context of free boundary problems.

The regularity theory for minimizers of $J_{0}$ was first significantly
developed by Alt and Caffarelli [2]. The topic has been extremely popular
since then and we do not make an attempt at a thorough recounting. We will
just point out the most relevant, which is the recent development of
regularity theory of almost minimizers. A series of papers by (various subsets
of) the authors David, Engelstein, Smit Vega Garcia, and Toro [15, 14, 13]
have initiated and made major developments in the study of almost minimizers
of the more general two-phase version of the energy $J$. David, Engelstein,
Smit Vega Garcia and Toro [13] (two-phase) and, independently, Trey [27]
(vector-valued) have proved a Lipschitz estimate for almost minimizers of
variable coefficient Bernoulli-type problems. De Silva and Savin [17] gave
another account of the regularity theory for almost minimizers of the one
phase problem, and, as mentioned before, we will be following their ideas.
Note that all these works are considering small scale regularity, while we are
considering large scales. In other words one can consider our work to be on
oscillatory coefficients $a(x/\varepsilon)$ which do not satisfy a uniform in
$\varepsilon>0$ Hölder regularity hypothesis.

There are very few works on the large scale regularity theory of free boundary
problems in heterogeneous media. The only other result we are aware of is by
Aleksanyan and Kuusi [1]. They obtain a result analogous to ours for a version
of the obstacle problem in random media. A major difficulty in studying the
Bernoulli free boundary problem, as far as we understand it, is that the
Euler-Lagrange equation (1.3) cannot be enough information by itself to get
regularity. This is already reflected in techniques for the homogeneous
problem which variously use the additional properties of being a minimizer,
almost minimizer, maximal subsolution, or minimal supersolution. As
established in [9, 23, 20, 18] the large scale limit of minimal
supersolutions, maximal subsolutions, and minimizers are all rather different
Bernoulli free boundary problems and thus regularity theory techniques would
need to be adapted to each case. In fact the regularity theory for the
homogenized problem for minimal supersolutions and maximal subsolutions is not
understood yet [18]. Thus we were led to consider first the case of minimizers
and almost minimizers as we do in this paper. In this case the large scale
effective problem is a standard constant coefficient minimization problem and
the regularity theory is well understood.

###  1.2. Acknowledgments

The author was supported by the NSF grant DMS-2009286. The author would like
to thank Farhan Abedin for extensive and motivating discussions on the topics
of the paper.

##  2\. Background results

This section introduces notations and conventions which will be used in the
paper and review some well known results from the literature about divergence
form elliptic homogenization.

###  2.1. Assumptions and conventions

As described in the introduction we will study energy functionals of the
following type

$J(u,U)=\int_{U}\nabla u\cdot a(x)\nabla u+Q(x)^{2}{\bf 1}_{\\{u>0\\}}\ dx.$

We make precise the assumptions on the coefficients which will be in place
throughout the paper, unless otherwise specified in some few locations.

  1. (Q1)

The coefficient $Q:\varmathbb{R}^{d}\to(0,\infty)$ is assumed to satisfy the
ellipticity condition

$\Lambda^{-1}\leqslant Q(x)\leqslant\Lambda.$

  2. (Q2)

The coefficient $Q$ is assumed to be $\varmathbb{Z}^{d}$-periodic and satisfy
the normalization condition

$\langle Q^{2}\rangle=\int_{[0,1]^{2}}Q(x)^{2}\ dx=1$

  1. (a1)

The coefficients $a:\varmathbb{R}^{d}\to M_{d\times d}^{sym}(\varmathbb{R})$
are $\varmathbb{Z}^{d}$-periodic and satisfy the ellipticity bounds

$\Lambda^{-1}I\leqslant a(x)\leqslant\Lambda I$

  2. (a2)

The coefficient field $a:\varmathbb{R}^{d}\to M_{d\times
d}^{sym}(\varmathbb{R})$ is Lipschitz.

See Remark 1.9 above for comments about the role of the microscopic regularity
Assumption (a1).

  1. $\bullet$

Constants $C$ and $c$ in the text below which depend at most on $d$,
$\Lambda$, $\beta$, $\|\nabla a\|_{\infty}$ are called _universal_. Such
constants may change meaning from line to line without mention. If we intend
to fix the value of such a universal constant for a segment of the argument we
may denote it as $c_{0}$, $C_{0}$, $c_{1}$ etc.

Notations:

  1. $\bullet$

Balls are denoted $B_{r}$ (if the center does not need denotation) or
$B_{r}(x)$. Boxes are denoted

$\square_{r}=[-r/2,r/2)^{d}\ \hbox{ and }\ \square_{r}(x)=x+\square_{r}.$

  2. $\bullet$

Lebesgue measure of a measurable set $U$ is denoted $|U|$. This notation may
also be used for lower dimensional measures if the set is obviously a
particular Hausdorff dimension, i.e. $|\partial B_{r}|$ is the surface area of
the sphere of radius $r$.

  3. $\bullet$

Averaged integrals

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{U}f\ dx=\frac{1}{|U|}\int_{U}f\ dx$

  4. $\bullet$

Averaged $L^{p}$ norms

$\|f\|_{{\underline{L}}^{p}(U)}=(\frac{1}{|U|}\int_{U}|f|^{p}\ dx)^{1/p}.$

###  2.2. Divergence form elliptic equations with periodic coefficients

We consider elliptic energy functionals of the type

$E(u,U)=\int_{U}\nabla u\cdot a(x)\nabla u\ dx$

evaluated on a domain $U$ of $\varmathbb{R}^{d}$.

Given $g\in W^{1,p}(U)$ for some $p\geqslant 2$ we can consider the Dirichlet
problem

$\min\\{E(u,U):u\in g+H^{1}_{0}(U)\\}$

with the associated Euler-Lagrange PDE

(2.1) $\begin{cases}-\nabla\cdot(a(x)\nabla u)=0&\hbox{in }U\\\ u=g&\hbox{on
}\partial U\end{cases}$

all interpreted in the appropriate weak sense.

We introduce the correctors $\chi_{q}$ which are the unique global
$\varmathbb{Z}^{d}$ periodic and mean zero solutions of the problem

$-\nabla\cdot(a(x)(q+\nabla\chi_{q}))=0\ \hbox{ in }\ \varmathbb{R}^{d}$

existing for each $q\in\varmathbb{R}^{d}$. It follows from the previous
properties that $\chi_{q}$ depends linearly on $q\in\varmathbb{R}^{d}$.

We make a few notes about corrector bounds which will be useful. By
multiplying the corrector equation by $q$ and integrating over a unit period
cell,

$\|\nabla\chi_{q}\|_{L^{2}([0,1)^{d})}\leqslant\Lambda^{2}|q|.$

By Poincaré and mean zero also

$\|\chi_{q}\|_{L^{2}([0,1)^{d})}\leqslant C|q|$

and using periodicity we can also conclude that for all $r\geqslant 1$

(2.2)
$\|\nabla\chi_{q}\|_{\underline{L}^{2}(B_{r})}+\|\chi_{q}\|_{\underline{L}^{2}(B_{r})}\leqslant
C|q|$

for a universal $C\geqslant 1$. By standard elliptic regularity theory (2.2)
can be upgraded to an $L^{\infty}$ estimate if we assume that $a\in
C^{0,\gamma}$, the new constant $C$ will also depend on $\gamma$ and
$[a]_{C^{0,\gamma}}$.

The homogenized matrix is defined

(2.3)
$\bar{a}_{ij}=\langle(e_{i}+\nabla\chi_{e_{i}}(x))a(x)(e_{j}+\nabla\chi_{e_{j}}(x))\rangle$

where $e_{i}$ are the standard basis vectors. We define the homogenized energy
functional

$E_{0}(u,U)=\int_{U}\nabla u\cdot\overline{a}\nabla u\ dx$

Given these definitions the following convergence result is classical. See [6,
4] for a proof in the more difficult context of random media.

######  Theorem 2.1 (Homogenized replacement in balls).

Suppose $r\geqslant 1$, $g\in W^{1,p}(B_{r})$ for some $p>2$, and $u$ and
$u_{0}$ respectively minimize $E$ and $E_{0}$ over $H^{1}_{0}(B_{r})+g$. Then
there are $\alpha\in(0,1)$ and $C\geqslant 1$ depending on $d$, $\Lambda$, and
$p$ so that

$|E(u,B_{r})-E_{0}(u_{0},B_{r})|+\frac{1}{r^{2}}\|u-u_{0}\|^{2}_{L^{2}(B_{r})}\leqslant
C\|\nabla g\|_{\underline{L}^{p}(U)}^{2}r^{-\alpha}|B_{r}|.$

This result does hold in the more general class of Lipschitz domains, and that
assumption can be pushed a bit further, but we will not use that generality in
our paper. We will use a slightly different homogenization result in the
positivity set $\\{u>0\\}$ which takes advantage of the a-priori Lipschitz
bound from Section 3, see Section 5.1 for that statement and more discussion.

From an initial quantitative homogenization result one can also prove a
version of elliptic regularity for $a$-harmonic functions. We will make use of
the following facts.

######  Theorem 2.2 (Interior regularity).

(1) (Lipschitz estimate [7, Lemma 16]) If $u$ is $a$-harmonic in $B_{r}$ then

$\|\nabla
u\|_{L^{\infty}(B_{r/2})}\leqslant\frac{C}{r}\|u\|_{L^{\infty}(B_{r})}$

where the constant $C$ is universal.

(2) ($C^{1,\alpha}$ estimate [7, Lemma 15]) If $u$ is $a$-harmonic in $B_{R}$
then for any $r_{0}\leqslant r\leqslant R$

$\sup_{B_{r}}|u(x)-(A+q\cdot x+\chi_{q}(x))|\leqslant
C(\frac{r}{R})^{1+\alpha}\|u\|_{\underline{L}^{2}(B_{R})}$

for some $q$ and $A$ depending on $r$ with $|A|+r|q|\leqslant
C\|u\|_{\underline{L}^{2}(B_{R})}$. And, since $A+q\cdot x+\chi_{q}$ is
$a$-harmonic,

$\|\nabla u-(q+\nabla\chi_{q})\|_{\underline{L}^{2}(B_{r})}\leqslant
C(\frac{r}{R})^{\alpha}\frac{1}{R}\|u\|_{\underline{L}^{2}(B_{R})}\,.$

An analogous $C^{1,1}$ estimate is also true, the general theory in [4]
explains this, but the statements there are for random media and in a less
convenient form for our purposes.

##  3\. Lipschitz estimate

In this section we prove the Lipschitz estimate Theorem 1.1 for almost
minimizers of energy functionals of the form

$J(u,U)=\int_{U}a(x)\nabla u\cdot\nabla u+Q(x)^{2}{\bf 1}_{\\{u>0\\}}\ dx.$

In this section $a(x)$ will be $\varmathbb{Z}^{d}$-periodic, and $a$ and $Q$
will satisfy the ellipticity bounds i.e. we are under Assumption (a1) and
Assumption (Q1). We do not need to assume any further structural assumptions
(e.g. periodicity, regularity) on $Q$ at this stage.

A competitor $u\in H^{1}(U)$ is called a $(R_{0},\beta)$-almost minimizer in a
domain $U$ if, for any $B_{r}\subset U$, and any $v\in u+H^{1}_{0}(B_{r})$

(3.1) $J(u,B_{r})\leqslant(1+(r/R_{0})^{\beta})J(v,B_{r}).$

######  Remark 3.1.

As mentioned in the statement of Theorem 1.1, for the purposes of the
Lipschitz estimate we only need the almost minimizing property to hold for the
$a$-harmonic replacement. That fact is convenient for making translations
between slightly different notions of almost minimality.

We also remark that the proofs go through with almost no change with an
additional additive term: for any $B_{r}\subset U$, and any $v\in
u+H^{1}_{0}(B_{r})$

(3.2)
$J(u,B_{r})\leqslant(1+(r/R_{0})^{\beta})J(v,B_{r})+(r/R_{0})^{\beta}|B_{r}|.$

We follow the approach of De Silva and Savin [17] which proceeds in three
steps:

  * •

Step 1 (Dichotomy): if an almost minimizer $u$ has large $L^{2}$-averaged
slope in a ball $B_{r}$ then either the $L^{2}$ averaged slope decreases by
half in a smaller ball $B_{\eta r}$ or $u$ is close to a corrected linear
$a$-harmonic function in $B_{\eta r}$.

  * •

Step 2 (Interior-like improvement): In the second case of the previous
dichotomy $u$ has interior $C^{1,\alpha}$-like improvement down to the unit
scale.

  * •

Step 3 (Iteration): Iterating the dichotomy either case results in a Lipschitz
bound.

The main point for us is to use the correctors and the Avellaneda-Lin [7]
interior estimates Theorem 2.2 in place of the corresponding results for
homogeneous problems. Although this sounds rather straightforward, we were
unable to make such an idea succeed in any of the other standard proofs of the
Lipschitz estimate from the literature, see [29] for a survey of such methods.

###  3.1. Step 1: Dichotomy

######  Lemma 3.2.

Let $u\in H^{1}(B_{r})$ and suppose that

$J(u,B_{r})\leqslant(1+\sigma)J(v,B_{r})\ \hbox{ for all }\ v\in
u+H^{1}_{0}(B_{r}).$

Let $\frac{1}{4}\geqslant\delta>0$ there exist constants $\eta$, $M$ and
$\sigma_{0}$ (depending on $\delta$ and universal constants) so that if
$\sigma\leqslant\sigma_{0}$, $\eta r\geqslant 1$, and

$\|\nabla u\|_{{\underline{L}}^{2}(B_{r})}\geqslant M$

then the following dichotomy holds: either

$\|\nabla u\|_{{\underline{L}}^{2}(B_{\eta r})}\leqslant\frac{1}{2}\|\nabla
u\|_{{\underline{L}}^{2}(B_{r})}$

or

$\|\nabla u-(q+\nabla\chi_{q}(x))\|_{{\underline{L}}^{2}(B_{\eta
r})}\leqslant\delta\|\nabla u\|_{{\underline{L}}^{2}(B_{r})}$

with some $q\in\varmathbb{R}^{n}$ satisfying

$C_{0}^{-1}\|\nabla u\|_{{\underline{L}}^{2}(B_{r})}\leqslant|q|\leqslant
C_{0}\|\nabla u\|_{{\underline{L}}^{2}(B_{r})}$

with a constant $C_{0}\geqslant 1$ universal (not depending on $\delta$).

###### Proof.

1\. Let $v$ be the $a$-harmonic replacement of $u$ in $B_{1}$. Then

$\int_{B_{r}}a(x)\nabla(u-v)\cdot\nabla(u-v)\ dx\leqslant
J(u,B_{r})+\int_{B_{r}}a(x)\nabla v\cdot\nabla v-2a(x)\nabla u\cdot\nabla v\
dx.$

Using the minimizing property of $u$ and ellipticity

$\Lambda^{-1}\|\nabla u-\nabla
v\|^{2}_{L^{2}(B_{r})}\leqslant(1+\sigma)J(v,B_{r})+\int_{B_{r}}a(x)\nabla
v\cdot\nabla v-2a(x)\nabla u\cdot\nabla v\ dx.$

Then combining the $|\nabla v|^{2}$ terms on the right

$\Lambda^{-1}\|\nabla u-\nabla v\|^{2}_{L^{2}(B_{r})}\leqslant\sigma\|\nabla
v\|^{2}_{L^{2}(B_{r})}+\Lambda|B_{r}|+\int_{B_{r}}2a(x)\nabla
v\cdot\nabla(v-u)\ dx.$

The last term is $0$ because $v$ is $a$-harmonic so

$\|\nabla u-\nabla v\|^{2}_{\underline{L}^{2}(B_{r})}\leqslant C\sigma\|\nabla
v\|^{2}_{\underline{L}^{2}(B_{r})}+C\leqslant C(\sigma\|\nabla
u\|^{2}_{\underline{L}^{2}(B_{r})}+1).$

2\. Since $v$ is $a$-harmonic in $B_{r}$ we can apply the interior
$C^{1,\alpha}$ estimate Theorem 2.2 to find that, as long as $\eta r\geqslant
1$,

$\|\nabla v-(q+\nabla\chi_{q})\|_{\underline{L}^{2}(B_{\eta r})}\leqslant
C\eta^{\alpha}\|\nabla v\|_{\underline{L}^{2}(B_{r})}\leqslant
C\eta^{\alpha}\|\nabla u\|_{\underline{L}^{2}(B_{r})}$

for some

$|q|\leqslant C\|\nabla u\|_{\underline{L}^{2}(B_{r})}.$

Note also, by the corrector estimates (2.2),

$\|\nabla\chi_{q}\|_{\underline{L}^{2}(B_{r})}\leqslant C|q|.$

3\. Now we combine the previous parts to estimate

$\displaystyle\|\nabla u-(q+\nabla\chi_{q}(x))\|_{\underline{L}^{2}(B_{\eta
r})}$ $\displaystyle\leqslant\|\nabla
v-(q+\nabla\chi_{q}(x))\|_{\underline{L}^{2}(B_{\eta r})}+\|\nabla u-\nabla
v\|_{\underline{L}^{2}(B_{\eta r})}$ $\displaystyle\leqslant
C\eta^{\alpha}\|\nabla
u\|_{\underline{L}^{2}(B_{r})}+C\sigma^{1/2}\eta^{-d/2}\|\nabla
u\|_{\underline{L}^{2}(B_{r})}+C\eta^{-d/2}$

and so, also,

$\|\nabla u\|_{\underline{L}^{2}(B_{\eta r})}\leqslant C\eta^{\alpha}\|\nabla
u\|_{\underline{L}^{2}(B_{r})}+C\sigma^{1/2}\eta^{-d/2}\|\nabla
u\|_{\underline{L}^{2}(B_{r})}+C\eta^{-d/2}+C|q|.$

Both estimates require that $\eta r\geqslant 1$.

Now, we first fix $\eta>0$ small depending on $\delta$, and then fix
$\sigma_{0}>0$ small and $M>1$ large depending on $\delta$ and the choice of
$\eta$ so that

$\displaystyle C\eta^{\alpha}\|\nabla u\|_{\underline{L}^{2}(B_{r})}+$
$\displaystyle C\sigma^{1/2}\eta^{-d/2}\|\nabla
u\|_{\underline{L}^{2}(B_{r})}+C\eta^{-d/2}$
$\displaystyle\leqslant\frac{1}{4}\delta\|\nabla
u\|_{\underline{L}^{2}(B_{r})}+C\eta^{-d/2}M^{-1}\|\nabla
u\|_{\underline{L}^{2}(B_{r})}$ $\displaystyle\leqslant\delta\|\nabla
u\|_{\underline{L}^{2}(B_{r})}.$

Now, as long as $\eta r\geqslant 1$, we have the two inequalities

$\|\nabla u-(q+\nabla\chi_{q}(x))\|_{\underline{L}^{2}(B_{\eta
r})}\leqslant\delta\|\nabla u\|_{\underline{L}^{2}(B_{r})}$

and, using $\delta\leqslant\frac{1}{4}$,

$\|\nabla u\|_{\underline{L}^{2}(B_{\eta r})}\leqslant\frac{1}{4}\|\nabla
u\|_{\underline{L}^{2}(B_{r})}+C_{1}|q|.$

4\. Finally we divide up into the dichotomy in the statement. If

$|q|\leqslant\frac{1}{4}C_{1}^{-1}\|\nabla u\|_{\underline{L}^{2}(B_{r})}$

then

$\|\nabla u\|_{\underline{L}^{2}(B_{\eta r})}\leqslant\frac{1}{2}\|\nabla
u\|_{\underline{L}^{2}(B_{r})}$

while, otherwise,

$\|\nabla u-(q+\nabla\chi_{q}(x))\|_{\underline{L}^{2}(B_{\eta
r})}\leqslant\delta\|\nabla u\|_{\underline{L}^{2}(B_{r})}$

with some

$\frac{1}{4}C_{1}^{-1}\|\nabla
u\|_{\underline{L}^{2}(B_{r})}\leqslant|q|\leqslant C\|\nabla
u\|_{\underline{L}^{2}(B_{r})}.$

∎

###  3.2. Step 2: Close to planar implies interior-like $C^{1,\alpha}$
improvement

The next lemma says that if $\nabla u$ is $\delta|q|$ close to a corrector
gradient $q+\nabla\chi_{q}$ in $\underline{L}^{2}$, then there is an interior
$C^{1,\alpha}$-like improvement of oscillation. Basically an energy argument
shows that the zero level $\\{u=0\\}$ must be relatively small and so the
oscillation improvement of the $a$-harmonic replacement carries over to $u$.

######  Lemma 3.3.

Let $u$ as in Lemma 3.2 and $r\geqslant 1$. Suppose that $|q|\geqslant 1$ and

$\|\nabla
u-(q+\nabla\chi_{q}(x))\|_{\underline{L}^{2}(B_{r})}\leqslant\delta|q|$

for some $1>\delta>0$.

There are $\alpha_{0}>0$, $\mu>0$, $1>\bar{\delta}>0$ and $c_{0}>0$ universal
such that if

$\delta\leqslant\bar{\delta},\ \sigma\leqslant c_{0}\delta^{2},\ \hbox{ and }\
\mu r\geqslant 1$

then

$\|\nabla
u-(q^{\prime}+\nabla\chi_{q^{\prime}}(x))\|_{\underline{L}^{2}(B_{\mu
r})}\leqslant\mu^{\alpha_{0}}\delta|q^{\prime}|$

with $q^{\prime}\in\varmathbb{R}^{d}$ satisfying

$|q^{\prime}-q|\leqslant C_{1}\delta|q|$

for a universal $C_{1}\geqslant 1$.

The result should be possible with any $\alpha\in(0,1)$ with parameters
depending on $\alpha$ using the $C^{1,1}$ version of Theorem 2.2.

###### Proof.

1\. (Harmonic replacement) Let $\bar{v}$ be the $a$-harmonic replacement of
$u$ in $B_{r/2}$ and let $v$ be

$v=u\ \hbox{ in }\ B_{r}\setminus B_{r/2}\ \hbox{ and }\ v=\bar{v}\ \hbox{ in
}\ B_{r/2}.$

Then

$J(u,B_{r})\leqslant(1+\sigma)J(v,B_{r})$

or

$J(u,B_{r/2})\leqslant\sigma J(u,B_{r}\setminus
B_{r/2})+(1+\sigma)J(v,B_{r/2}).$

Also note, using the corrector estimate
$\|\nabla\chi_{q}\|_{\underline{L}^{2}(B_{r})}\leqslant C|q|$ from (2.2),

$\|\nabla u\|_{\underline{L}^{2}(B_{r})}\leqslant C|q|.$

Thus

$\int_{B_{r/2}}a(x)\nabla u\cdot\nabla u-a(x)\nabla v\cdot\nabla v\
dx+\int_{B_{r/2}}Q(x){\bf 1}_{\\{u>0\\}}\
dx\leqslant\sigma(C|q|^{2}+\Lambda)|B_{r}|+\int_{B_{r/2}}Q(x)\ dx.$

Since $v$ is the $a$-harmonic replacement for $u$ in $B_{r/2}$

$\int_{B_{r/2}}a(x)\nabla u\cdot\nabla u-a(x)\nabla v\cdot\nabla v\
dx=\int_{B_{r/2}}a(x)\nabla(u-v)\cdot\nabla(u-v)\ dx$

so, plugging that in, re-arranging, and using ellipticity,

$\|\nabla u-\nabla v\|_{\underline{L}^{2}(B_{r/2})}^{2}\leqslant
C\sigma(|q|^{2}+1)+\frac{1}{|B_{r/2}|}\int_{B_{r/2}}Q(x){\bf 1}_{\\{u=0\\}}\
dx$

or

(3.3) $\|\nabla u-\nabla v\|_{\underline{L}^{2}(B_{r/2})}^{2}\leqslant
C\sigma|q|^{2}+\Lambda\frac{|\\{u(x)=0\\}\cap B_{r/2}|}{|B_{r/2}|}$

2\. (Large slope implies small measure of zero level) Next we claim that

$\frac{|\\{u(x)=0\\}\cap B_{r/2}|}{|B_{r/2}|}\leqslant C\delta^{2+\beta}$

for some universal $C,\beta>0$. Actually it will hold for $2+\beta\leqslant
2^{*}=\frac{2d}{d-2}=2+\frac{4}{d-2}$ the Sobolev embedding exponent for
$L^{2}$ (or $2+\beta<2^{*}=+\infty$ in $d=2$).

Note that

$\int_{B_{r}}(u-\ell)\ dx=0\ \hbox{ where }\ \ell(x)=q\cdot
x+\chi_{q}(x)+\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}}u-\chi_{q}\ dx.$

So, by Poincaré,

$c\|u-\ell\|_{\underline{L}^{2}(B_{r})}\leqslant r\|\nabla
u-\nabla\ell\|_{\underline{L}^{2}(B_{r})}\leqslant\delta|q|r.$

Since $u\geqslant 0$

$c\|(\ell)_{-}\|_{\underline{L}^{2}(B_{r})}\leqslant\delta|q|r.$

If $q\cdot x+\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}}(u-\chi_{q})$ were not positive in
$B_{3r/4}$ then $|q\cdot x|\geqslant\frac{1}{8}|q|r$ in some ball of radius
$r/16$ contained in $B_{r}$ so, using
$\|\chi_{q}\|_{\underline{L}^{2}(B_{r})}\leqslant K|q|$ from (2.2),

$\|(\ell)_{-}\|_{\underline{L}^{2}(B_{r})}\geqslant
c|q|r-\|\chi_{q}\|_{\underline{L}^{2}(B_{r})}\geqslant c|q|(r-c^{-1}K)$

for a universal constant $c$. This is a contradiction for $r\geqslant
2c^{-1}K$ and $\delta$ sufficiently small depending on $K$ and other universal
constants. Note that, given the hypothesis $\mu r\geqslant 1$, we can think of
this as a requirement on the choice of $\mu$, that we must choose
$\mu\leqslant\frac{1}{2}cK^{-1}$.

Thus

$\ell(x)\geqslant\frac{1}{4}|q|r-\|\chi_{q}\|_{\infty}\geqslant\frac{1}{4}|q|(r-4K)\geqslant
c|q|r\ \hbox{ in }\ B_{r/2}$

for a universal $c>0$ and using $r\geqslant 8K$ (imposing additionally
$\mu\leqslant\frac{1}{8K}$).

By Poincaré-Sobolev

$\|u-\ell\|_{\underline{L}^{2^{*}}(B_{r})}\leqslant Cr\|\nabla
u-\nabla\ell\|_{\underline{L}^{2}(B_{r})}\leqslant C\delta|q|r$

but also

$\|u-\ell\|_{\underline{L}^{2^{*}}(B_{r})}\geqslant c|q|r\frac{|\\{u=0\\}\cap
B_{r/2}|^{1/2^{*}}}{|B_{r/2}|^{1/2^{*}}}.$

So the claim follows combining the previous two inequalities.

3\. Note that $v(x)-(q\cdot x+\chi_{q}(x))$ is the $a$-harmonic replacement of
$u(x)-(q\cdot x+\chi_{q}(x))$. So

$\|\nabla
v-(q+\nabla\chi_{q})\|_{\underline{L}^{2}(B_{r/2})}\leqslant\Lambda^{2}\|\nabla
u-(q+\nabla\chi_{q})\|_{\underline{L}^{2}(B_{r/2})}\leqslant\delta\Lambda^{2}|q|.$

By the Avellaneda-Lin interior $C^{1,\alpha}$ estimates for $a$-harmonic
functions Theorem 2.2 applied to $v(x)-(q\cdot x+\chi_{q}(x))$, for any
$0<\mu<1/2$ such that $\mu r\geqslant 1$,

$\|\nabla
v-(q+\nabla\chi_{q})-(\bar{q}+\nabla\chi_{\bar{q}})\|_{\underline{L}^{2}(B_{\mu
r})}\leqslant C\mu^{\alpha}\|\nabla
v-(q+\nabla\chi_{q})\|_{\underline{L}^{2}(B_{r/2})}\leqslant
C\mu^{\alpha}\delta|q|$

for some $\bar{q}\in\varmathbb{R}^{d}$ with

$|\bar{q}|\leqslant C\|\nabla
v-(q+\nabla\chi_{q})\|_{\underline{L}^{2}(B_{r/2})}\leqslant C\delta|q|.$

Write $q^{\prime}=q+\bar{q}$ and recall the linearity of the corrector
$\chi_{q}+\chi_{\bar{q}}=\chi_{q+\bar{q}}$. Assuming
$C\delta\leqslant\frac{1}{2}$ we have $2|q^{\prime}|\geqslant|q|\geqslant 1$.

Combining this with estimate (3.3)

$\|\nabla
u-(q^{\prime}+\nabla\chi_{q^{\prime}})\|^{2}_{\underline{L}^{2}(B_{\mu
r})}\leqslant
C\sigma\mu^{-d}|q|^{2}+C\mu^{-d}\delta^{2+\beta}+C\mu^{2\alpha}\delta^{2}|q|^{2}.$

Now choose parameters in the following order: let $\alpha_{0}=\alpha/2$ (or
anything smaller than $\alpha$ works), then choose $\mu$ small enough so that
$C\mu^{2(\alpha-\alpha_{0})}\leqslant\frac{1}{6}$ for the third term (in
addition to the previous requirements), then choose
$\delta\leqslant\bar{\delta}$ with
$\bar{\delta}^{\beta}=\frac{1}{6}C^{-1}\mu^{d+2\alpha_{0}}$ for the second
term. Finally the condition
$\sigma\leqslant\frac{1}{6}C^{-1}\delta^{2}\mu^{d+2\alpha_{0}}$ gives that all
three terms above are smaller than
$\frac{1}{6}\mu^{2\alpha_{0}}\delta^{2}|q^{\prime}|^{2}$ (note we are using
$|q^{\prime}|\geqslant\frac{1}{2}|q|\geqslant\frac{1}{2}$).

∎

######  Lemma 3.4.

Suppose that $u$ is a $(R_{0},\beta)$-almost minimizer of $J$ in $B_{R}$ for
some $R_{0}\geqslant R\geqslant 1$ and for some $|q|\geqslant 2$

$\|\nabla
u-(q+\nabla\chi_{q}(x))\|_{\underline{L}^{2}(B_{R})}\leqslant\delta|q|.$

There is universal $\delta_{*}>0$ so that if $\delta\leqslant\delta_{*}$ then

$\|\nabla u\|_{\underline{L}^{2}(B_{r})}\leqslant C|q|\ \hbox{ for all }\
1\leqslant r\leqslant R.$

###### Proof.

We iterate Lemma 3.3 with $\alpha=\beta/2$. Pick $\delta_{*}$ so that

$\delta_{*}\leqslant\bar{\delta}\ \hbox{ and }\
4C_{1}\frac{1}{1-\mu^{\alpha}}\delta_{*}\leqslant 1$

where $\bar{\delta}$ is the universal constant from Lemma 3.3.

It suffices to consider the case

(3.4) $\sigma_{0}:=(R/R_{0})^{\beta}\leqslant c_{0}\delta_{*}^{2}$

where $c_{0}$ is the universal constant from Lemma 3.3 and $\delta_{*}$ is
defined above.

We claim that for all $j$ such that $\mu^{j}R\geqslant 1$ there are $q_{j}$
such that

(3.5) $\|\nabla
u-(q_{j}+\nabla\chi_{q_{j}}(x))\|_{\underline{L}^{2}(B_{\mu^{j}R})}\leqslant\delta_{*}\mu^{\alpha
j}|q_{j}|$

with

(3.6) $|q_{j+1}-q_{j}|\leqslant C_{1}\delta_{*}\mu^{\alpha j}|q_{j}|\ \hbox{
and }\ 1\leqslant|q_{j}|\leqslant 2|q|.$

We will prove this by induction. By the hypothesis of the theorem (3.5) and
(3.6) hold for $j=0$. Suppose that (3.5) and (3.6) hold for all $0\leqslant
j\leqslant k$.

Note that we have

$|q_{k+1}-q|\leqslant\sum_{j=0}^{k}|q_{j+1}-q_{j}|\leqslant
2C_{1}\frac{1}{1-\mu^{\alpha}}\delta_{*}|q|.$

In particular

$|q_{k+1}|\leqslant(1+2C_{1}\frac{1}{1-\mu^{\alpha}}\delta_{*})|q|\leqslant
2|q|$

because of the choice of $\delta_{*}$, and, similarly,

$|q_{k+1}|\geqslant(1-2C_{1}\frac{1}{1-\mu^{\alpha}}\delta_{*})|q|\geqslant\frac{1}{2}|q|\geqslant
1.$

Thus we have established the second part of (3.6) for $k+1$.

If $\mu^{k+1}R\leqslant 1$ then we are already done. Otherwise $u$ is in the
set up of Lemma 3.3 in $B_{\mu^{k}R}$ with (3.5) and

(3.7) $\delta_{k}=\delta_{*}\mu^{\alpha k}\ \hbox{ and }\
\sigma_{k}=(r_{k}/R_{0})^{\beta}=\mu^{k\beta}\sigma_{0}.$

Note that $\delta_{k}\leqslant\delta_{*}\leqslant\bar{\delta}$ and

(3.8)
$\sigma_{k}=\mu^{2k\alpha}\sigma_{0}=\delta_{k}^{2}\delta_{*}^{-2}\sigma_{0}\leqslant
c_{0}\delta_{k}^{2}$

by (3.4). Thus all the assumptions of Lemma 3.3 hold and we find that there is
$q_{k+1}$ such that

$\|\nabla
u-(q_{k+1}+\nabla\chi_{q_{k+1}}(x))\|_{\underline{L}^{2}(B_{\mu^{k+1}R})}\leqslant\delta\mu^{\alpha(k+1)}|q_{k+1}|$

and with

$|q_{k+1}-q_{k}|\leqslant C_{1}\delta_{k}|q_{k}|=C_{1}\mu^{\alpha
k}\delta_{*}|q_{k}|.$

Then we have the result of the theorem because (3.5) and (3.6) imply

$\|\nabla u\|_{\underline{L}^{2}(B_{\mu^{k}R})}\leqslant C|q_{j}|\leqslant
C|q|$

and the estimate for intermediate values $\mu^{k+1}R\leqslant r<\mu^{k}R$ also
holds up to an additional factor of $\mu^{-d/2}$ which is also universal.

∎

######  Remark 3.5.

The previous proof will also go through when the almost minimality hypothesis
(3.1) is replaced by

(3.9)
$J(u,B_{r})\leqslant(1+\omega(r/R_{0}))J(v,B_{r})+\omega(r/R_{0})|B_{r}|$

(for $v$ the $a$-harmonic replacement in $B_{r}$) as long as
$\omega^{\frac{1}{2}}$ is a Dini modulus, in the sense that
$\int_{0}^{1}\omega(t)^{\frac{1}{2}}\frac{dt}{t}<+\infty$, with
$\omega(1)\leqslant 1$. The inductive hypotheses (3.5) and (3.6) with
$\delta_{k}=\delta_{*}\mu^{k\alpha}$ are replaced by

$\delta_{k}=\delta_{*}\omega(R/R_{0})^{-\frac{1}{2}}\omega(\mu^{k}R/R_{0})^{\frac{1}{2}}.$

This choice is so that later in (3.7), when
$\sigma_{k}=\omega(\mu^{k}R/R_{0})$, we will have

$\sigma_{k}=\delta_{k}^{2}\delta_{*}^{-2}\omega(R/R_{0})\leqslant
c_{0}\delta_{k}^{2}$

as long as $R/R_{0}$ small enough depending on $\omega$ so that
$\omega(R/R_{0})\leqslant c_{0}\delta_{*}^{2}$. The purpose of the Dini
modulus assumption is to guarantee that
$\sum_{k=1}^{\infty}\delta_{k}<+\infty$.

###  3.3. Step 3: Iteration

######  Theorem 3.6.

If $u$ is a minimizer of $J$ in $B_{R}(0)$ then for all $1\leqslant r\leqslant
R$

$\|\nabla u\|_{\underline{L}^{2}(B_{r}(0))}\leqslant C(\|\nabla
u\|_{\underline{L}^{2}(B_{R}(0))}+1).$

###### Proof.

Let $\eta>0$ small enough and $M\geqslant 2C_{0}$ large so that the statement
of Lemma 3.2 holds with $\delta=C_{0}^{-1}\delta_{*}$ constants $C_{0}$ from
Lemma 3.2 and $\delta_{*}$ from Lemma 3.4. Define

$b_{k}=\|\nabla u\|_{\underline{L}^{2}(B_{\eta^{k}R})}.$

Consider the inequality

(3.10) $b_{k}\leqslant\eta^{-d/2}M+2^{-k}b_{0}.$

This is true for $k=0$, suppose that the statement holds for $0\leqslant
k\leqslant j$. If $\eta^{j+1}R\leqslant 1$ we are done so we can assume
$\eta^{j+1}R\geqslant 1$.

If $b_{j}\leqslant M$ then

$b_{j+1}\leqslant\eta^{-d/2}b_{j}\leqslant\eta^{-d/2}M$

so (3.10) holds for $k=j+1$.

Otherwise $b_{j}\geqslant M$ and so Lemma 3.2 applies and either

$b_{j+1}\leqslant\frac{1}{2}b_{j}\leqslant\eta^{-d/2}M+2^{-(j+1)}b_{0},$

so (3.10) holds for $k=j+1$, or

$\|\nabla u-(q+\nabla\chi_{q})\|_{\underline{L}^{2}(B_{\eta^{j+1}R})}\leqslant
C_{0}^{-1}\delta_{*}b_{j}$

for some

$C_{0}^{-1}b_{j}\leqslant|q|\leqslant C_{0}b_{j}.$

Since $b_{j}\geqslant M\geqslant 2C_{0}$ we can apply Lemma 3.4 to find

$\|\nabla u\|_{\underline{L}^{2}(B_{r})}\leqslant Cb_{j}\leqslant
C(\eta^{-d/2}M+2^{-j}b_{0})$

for all $1\leqslant r\leqslant\eta^{j+1}R$. ∎

###  3.4. Local regularity

Finally we mention that when the $C^{0,\gamma}$ hypothesis is added,
Assumption (a2), we can also recover a true Lipschitz estimate from Theorem
1.1.

######  Corollary 3.7.

Suppose Assumption (Q1), Assumption (a1), and $a\in C^{0,\gamma}$. If $u$ is
$(R_{0},\beta)$-almost minimal in $B_{R}(0)$ for some $R_{0}\geqslant R>0$
then

$|\nabla u(0)|\leqslant C(1+\|\nabla u\|_{\underline{L}^{2}(B_{R})})$

for $C\geqslant 1$ depending on $\Lambda$, $d$, $\beta$, $\gamma$, and
$[a]_{C^{0,\gamma}}$.

###### Proof.

If $R\geqslant 2r_{0}$ then Theorem 1.1 implies that

$\|\nabla u\|_{\underline{L}^{2}(B_{r_{0}}(x_{0}))}\leqslant
C(d,\Lambda,\beta)(1+\|\nabla u\|_{\underline{L}^{2}(B_{R})})$

for all $x_{0}\in B_{R/2}$. If $R\leqslant 2r_{0}$ we can just do the below
argument in $B_{R}$ instead of in $B_{r_{0}}$.

Without loss we can consider the case $x_{0}=0$, define the functional
centered at $0$

$\tilde{J}_{0}(v,U)=\int_{U}a(0)\nabla v\cdot\nabla v+Q(x)^{2}{\bf
1}_{\\{v>0\\}}\ dx.$

Note that

$|\tilde{J}_{0}(v,B_{r}(0))-J(v,B_{r}(0))|\leqslant[a]_{C^{0,\gamma}}r^{\gamma}\|\nabla
v\|_{L^{2}(B_{r}(0)\cap\\{v>0\\})}^{2}$

by ellipticity the gradient norm on the right hand side can be bounded above
by $J(v,B_{r})$ or by $\tilde{J}_{0}(v,B_{r})$.

So we can compute writing $B_{r}$ for $B_{r}(0)$

$\displaystyle\tilde{J}_{0}(u,B_{r})$
$\displaystyle=J(u,B_{r})+(\tilde{J}_{0}(u,B_{r})-J(u,B_{r}))$
$\displaystyle\leqslant(1+(r/R_{0})^{\beta})J(v,B_{r})+Cr^{\gamma}J(u,B_{r})$
$\displaystyle\leqslant(1+(r/R_{0})^{\beta})(1+Cr^{\gamma})J(v,B_{r})$
$\displaystyle\leqslant(1+(r/R_{0})^{\beta})(1+Cr^{\gamma})^{2}\tilde{J}_{0}(v,B_{r})$

on the second inequality we used the almost minimality of $u$. Thus $u$ has an
almost minimality property for $\tilde{J}_{0}$ and so we can apply
[17][Theorem 1.1] to get

$|\nabla u(0)|\leqslant C(1+\|\nabla
u\|_{\underline{L}^{2}(B_{r_{0}})})\leqslant C(1+\|\nabla
u\|_{\underline{L}^{2}(B_{R})}).$

∎

##  4\. Initial free boundary regularity

In this section we explain the non-degeneracy at the free boundary of almost
minimizers, see Section 4.1. Some inputs from homogenization theory are
needed, and since we want to consider almost minimizers we again follow the
arguments introduced by De Silva and Savin [17]. Non-degeneracy then implies a
Hausdorff dimension $(d-1)$ estimate of the free boundary, see Section 4.2.
Together with the Lipschitz estimate proved previously this is sufficient
domain regularity to apply quantitative homogenization estimates in
$\\{u>0\\}$, see Section 5.1 later.

The following hypotheses on $u\in H^{1}(U)$ non-negative will be used in this
section.

  1. $\bullet$

(Lipschitz estimate)

(4.1) $\|\nabla u\|_{L^{\infty}(U)}\leqslant L.$

  2. $\bullet$

($\sigma$-almost minimal) There is $\sigma\geqslant 0$ so that for any ball
$B_{\rho}\subset U$

(4.2) $J(u,B_{\rho})\leqslant J(v,B_{\rho})+\sigma|B_{\rho}|\ \hbox{ for all
}\ v\in u+H^{1}_{0}(B_{\rho}).$

The interior Lipschitz estimate of almost minimizers has been proved already
in Section 3, and will be considered a hypothesis in this section.

Note that given the Lipschitz estimate the almost minimality condition above
follows from the almost minimality of the type (3.1) via

$J(u,B_{r})\leqslant(1+(r/R_{0})^{\beta})J(v,B_{r})\leqslant
J(v,B_{r})+(r/R_{0})^{\beta}(\Lambda^{2}+L^{2})|B_{r}|$

for all $v\in u+H^{1}_{0}(B_{r})$.

###  4.1. Non-degeneracy

Under (4.1) and (4.2) we will show that $u$ is non-degenerate at its free
boundary. More precisely, if $\sigma$ is sufficiently small, then for any
$x\in\partial\\{u>0\\}$ and $r>0$ so that $B_{r}(x)\subset U$ then

$\sup_{B_{r}(x)}u\geqslant cr.$

If we were considering minimizers we could follow the original argument of Alt
and Caffarelli [2] with some small inputs from homogenization for the
oscillatory operator. Specifically we would use a Lipschitz estimate up to the
boundary for the $a$-harmonic function interpolating between $1$ on $\partial
B_{r}$ and $0$ on $\partial B_{r/2}$ in the annulus $B_{r}\setminus B_{r/2}$.

To consider almost minimizers we are instead following the line of arguments
by De Silva and Savin [17]. Still we view the adaptations as fairly natural,
if a bit technical, using $a$-harmonic and $\overline{a}$-harmonic
replacements and quantitative homogenization theory in $B_{r}$. Note that,
importantly, we are not using quantitative homogenization theory for the
domain $\\{u>0\\}$ yet.

######  Lemma 4.1 (Weak non-degeneracy).

Suppose $u>0$ and $L$-Lipschitz in $B_{r}$ and

$J(u,B_{r})\leqslant J(v,B_{r})+\sigma|B_{r}|\ \hbox{ for all }\ v\in
u+H^{1}_{0}(B_{r})$

then there is $\sigma_{0}$, depending on $L$ and universal parameters, and
$c>0$, universal, so that if $\sigma\leqslant\sigma_{0}$ then

$u(0)\geqslant cr.$

Furthermore if there is $x\in\partial B_{r}(0)\cap\partial\\{u>0\\}$ then

$\sup_{\partial B_{(1-\eta)r}}u\geqslant(1+\delta)u(0)$

for some constants $\delta$ and $\eta>0$ depending on $L$ and universal
parameters.

For the second part we will need the following technical Lemma.

######  Lemma 4.2.

Given $r_{0}\geqslant 1$ there is a constant $c$ depending on $r_{0}$ and
universal parameters so that the Poisson kernel at $0$, $p(y)=P_{B_{r}}(0,y)$,
for the operator $-\nabla\cdot(a(x)\nabla\cdot)$ in $B_{r}(0)$ for any
$0<r\leqslant r_{0}$ has

$p(y)\geqslant c\frac{1}{|\partial B_{r}|}.$

The proof of is postponed to the Appendix, Section A.1. This is the only place
that we use the assumption that $a\in C^{0,1}$ as opposed to $C^{0,\gamma}$.

###### Proof of Lemma 4.1.

Note: constants in this proof may depend on universal parameters and also on
$L$. We will try to explicitly indicate the $L$ dependence each time it
appears, afterwards constants which depend on $L$ will be denoted with $k$ and
$K$ and constants which are universal and don’t depend on $L$ will be denoted
$c$ and $C$. The values of $k$ and $K$ may change from line to line as is the
convention with $c$ and $C$.

1\. Let $h$ be the $a$-harmonic replacement of $u$ in $B_{r}$. Then, since
$u>0$ in $B_{r}$ and $h>0$ in $B_{r}$ (strong maximum principle),

$J(h,B_{r})=E(h,B_{r})+\int_{B_{r}}Q(x)^{2}\ dx\leqslant
E(u,B_{r})+\int_{B_{r}}Q(x)^{2}\ dx=J(u,B_{r})\leqslant
J(h,B_{r})+\sigma|B_{r}|$

so subtracting the two energies and using the $a$-harmonic replacement
property we find

$\int_{B_{r}}|\nabla u-\nabla h|^{2}dx\leqslant C\sigma|B_{r}|.$

By Poincaré

$\int_{B_{r}}(u-h)^{2}\ dx\leqslant C\sigma r^{2}|B_{r}|.$

By the interior Lipschitz estimate Theorem 2.2 for $a$-harmonic functions and
maximum principle

$\|\nabla
h\|_{L^{\infty}(B_{\frac{3r}{4}})}\leqslant\frac{C}{r}\textup{osc}_{B_{r}}h\leqslant\frac{C}{r}\textup{osc}_{B_{r}}u\leqslant
CL.$

So

(4.3) $\|u-h\|_{L^{\infty}(B_{r/2})}^{d+2}\leqslant K\sigma r^{2}|B_{r}|.$

Let $\phi$ be a standard smooth cutoff $\equiv 1$ outside $\partial B_{r/2}$
and $\equiv 0$ in $B_{r/4}$ with $|\nabla\phi|\leqslant C/r$. Using, as
earlier, that $h$ and $u$ are positive in $B_{r}$

$J(h,B_{r})\leqslant J(u,B_{r})\leqslant J(h\phi,B_{r})+\sigma|B_{r}|.$

From this point we argue differently for large and small $r>0$.

For $r\geqslant r_{0}$ large to be specified let $\overline{h}$ be the
$\overline{a}$-harmonic replacement of $h$ in $B_{r}$ then, by Theorem 2.1,

$|E(h,B_{r})-E_{0}(\overline{h},B_{r})|+\frac{1}{r^{2}}\|h-\overline{h}\|_{L^{2}(B_{r})}^{2}\leqslant
CL^{2}r^{-\alpha}|B_{r}|.$

Note the same inequality holds for $J(h,B_{r})-J_{0}(\overline{h},B_{r})$ by
positivity. By similar arguments to above for (4.3), using the interior
Lipschitz estimate,

(4.4) $\|h-\overline{h}\|_{L^{\infty}(B_{r/2})}^{d+2}\leqslant
Kr^{2-\alpha}|B_{r}|.$

Since $\overline{h}\geqslant 0$ solves a constant coefficient equation in
$B_{r}$ (i.e. it is harmonic under a linear non-degenerate change of
variables) we have by Harnack inequality

$\|\bar{h}\|_{L^{\infty}(B_{3r/4})}\leqslant C\bar{h}(0)$

and so for $h$ we have

$\|h\|_{L^{\infty}(B_{3r/4})}\leqslant Ch(0)+Kr^{1-\frac{\alpha}{d+2}}.$

and also using the interior Lipschitz estimate for $a$-harmonic functions
Theorem 2.2

$\|\nabla h\|_{L^{\infty}(B_{r/2})}\leqslant
Cr^{-1}\|h\|_{L^{\infty}(B_{2r/4})}\leqslant
Cr^{-1}h(0)+Kr^{-\frac{\alpha}{d+2}}.$

So, using several of the previously established inequalities,

$\displaystyle E(h,B_{r})+\int_{B_{r}}Q(x)\ dx$ $\displaystyle\leqslant
J(h,B_{r})$ $\displaystyle\leqslant J(h\phi,B_{r})+\sigma|B_{r}|$
$\displaystyle=E(h\phi,B_{r})+\int_{B_{r}\setminus B_{r/4}}Q(x)\
dx+\sigma|B_{r}|$ $\displaystyle\leqslant E(h,B_{r})+\int_{B_{r}\setminus
B_{r/4}}Q(x)\ dx$
$\displaystyle\quad\quad\cdots+\left[C(\frac{1}{r^{2}}\|h\|_{L^{\infty}(B_{r/2})}^{2}+\|\nabla
h\|^{2}_{L^{\infty}(B_{r/2})})+\sigma\right]|B_{r}|$ $\displaystyle\leqslant
E(h,B_{r})+\int_{B_{r}\setminus B_{r/4}}Q(x)\
dx+\left(C\frac{1}{r^{2}}h(0)^{2}+Kr^{-\frac{\alpha}{d+2}}+\sigma\right)|B_{r}|$

Rearranging this inequality we find

$h(0)^{2}\geqslant cr^{2}(\frac{1}{|B_{r}|}\int_{B_{r/4}}Q(x)dx-
Kr^{-\frac{\alpha}{d+2}}-\sigma)\geqslant
cr^{2}(\Lambda^{-1}4^{-d}-Kr^{-\frac{\alpha}{d+2}}-\sigma)$

So as long as $Kr^{-\frac{\alpha}{d+2}}+\sigma$ is sufficiently small
depending on universal parameters we find

$h(0)\geqslant cr.$

Then plugging in (4.3) and (4.4)

$u(0)\geqslant(c-K\sigma^{\frac{1}{d+2}}-Kr^{-\frac{\alpha}{d+2}})r\geqslant\frac{1}{2}cr$

This holds for $r\geqslant r_{0}$ and $\sigma\leqslant\sigma_{0}$ where
$r_{0}$ and $\sigma_{0}$ depend on universal constants and on the Lipschitz
constant $L$.

Next we argue for $r\leqslant r_{0}$. Note that we can actually argue for
$r\leqslant r_{1}$ small, to be determined, and then for $r_{1}\leqslant
r\leqslant r_{0}$ we simply use that $B_{r_{1}}\subset B_{r}$ and so

$u(0)\geqslant cr_{1}\geqslant c\tfrac{r_{1}}{r_{0}}r$

and the additional factor $r_{1}/r_{0}$ still depends only on $L$ and
universal parameters.

The small $r_{1}$ case follows a very similar line of argument to the previous
using the $a(0)$-harmonic replacement $\tilde{h}$ in $B_{r}$ (instead of
$\overline{a}$-harmonic replacement) along with the estimate

$\|h-\tilde{h}\|_{\underline{L}^{2}(B_{r})}^{2}\leqslant
Cr^{2}\|\nabla(h-\tilde{h})\|_{\underline{L}^{2}(B_{r})}^{2}\leqslant
CL^{2}r^{2}r^{2\gamma}|B_{r}|$

which follows from the standard energy estimate

$\|\nabla(h-\tilde{h})\|_{L^{2}(B_{r})}\leqslant
C\|a-a(0)\|_{L^{\infty}(B_{r})}\|\nabla\tilde{h}\|_{L^{2}(B_{r})}\leqslant
Cr^{\gamma}(\int_{\partial B_{r}}|u||\nabla u|)^{1/2}\leqslant
Cr^{\gamma}L|B_{r}|^{1/2}$

using that $a$ is $C^{0,\gamma}$.

2\. Since $u$ is Lipschitz in $B_{r}$, $\inf_{B_{r}}u=0$, and $u(0)\geqslant
cr$ (from the first part of the proof) we can also conclude that
$u(x)\leqslant\frac{1}{4}u(0)$ on a constant $k(L)$ fraction of $\partial
B_{(1-\eta)r}$ for some $\eta(L)>0$. Since

$\sup_{B_{(1-\eta)r}}|u-\bar{h}|\leqslant
K(\sigma^{\frac{1}{d+2}}+r^{-\frac{\alpha}{d+2}})r\leqslant
K(\sigma^{\frac{1}{d+2}}+r^{-\frac{\alpha}{d+2}})u(0)$

when $\sigma\leqslant\sigma_{0}(L)$ and $r\geqslant r_{0}(L)$ also
$\bar{h}(x)\leqslant\frac{1}{2}\bar{h}(0)$ on a constant fraction of $\partial
B_{(1-\eta)r}$. Then by the Poisson kernel formula and the argument in
Caffarelli [11][Lemma 7] we must have

$\sup_{x\in B_{(1-\eta)r}}\bar{h}(x)\geqslant(1+2\delta)\bar{h}(0)$

and choosing $\sigma_{0}$, and $r_{0}$ smaller if necessary we get the same
for $u$ via

$\displaystyle\sup_{x\in B_{(1-\eta)r}}u(x)$ $\displaystyle\geqslant\sup_{x\in
B_{(1-\eta)r}}\bar{h}(x)-K(\sigma^{\frac{1}{d+2}}+r^{-\frac{\alpha}{d+2}})u(0)$
$\displaystyle\geqslant(1+2\delta)\bar{h}(0)-K(\sigma^{\frac{1}{d+2}}+r^{-\frac{\alpha}{d+2}})u(0)$
$\displaystyle\geqslant(1+2\delta)u(0)-K(\sigma^{\frac{1}{d+2}}+r^{-\frac{\alpha}{d+2}})u(0)$
$\displaystyle\geqslant(1+\delta)u(0).$

For $r\leqslant r_{0}$ we just do $a$-harmonic replacement $h$ with the
estimate

$\sup_{B_{(1-\eta)r}}|u-h|\leqslant K\sigma^{\frac{1}{d+2}}r\leqslant
K\sigma^{\frac{1}{d+2}}u(0)$

and then apply the Poisson kernel lower bound Lemma 4.2, depending on
$r_{0}(L)$, and the same argument from [11][Lemma 7]

$\sup_{x\in B_{(1-\eta)r}}h(x)\geqslant(1+2\delta)h(0)$

where $\eta$ is the same as above, and $\delta$ will now depend on the Poisson
kernel lower bound from Lemma 4.2. Using the supremum $|u-h|$ estimate as
before and choosing $\sigma_{0}$ smaller again if necessary we conclude. ∎

######  Lemma 4.3 (Strong non-degeneracy).

Suppose $u$ is $L$-Lipschitz (4.1), and satisfies (4.2) in $B_{r}(0)$. There
are $\sigma_{0},c>0$ depending on $L$ and universal parameters so that if
$0<\sigma\leqslant\sigma_{0}$, and if $0\in\partial\\{u>0\\}\cap U$ then

$\sup_{B_{r}(0)\cap U}u\geqslant cr.$

###### Proof.

As in [17][Lemma 3.5] the proof follows a standard argument applying Lemma
4.1, see originally [11][Lemma 7]. We just need the following claim
constructing a polygonal chain along which $u$ grows linearly.

Claim: There are $\delta>0$ small and $C\geqslant 1$ depending on $L$ and
universal constants so that: given $x_{0}\in\\{u>0\\}\cap B_{r}$ (near the
origin) there is a sequence $x_{k}\in B_{r}\cap\\{u>0\\}$ with

$u(x_{k+1})\geqslant(1+\delta)u(x_{k})$

and

$|x_{k+1}-x_{k}|=cd(x_{k},\partial\\{u>0\\}).$

This is exactly what we proved in the second part of Lemma 4.1. ∎

######  Remark 4.4.

Note that Caffarelli’s Harnack chain argument requires starting from a
possibly arbitrarily small radius ball so we do need Lemma 4.1 for all values
of $r>0$ and so we do need the $a\in C^{0,\gamma}$ assumption here. Maybe that
assumption could be removed here by using some stronger notion of “bulk" free
boundary point.

###  4.2. Hausdorff dimension of the free boundary

In this section we recall estimates of the Hausdorff dimension / measure of
$\partial\\{u>0\\}$.

######  Lemma 4.5 (Free boundary strip energy bound).

Suppose that $u\in H^{1}(B_{2r})$ with $r\geqslant 1$ has

$J(u,B_{2r})\leqslant J(v,B_{2r})+\sigma|B_{2r}|\ \hbox{ for all }\ v\in
u+H^{1}_{0}(B_{2r})$

Then for $1\leqslant t\leqslant r$

$J(u,\\{0<u\leqslant t\\}\cap B_{r})\leqslant
C\left[\sigma+\frac{t}{r}(1+\|\nabla
u\|_{\underline{L}^{2}(B_{2r})})\right]|B_{r}|$

with $C\geqslant 1$ universal.

###### Proof.

The proof is standard, see [29][Lemma 5.6], and only uses the ellipticity of
the coefficient fields $a(x)$ and $Q(x)$. The idea is to use the following
energy competitor: take $0\leqslant\phi\leqslant 1$ be a smooth cutoff
function $\phi=0$ in $B_{r}$, $\phi=1$ in a neighborhood of $\partial B_{2r}$
and $|\nabla\phi|\leqslant C/r$ and define

$v(x)=(1-\phi(x))(u(x)-t)_{+}+\phi(x)u(x).$

Using the almost minimality property and some computations give the result. ∎

######  Lemma 4.6 (Free boundary Hausdorff dimension).

Suppose that $u$ Lipschitz with constant $L\geqslant 1$ and $\ell$-non-
degenerate in $B_{2r}$. Then there is a universal $C\geqslant 1$ so that for
any covering $(\square_{i})_{i\in I}$ of $\partial\\{u>0\\}\cap B_{r}$ by
almost disjoint boxes of side length $1\leqslant t\leqslant c(d)r$

$\\#I\leqslant
C(d,\Lambda)\left(\tfrac{L}{\ell}\right)^{d}t^{-d}J(u,\\{0<u\leqslant
c(d)tL\\}\cap B_{2r})$

Of course Lemma 4.5 can then be used to bound the right hand side in the
inequality obtained here.

###### Proof.

Let $\square$ be a cube with side length $t$ such that
$\partial\\{u>0\\}\cap\square\neq\emptyset$ and $2\square$ be the cube with
the same center dilated by a factor of $2$. Then by the non-degeneracy
hypothesis

$\sup_{x\in 2\square}u(x)\geqslant c(d)\ell t$

By the Lipschitz assumption there is a ball of radius $c(d)\frac{\ell}{L}t$
inside of $3\square$ with

$u(x)\geqslant c(d)\ell t\ \hbox{ in }\ B_{c(d)\frac{\ell}{L}t}\subset
3\square.$

For each $\square_{i}$ from the collection in the statement let
$B_{\frac{\ell}{2L}}(y_{i})\subset 3\square_{i}$ be the corresponding cube
described above. Since $\square_{i}$ are almost disjoint there is a
dimensional constant $C(d)$ so that each $x\in B_{r}$ is in at most $C(d)$ of
the $(B_{\frac{\ell}{2L}}(y_{i}))_{i\in I}$. Thus

$\\#I=\sum_{i\in I}1=\sum_{i\in I}t^{-d}|\square_{i}|\leqslant
C(d)(\tfrac{L}{\ell t})^{d}\sum_{i\in
J}|B_{c(d)\frac{\ell}{L}t}(y_{i})|\leqslant C(d)(\tfrac{L}{\ell
t})^{d}|\cup_{i\in I}B_{c(d)\frac{\ell}{L}t}(y_{i})|$

and

$|\cup_{i\in I}B_{c(d)\frac{\ell}{L}t}(y_{i})|\leqslant\Lambda
J(u,\cup_{i}3\square_{i})\leqslant\Lambda J(u,\\{0<u<c(d)Lt\\}\cap
B_{r+3\sqrt{d}t})$

since $t\leqslant c(d)r$ we have $B_{r+3\sqrt{d}t}\subset B_{2r}$. ∎

##  5\. Hölder rate of homogenization

In this section we combine the previous regularity results to show a sub-
optimal quantitative homogenization rate for almost minimizers. We will apply
quantitative homogenization results for Dirichlet boundary value problems.

###  5.1. Quantitative homogenization of Dirichlet problems

We recall some results about quantitative homogenization results for Dirichlet
boundary value problems in bounded domains $U\subset\varmathbb{R}^{n}$. Recall
the energy functionals

$E(u,U)=\int_{U}a(x)\nabla u\cdot\nabla u\ dx\ \ \hbox{ and }\
E_{0}(u,U)=\int_{U}\bar{a}\nabla u\cdot\nabla u\ dx$

originally introduced in Section 2.2.

Given a bounded domain $U$ in $\varmathbb{R}^{d}$ we consider a function $u\in
H^{1}(U)$ satisfying

$\|\nabla u\|_{L^{\infty}(U)}\leqslant L$

in our application $u$ will be an almost minimizer of the functional $J$ and
$U$ will be $\\{u>0\\}$, but that information is not needed for the present
statements.

We also consider a function $u_{0}\in H^{1}(U)$ satisfying

$\|\nabla u_{0}\|_{L^{\infty}(U)}\leqslant L$

and

$E_{0}(u_{0},U)\leqslant E(v,U)\ \hbox{ for all }\ v\in u_{0}+H^{1}_{0}(U).$

The function $u_{0}$ will arise by doing a $J_{0}$ minimizer replacement of an
almost minimizer and $U$ will be $\\{u_{0}>0\\}$, but that information is not
needed for the present statements.

We will take advantage of the a-priori Lipschitz estimate. It is convenient to
work directly with the explicit upscalings/downscalings, which usually appear
as intermediate tools in quantitative homogenization proofs, instead of with
the usual comparison with the $\bar{a}$-harmonic replacement. See Remark 5.2
below.

$\circ$ Specific choice of upscaling. Let $\varphi:\varmathbb{R}\to[0,1]$ be
the piecewise linear continuous cutoff function with $\varphi(t)\equiv 0$ for
$t\leqslant 1$, $\varphi\equiv 1$ for $t\geqslant 2$, and
$|\varphi^{\prime}|\leqslant 2$. Let $t\ll r$ be a scale to be determined
define

$\xi_{t}(x)=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{x+B_{t}}u(y)\ dy$

and

(5.1)
$\bar{u}(x)=\varphi(\tfrac{d(x)}{t})\xi_{t}(x)+(1-\varphi(\tfrac{d(x)}{t}))u(x)$

where $d(x)$ is the distance function to the complement of $U$. It is easy to
check

$\|\nabla\xi_{t}\|_{L^{\infty}(U)}\leqslant\|\nabla u\|_{L^{\infty}(U)}$

and

$\|u-\bar{u}\|_{L^{\infty}(U)}\leqslant\|\nabla u\|_{L^{\infty}(U)}t.$

Since

$\nabla\bar{u}=\varphi(\tfrac{d(x)}{t})\nabla\xi_{t}(x)+(1-\varphi(\tfrac{d(x)}{t}))\nabla
u(x)+\frac{1}{t}\nabla d(x)\varphi^{\prime}(\tfrac{d(x)}{t})(\xi_{t}(x)-u(x))$

and $|\nabla d|\leqslant 1$

$\|\nabla\bar{u}\|_{L^{\infty}(U)}\leqslant 2\|\nabla u\|_{L^{\infty}(U)}.$

$\circ$ Specific choice of downscaling. Let $\varphi:\varmathbb{R}\to[0,1]$ be
a fixed smooth cutoff function with $\varphi(t)\equiv 0$ for $t\leqslant 1$,
$\varphi\equiv 1$ for $t\geqslant 2$, and $|\varphi^{\prime}|\leqslant C$. Let
$t\ll r$ be a scale to be determined and

(5.2) $\tilde{v}(x)=\varphi(\tfrac{d(x)}{t})[v(x)+\chi_{\nabla v(x)}(x)\nabla
v(x)]+(1-\varphi(\tfrac{d(x)}{t}))v(x)$

where $d(x)$ is the distance function to the complement of $U$ and $\chi_{q}$
are the correctors defined in Section 2.2.

######  Proposition 5.1 (Quantitative homogenization).

Suppose that $U$ is a bounded domain and let $u$, $\bar{u}$, $u_{0}$, and
$\tilde{u}_{0}$ as above. Let $t\geqslant 1$ be an integer, and let $U_{t}$ be
the union of the $t\varmathbb{Z}^{d}$ lattice cubes which are contained in
$U$. There is $C\geqslant 1$ depending on $d$ and $\Lambda$ so that

$E_{0}(\bar{u},U)\leqslant
E(u,U)+CL^{2}\left[\frac{1}{t^{\frac{1}{2}}}|U|+|U\setminus U_{t}|\right]$

and

$\left|E(\tilde{u}_{0},U)-E_{0}(u_{0},U)\right|\leqslant
CL^{2}\left[\frac{1}{t}|U|+|U\setminus U_{t}|\right].$

######  Remark 5.2.

We are presenting a somewhat atypical scenario for quantitative Dirichlet data
homogenization because we have assumed an a-priori Lipschitz estimate up to
the domain boundary. This usually would require significant domain regularity
to prove, but we will know it from Theorem 1.1 for $J$ almost minimizers.

On the other hand we also know _less_ domain regularity than one typically
assumes for such quantitative homogenization results. A typical assumption is
that the domain $U$ is Lipschitz, see for example [4][Theorem 1.12], but we
are aiming to prove this. Actually an inner and outer density bound is
sufficient when the equation does not have a right hand side, see [4]. We can
indeed prove inner and outer density bounds at this stage, but the result is
not strictly needed for our main theorems so we have postponed it to the
appendix in Section A.4.

Results in the literature also typically estimate the difference with the
$\bar{a}$-harmonic replacement. With the amount of domain regularity we
currently know, the $\bar{a}$-harmonic replacement is actually less regular up
to the domain boundary. It will not necessarily be Lipschitz up to the
boundary of $U$ (insufficient domain regularity) and we won’t have an
$L^{\infty}$ convergence estimate (insufficient domain regularity again).

The proof follows standard lines. In particular, the statements and arguments
in [26][Ch. 3.2 and 3.3] are very similar to what we will do here. We will
refer there in some places but we didn’t find any particular statement there
to cover the entirety of Proposition 5.1. We also emphasize that we are not
making any attempt to find optimal rate here, it is not needed for our
purposes.

###### Proof of Proposition 5.1.

We will make use of the dual energy introduced by [6]

(5.3) $\mu(U,q)=\inf\\{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{U}\tfrac{1}{2}\nabla v\cdot a(x)\nabla
v-q\cdot\nabla v\ dx:\ v\in H^{1}(U)\\}.$

This quantity has the following large scale limit

$|\mu(B_{t},q)+\frac{1}{2}q\cdot\bar{a}^{-1}q|\leqslant
C(\Lambda,d)|q|^{2}\frac{1}{t^{1/2}}$

which we now justify. We only are interested in the lower bound, the upper
bound is immediate from testing the appropriate corrected linear function
(below).

For $\tau\in[0,1)^{d}$ let $v$ be the minimizer for (5.3) solving

$\begin{cases}-\nabla(a(x)\nabla v)=0&\hbox{ in }\ B_{t}\\\ n\cdot(a(x)\nabla
v_{\tau}-q)=0&\hbox{ on }\ \partial B_{t}\end{cases}$

with mean zero on $B_{t}$ and let

$v_{0}=p\cdot x+\chi_{p}(x)\ \hbox{ with }\ \bar{a}p=q$

note that $p\cdot x$ solves the effective version for the previous Neumann
problem. Note

$|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{t}}\tfrac{1}{2}\nabla v_{0}\cdot a(x)\nabla
v_{0}-q\cdot\nabla v_{0}\ dx-(\frac{1}{2}p\cdot\bar{a}p-q\cdot p)|\leqslant
Ct^{-1}|q|^{2}.$

Then call

$w=v(x)-p\cdot x-\chi_{p}(x)\phi(x)$

where $\phi$ is a standard cutoff which goes from $1$ to $0$ in a unit
neighborhood of $\partial B_{t}$. By [26][Theorem 3.3.4]

$\|\nabla w\|_{L^{2}(B_{t})}\leqslant Ct^{-1/2}|q|$

and so

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{t}}\tfrac{1}{2}\nabla v\cdot a(x)\nabla
v-q\cdot\nabla v\ dx\geqslant\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{t}}\tfrac{1}{2}\nabla v_{0}\cdot a(x)\nabla
v_{0}-q\cdot\nabla v_{0}\ dx-Ct^{-1/2}|q|.$

Part 1 (estimate of the upscaling). Define, for $x\in U_{t}$

$\xi(x)=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{x+B_{t}}u(y)\ dy$

and

$\bar{u}(x)=\varphi_{t}(x)\xi(x)+(1-\varphi_{t}(x))u(x).$

Call

$p(x)=\nabla\xi(x)=\int_{x+B_{t}}\nabla u(y)\ dy$

and

$q(x)=\bar{a}p(x)$

Note that $|q(y)|\leqslant\Lambda L$. Now the dual quantity $\mu$ is exactly
useful for lower bounds of the energy

$\displaystyle\int_{U}\tfrac{1}{2}\nabla u\cdot a(x)\nabla u\ dx$
$\displaystyle\geqslant\int_{U_{t}}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{y+B_{t}}\tfrac{1}{2}\nabla u(x)\cdot a(x)\nabla
u(x)dxdy$ $\displaystyle\geqslant\int_{U_{t}}\mu(B_{t},q(y))+q(y)\cdot p(y)dy$
$\displaystyle=\int_{U_{t}}\mu(B_{t},q(y))+p(y)\cdot\bar{a}p(y)dy$
$\displaystyle\geqslant\int_{U_{t}}\tfrac{1}{2}p(y)\cdot\bar{a}p(y)dy+\int_{U_{t}}\left[\frac{1}{2}q(y)\cdot\bar{a}^{-1}q(y)+\mu(B_{t},q(y))\right]\
dy$ $\displaystyle=E_{0}(\bar{u},U_{t})-CL^{2}t^{-1/2}|U|$

Then we need to estimate the difference between the homogenized energy of
$\bar{u}$ on $U_{t}$ and on $U$

$\displaystyle\int_{U_{t}}\tfrac{1}{2}p(x)\cdot\bar{a}p(x)dx-\int_{U}\tfrac{1}{2}\nabla\bar{u}\cdot\bar{a}\nabla\bar{u}dx$
$\displaystyle=-\int_{U\setminus
U_{t}}\nabla\bar{u}\cdot\bar{a}\nabla\bar{u}dx$
$\displaystyle\geqslant-\Lambda L^{2}|U\setminus U_{t}|$

Together we have

$E_{0}(\bar{u},U)\leqslant E(u,U)+CL^{2}\left[t^{-1}|U|+|U\setminus
U_{t}|\right].$

Part 2 (estimate of the downscaling). This is basically the same as
[26][Theorem 3.3.2]. Recall that $u_{0}$ is a minimizer of $E_{0}$ on $U$, so
$\nabla u_{0}$ is $\bar{a}$-harmonic and we have the $C^{1,1}$ estimate

$\|D^{2}u_{0}\|_{L^{\infty}(U_{t})}\leqslant Ct^{-1}L.$

The downscaling was defined, we recall,

$\tilde{u}_{0}(x)=u_{0}(x)+\chi(x)\cdot\nabla u_{0}(x)\varphi_{t}(x).$

The gradient is, with the primary term listed first and error terms listed
after,

$\nabla\tilde{u}_{0}=[\nabla u_{0}+\nabla\chi\nabla
u_{0}]+(1-\varphi_{t})[\nabla\chi\nabla u_{0}]+\chi
D^{2}u_{0}\varphi_{t}+(\chi\cdot\nabla u_{0})\nabla\varphi_{t}.$

We make note of the following bounds following from the $L$-Lipschitz
hypothesis of $u_{0}$ and corrector bounds (2.2)

$|\nabla u_{0}+\nabla\chi\nabla u_{0}|\leqslant
CL,\quad|(1-\varphi_{t})[\nabla\chi\nabla u_{0}]|\leqslant
L(1-\varphi_{t}),\quad|\chi D^{2}u_{0}\varphi_{t}|\leqslant CLt^{-1}$

and

$(\chi\cdot\nabla u_{0})\nabla\varphi_{t}\leqslant CLt^{-1}.$

Let $p(x)$ be the piecewise constant function which is the average of $\nabla
u_{0}$ over the $t\varmathbb{Z}^{d}$ translations of $[0,t)^{d}$ which
contains $x$. Then

$|p-\nabla u_{0}|\varphi_{t}\leqslant CLt^{-1}$

by the previous $C^{1,1}$ estimate.

Thus, by standard algebra with the quadratic energy,

$\left|E(\tilde{u}_{0},U)-\int_{U_{t}}(p(x)+\nabla\chi(x)p(x))\cdot
a(x)(p(x)+\nabla\chi(x)p(x))\ dx\right|\leqslant CL^{2}[t^{-1}|U|+|U\setminus
U_{t}|].$

Since $t\geqslant 1$ is an integer and by the definition of $\bar{a}$, (2.3),

$\int_{U_{t}}(p(x)+\nabla\chi(x)p(x))\cdot a(x)(p(x)+\nabla\chi(x)p(x))\
dx=\int_{U_{t}}p(x)\cdot\bar{a}p(x)\ dx$

and

$|\int_{U_{t}}p(x)\cdot\bar{a}p(x)\ dx-\int_{U_{t}}\nabla
v(x)\cdot\bar{a}\nabla u_{0}(x)\ dx|\leqslant CL^{2}t^{-1}|U|$

so finally, using one more time the Lipschitz estimate for
$E_{0}(u_{0},U\setminus U_{t})\leqslant CL^{2}|U\setminus U_{t}|$,

$|E(\tilde{u}_{0},U)-E_{0}(u_{0},U)|\leqslant CL^{2}[t^{-1}|U|+|U\setminus
U_{t}|].$

∎

###  5.2. Rate of homogenization for $J$

In this section we combine all the previous regularity results to obtain a
rate of homogenization in the energy. We will consider $u\in H^{1}(U)$ non-
negative satisfying (4.1) and (4.2).

Note that if $u$ is $L$-Lipschitz and satisfies (4.2) for $\sigma>0$
sufficiently small depending on $L$ and universal constants then $u$ satisfies
the hypotheses of Lemma 4.3, Lemma 4.5, and Lemma 4.6.

######  Lemma 5.3 (Energy convergence).

Let $r\geqslant 1$.

  1. (a)

Suppose that $u\in H^{1}(B_{2r})$ non-negative satisfies (4.1) and (4.2) with
$0<\sigma\leqslant\bar{\sigma}(L)$ sufficiently small so that Lemma 4.3 holds.
There is a regularization $\bar{u}$ of $u$, $\overline{u}\in
u+H^{1}_{0}(\\{u>0\\}\cap B_{r})$ with

$\|\nabla\bar{u}\|_{L^{\infty}(B_{r})}\leqslant L,\
\|u-\bar{u}\|_{L^{\infty}(B_{r})}\leqslant Lr^{2/3}$

and

$J_{0}(\bar{u},B_{r})\leqslant
J(u,B_{r})+C(d,\Lambda,L)[\sigma+r^{-1/3}]|B_{r}|.$

  2. (b)

Suppose that $v$ is a $J_{0}$-minimizer in $B_{2r}$ satisfying (4.1) in
$B_{2r}$. There is $\tilde{v}\in v+H^{1}_{0}(\\{v>0\\}\cap B_{r})$ such that

$J(\tilde{v},B_{r})\leqslant
J_{0}(v,B_{r})+C(d,\Lambda)L^{\frac{d+5}{2}}r^{-1/2}|B_{r}|.$

  3. (c)

Let $u$ and $\bar{u}$ as in part $(a)$. For any $w\in
u+H^{1}_{0}(\\{u>0\\}\cap B_{r})$

$J_{0}(\bar{u},B_{r})\leqslant J_{0}(w,B_{r})+C(L)[r^{-\omega}+\sigma]|B_{r}|$

where $\omega$ is a dimensional constant and can be taken to be
$\omega=\frac{1}{4d(d+5)+6}$ .

######  Remark 5.4.

This is the only place in the paper where the periodicity of $Q$ is used.
Other assumptions on the convergence of the spatial averages of $Q$ could be
easily slotted in here as long as they result in an error rate which is
summable over geometric sequences of scales (that property is used in the
following section).

######  Remark 5.5.

The constant $\omega=\frac{1}{4d(d+5)+6}$ is certainly suboptimal, we compute
it just to be careful that we indeed have a Hölder rate. The dependence on $L$
in part (b) is followed for the sake of computing $\omega$, so, again, it is
not that important.

For the proof of Lemma 5.3 we will need one more technical Lemma about $J_{0}$
minimizers in a domain with Lipschitz Dirichlet data. The claim is that there
is not too much energy concentrated near the domain boundary.

######  Lemma 5.6 (Domain boundary strip energy bound).

Suppose that $u\in g+H^{1}_{0}(B_{r})$, $g$ is $L$-Lipschitz, and $u$
minimizes $J_{0}$ over $g+H^{1}_{0}(B_{r})$ then for every $\alpha<1/2$

$J_{0}(u,B_{r}\setminus B_{(1-\eta)r})\leqslant
C(\alpha,L)\eta^{\alpha}(1+\frac{1}{r})|B_{r}|.$

The same result is probably true for all $\alpha<1$ but we did not need this
and it would require some more work to prove. The result would also be true
for a regular domain $U$ instead of a ball, but again we did not need that.
The proof is technical and is postponed to Section A.2. There is a barrier
argument to get Hölder continuity up to $\partial B_{r}$ and then an energy
comparison argument.

###### Proof of Lemma 5.3.

In the proof throughout we will assume $L\geqslant 1$, constants which depend
on $L$ and other universal constants will be denoted $C(L)$ or $c(L)$,
universal constants which do not depend on $L$ will be denoted $C$, $c$ etc.
For parts (b) and (c) we need to be a bit careful keeping track of $L$
dependence.

Part (a). Let $\bar{\sigma}>0$ depending on universal constants and on $L$ so
that Lemma 4.3 holds. Let $U=\\{u>0\\}\cap B_{r}$ and let $\bar{u}$ be the
upscaling of $u$ defined in Section 5.1 with parameter $t\geqslant 1$ to be
chosen shortly. Let $U_{t}$ be the union of the $t\varmathbb{Z}^{d}$ lattice
cubes contained in $U$. By the perimeter estimate Lemma 4.6 and Lemma 4.5

$|U\setminus U_{t}|\leqslant C(L)(\sigma+\frac{t}{r})|B_{r}|$

Thus to infimize the error term from Proposition 5.1 we should choose
$t=r^{2/3}$

$\inf_{t>0}[\frac{1}{t^{\frac{1}{2}}}|U|+|U\setminus U_{t}|]\leqslant
C(L)[\sigma+r^{-1/3}]|B_{r}|$

Call $\mathcal{Q}$ to be the cubes of the $\varmathbb{Z}^{d}$ lattice

$\mathcal{Q}=\\{k+[0,1)^{d}:k\in\varmathbb{Z}^{d}\\}$

and then define

$\mathcal{A}=\\{\square\in\mathcal{Q}:\ \square\subset(\\{u>0\\}\cap
B_{r})\\}$

and

$\mathcal{B}=\\{\square\in\mathcal{Q}:\ \square\cap\partial(\\{u>0\\}\cap
B_{r})\neq\emptyset\\}.$

Since $u$ satisfies the conclusion of Lemma 4.6,

$\\#\mathcal{B}\leqslant Cr^{d-1}\leqslant C(L)[\sigma+\frac{1}{r}]|B_{r}|.$

Note that for any $\square\in\mathcal{A}$

$\int_{(\\{u>0\\}\cap B_{r})\cap\square}Q(x)^{2}\ dx=\int_{\square}Q(x)^{2}\
dx=\langle Q^{2}\rangle=\int_{(\\{\overline{u}>0\\}\cap
B_{r})\cap\square}\langle Q^{2}\rangle\ dx$

using, respectively, that $\square\subset\\{u>0\\}\cap B_{r}$, $\square$ is a
period cell for $Q$, and that $u$ and $\overline{u}$ have the same positivity
set.

Then applying Proposition 5.1

$\displaystyle J_{0}(\overline{u},B_{r})-J(u,B_{r})$
$\displaystyle=E_{0}(\overline{u},\\{u>0\\}\cap B_{r})-E(u,\\{u>0\\}\cap
B_{r})$
$\displaystyle\quad\quad\cdots+\sum_{\square\in\mathcal{A}\cup\mathcal{B}}\int_{\\{u>0\\}\cap\square\cap
B_{1}}\langle Q^{2}\rangle-Q(x)^{2}\ dx$ $\displaystyle\leqslant
C(L)[\sigma+r^{-1/3}]|B_{r}|+\Lambda|\mathcal{B}|$ $\displaystyle\leqslant
C(L)(\sigma+r^{-1/3})|B_{r}|.$

Part (b). The proof follows an analogous line of arguments to part (a). Since
we need to keep track of the dependence on the Lipschitz constant for the
purpose of part (c). Note that the non-degeneracy constant does _not_ depend
on the Lipschitz constant in the case of $J_{0}$ minimizers, it is universal.
Let $V=\\{v>0\\}\cap B_{r}$ and let $\tilde{v}$ be the downscaling of $v$ in
$V$ defined in Section 5.1. By the perimeter estimates Lemma 4.6 and Lemma 4.5
(with $\sigma=0$), now keeping more carefully the dependence on $L$,

$|U\setminus U_{t}|\leqslant CL^{d+2}t^{2}r^{-1}|B_{r}|$

Thus to infimize the error term from Proposition 5.1 we should choose
$t=L^{-(d+1)/2}r^{1/2}$

$\inf_{t>0}[\frac{1}{t}|U|+|U\setminus U_{t}|]\leqslant
CL^{\frac{d+1}{2}}r^{-1/2}|B_{r}|.$

The remainder of the argument is similar and we end up with the estimate

$J(\tilde{v},B_{r})\leqslant J_{0}(v,B_{r})+CL^{\frac{d+5}{2}}r^{-1/2}|B_{r}|$

with the additional factor of $L^{2}$ coming from Proposition 5.1.

Part (c). Let $\overline{u}$ be the upscaling of $u$ in $B_{r}\cap\\{u>0\\}$
defined earlier. Let $v$ be a minimizer of $J_{0}$ in the class
$u+H^{1}_{0}(\\{u>0\\}\cap B_{r})$. Then, since $v$ is a minimizer for $J_{0}$
in $B_{r}$,

$J_{0}(v,B_{r})\leqslant J_{0}(\overline{u},B_{r}).$

In particular

$\|\nabla v\|_{\underline{L}^{2}(B_{r})}\leqslant C(\|\nabla
u\|_{\underline{L}^{2}(B_{r})}+1)\leqslant CL.$

Let $1>\eta>0$ to be chosen small and $\mathcal{Q}$ be the class of $\eta
r$-lattice cubes which are contained in $B_{(1-\eta)r}$. Note that for every
$Q\in\mathcal{Q}$ we have $2Q\subset B_{r}$. In particular by the interior
Lipschitz estimate of $J_{0}$ minimizers $v$ is Lipschitz in $B_{(1-\eta)r}$
with

(5.4) $|\nabla v(x)|\leqslant C(1+\|\nabla v\|_{\underline{L}^{2}(B_{\eta
r})})\leqslant C\eta^{-d/2}L\ \hbox{ for }\ x\in B_{(1-\eta)r}.$

Let $\tilde{v}_{Q}$ be the downscaling of $v$ in $Q\cap\\{v>0\\}$ defined in
Section 5.1 with scale $t=L^{-(d+1)/2}\eta^{\frac{d(d+1)}{2}}(\eta r)^{1/2}$,
the scale is chosen as in (b) taking into account the Lipschitz constant of
$v$ from (5.4). Note that at this point it is really the $\eta$ dependence and
not the $L$ dependence that we want to track. We have the estimate from (b)

(5.5) $J(\tilde{v}_{Q},Q)\leqslant
J_{0}(v,Q)+C(L)\eta^{-\frac{(d+5)d}{2}}(\eta r)^{-1/2}|Q|$

We will make an energy competitor

$w=v{\bf
1}_{B_{r}\setminus\cup_{Q\in\mathcal{Q}}Q}+\sum_{Q\in\mathcal{Q}}\tilde{v}_{Q}{\bf
1}_{Q}$

which does have $w\in u+H^{1}_{0}(B_{r})$ due to the matching of appropriate
traces on each cube boundary. Since $u$ is a minimizer for $J$ in $B_{r}$,

$J(u,B_{r})\leqslant
J(w,B_{r})=\sum_{Q\in\mathcal{Q}}J(\tilde{v}_{Q},Q)+J(v,B_{r}\setminus\cup_{Q\in\mathcal{Q}}Q).$

The boundary layer term we will bound by applying Lemma 5.6, using that
$\|\nabla u\|_{L^{\infty}(B_{r})}\leqslant L$,

$J(v,B_{r}\setminus\cup_{Q\in\mathcal{Q}}Q)\leqslant J(v,B_{r}\setminus
B_{(1-C(d)\eta)r})\leqslant C(L)\eta^{1/4}|B_{r}|.$

We also have, from part (a) and the above,

$\displaystyle J_{0}(\overline{u},B_{r})$ $\displaystyle\leqslant
J(u,B_{r})+C(L)[r^{-1/2}+\sigma]|B_{r}|$
$\displaystyle\leqslant\sum_{Q\in\mathcal{Q}}J(\tilde{v}_{Q},Q)+C(L)[r^{-1/3}+\eta^{1/4}+\sigma]|B_{r}|$

and by part (b)

$\displaystyle\sum_{Q\in\mathcal{Q}}J(\tilde{v}_{Q},Q)$
$\displaystyle\leqslant\sum_{Q\in\mathcal{Q}}\left\\{J_{0}(v,Q)+C(L)\eta^{-\frac{d(d+5)+1}{2}}r^{-1/2}|Q|\right\\}$
$\displaystyle\leqslant
J_{0}(v,B_{r})+C(L)\eta^{-\frac{d(d+5)+1}{2}}r^{-1/2}|B_{r}|.$

Combining all the above gives

$J_{0}(\overline{u},B_{r})\leqslant
J_{0}(v,B_{r})+C(L)(r^{-1/3}+\eta^{-\frac{d(d+5)+1}{2}}r^{-1/2}+\eta^{1/4}+\sigma)|B_{r}|$

then choose $\eta=r^{-\frac{1}{2}\frac{4}{2d(d+5)+3}}$ to match the middle two
error terms on the right.

∎

##  6\. Flat free boundaries are (large scale) Lipschitz

In this section we will show that the free boundaries of flat minimizers of
$J$ are $C^{1,\alpha}$ at intermediate scales and Lipschitz down to the unit
scale. This will prove Theorem 1.2 as a consequence of Lemma 6.4 and Corollary
6.5 below. We will also show, in Section 6.1, a Liouville result for large
scale flat minimizers of $J$ on $\varmathbb{R}^{d}$ Theorem 6.7.

We will combine the homogenization result Lemma 5.3 with the following
$C^{1,\alpha}$ flatness improvement result of De Silva and Savin [17] for
almost minimizers of $J_{0}$:

######  Lemma 6.1 (De Silva and Savin [17] Lemma 4.5).

Let $u$ satisfy

(6.1) $\|\nabla u\|_{L^{\infty}(B_{1})}\leqslant L\ \hbox{ and }\
J_{0}(u,B_{1})\leqslant J_{0}(v,B_{1})+\sigma\ \hbox{ for all }\ v\in
u+H^{1}_{0}(B_{1}).$

For any $0<\alpha<1$ there exist constants $\bar{\delta},\eta>0$ and
$C\geqslant 1$ depending on $L$ and $\alpha$ so that if
$0\in\partial\\{u>0\\}$ and, for some $\nu\in S^{d-1}$,

$|u(x)-\tfrac{1}{(\nu\cdot\overline{a}\nu)^{1/2}}(x\cdot\nu)_{+}|\leqslant\delta\
\hbox{ in }\ B_{1}$

for some $0<\delta\leqslant\bar{\delta}$ then there is $\nu^{\prime}\in
S^{d-1}$ with $|\nu^{\prime}-\nu|\leqslant C(\delta+\sigma^{\frac{1}{d+4}})$

$|u(x)-\tfrac{1}{(\nu^{\prime}\cdot\overline{a}\nu^{\prime})^{1/2}}(x\cdot\nu^{\prime})_{+}|\leqslant\eta^{1+\alpha}(\delta+C\sigma^{\frac{1}{d+4}})\
\hbox{ in }\ B_{\eta}(0).$

######  Remark 6.2.

Note that [17] works with the standard Dirichlet energy, but we can transform
to this case by looking at $v(x)=u(\bar{a}^{1/2}x)$.

We apply Lemma 6.1 to minimizers of $J$ by doing an upscaling and applying
Lemma 5.3 to see that the upscaling has (almost) minimal energy for $J_{0}$.

First we give the one step flatness improvement.

######  Lemma 6.3 (Improvement of flatness).

Let $u$ satisfy (4.1) and (4.2) in $B_{2r}$ with
$0<\sigma\leqslant\bar{\sigma}$ sufficiently small so that Lemma 4.3 holds.
For any $0<\alpha<1$ there exist constants $\bar{\delta},\eta>0$ depending on
$L$ and $\alpha$ so that if $0\in\partial\\{u>0\\}$ and, for some $\nu\in
S^{d-1}$,

$|u(x)-\tfrac{1}{(\nu\cdot\overline{a}\nu)^{1/2}}(x\cdot\nu)_{+}|\leqslant\delta
r\ \hbox{ in }\ B_{r}$

for some $0\leqslant\delta\leqslant\bar{\delta}$ then there is
$\nu^{\prime}\in S^{d-1}$ with $|\nu^{\prime}-\nu|\leqslant
C[\delta+\sigma^{\frac{1}{d+4}}+r^{-\omega}]$ and

$|u(x)-\tfrac{1}{(\nu^{\prime}\cdot\overline{a}\nu^{\prime})^{1/2}}(x\cdot\nu^{\prime})_{+}|\leqslant\eta^{1+\alpha}[\delta+C\sigma^{\frac{1}{d+4}}+Cr^{-\omega}]r\
\hbox{ in }\ B_{\eta r}$

where $\omega>0$ is a dimensional constant.

Then Lemma 6.3 can be iterated to obtain the following $C^{1,\alpha}$
iteration for $(R_{0},\beta)$-almost minimizers. Note that flatness improves
down to a certain intermediate length scale, then gets worse until the
iteration is forced to stop at scale $1$.

######  Lemma 6.4 (Flat implies $C^{1,\alpha}$ iteration for heterogeneous
problem).

Suppose that $u:B_{R}\to[0,\infty)$ has $\|\nabla
u\|_{L^{\infty}(B_{R})}\leqslant L$ and $u$ is an $(R_{0},\beta)$-almost
minimizer of $J$ in $B_{R}$.

There is $\bar{\delta}(d,\Lambda,L)>0$ and $c(d,\Lambda,L,\beta)>0$ so that if
$0<\delta\leqslant\bar{\delta}$ and

$|u(x)-\tfrac{1}{(\nu\cdot\overline{a}\nu)^{1/2}}(x\cdot\nu)_{+}|\leqslant\delta
R\ \hbox{ in }\ B_{R}\ \hbox{ for some }\ 1\leqslant R\leqslant cR_{0}$

and $0\in\partial\\{u>0\\}$ then for any $1\leqslant r\leqslant R$ there is a
unit direction $\nu^{\prime}$ so that

$|u(x)-\tfrac{1}{(\nu^{\prime}\cdot\overline{a}\nu^{\prime})^{1/2}}(x\cdot\nu^{\prime})_{+}|\leqslant
C\left[\left(\frac{r}{R}\right)^{\alpha}(\delta+(R/R_{0})^{\frac{\beta}{d+4}})+r^{-\omega}\right]r\
\hbox{ for }\ x\in B_{r}$

and

$|\nu^{\prime}-\nu|\leqslant
C\left[\delta+(R/R_{0})^{\frac{\beta}{d+4}}+r^{-\omega}\right]$

where $\omega>0$ is a dimensional constant.

This maintenance of flatness result down to the microscale directly implies
that the free boundary stays a unit distance from a Lipschitz graph over the
approximate normal direction $\nu$.

######  Corollary 6.5.

Under the hypotheses of Lemma 6.4 there is a function
$\psi:\\{x\cdot\nu=0\\}\cap B_{R/8}\to\varmathbb{R}$ with $\psi(0)=0$ which is
$2$-Lipschitz and

$\\{x\cdot\nu\geqslant\psi(x^{\prime})-C\\}\cap B_{R/8}\subset\\{u>0\\}\cap
B_{r/2}\subset\\{x\cdot\nu\geqslant\psi(x^{\prime})+C\\}\cap B_{R/8}$

where $x^{\prime}=x-(x\cdot\nu)\nu$, and $C=C(\Lambda,d,L)$.

Since this Corollary is just a technical reinterpretation of Lemma 6.4 we will
postpone the proof to Section A.3.

Now we proceed to the proofs of Lemma 6.3 and Lemma 6.4.

###### Proof of Lemma 6.3.

Let $u$ as in the statement, and $\overline{u}\in u+H^{1}_{0}(\\{u>0\\}\cap
B_{r})$ be the upscaling of $u$ defined in Lemma 5.3 then, by Lemma 5.3,

$J_{0}(\bar{u},B_{r})\leqslant
J_{0}(v,B_{r})+C(L)[\sigma+r^{-\omega_{0}}]|B_{r}|\ \hbox{ for all $v\in
u+H^{1}_{0}(B_{r})$}$

where $\omega_{0}$ is the dimensional constant from Lemma 5.3. Also

(6.2) $\|\nabla\bar{u}\|_{L^{\infty}(B_{r})}\leqslant 2L\ \hbox{ and }\
\frac{1}{r}\|u-\bar{u}\|_{L^{\infty}(B_{r})}\leqslant Lr^{-1/3}.$

Thus $\overline{u}$ has $\|\nabla\bar{u}\|_{L^{\infty}(B_{r})}\leqslant 2L$,

$|\bar{u}(x)-\tfrac{1}{(\nu\cdot\overline{a}\nu)^{1/2}}(x\cdot\nu)_{+}|\leqslant[\delta+Lr^{-1/3}]r\
\hbox{ in }\ B_{r},$

and satisfies

$J_{0}(\bar{u},B_{r})\leqslant J_{0}(v,B_{r})+\sigma_{0}|B_{r}|\ \hbox{ for
all }\ v\in u+H^{1}_{0}(B_{r})$

with

$\sigma_{0}=C(L)[\sigma+r^{-\omega_{0}}].$

Applying Lemma 6.1 we find, absorbing the $Cr^{-1/2}$ term into the
$Cr^{-\frac{\omega_{0}}{(d+4)}}$ term,

$|\overline{u}(x)-\tfrac{1}{(\nu^{\prime}\cdot\overline{a}\nu^{\prime})^{1/2}}(x\cdot\nu^{\prime})_{+}|\leqslant\eta^{1+\alpha}[\delta+C\sigma^{\frac{1}{d+4}}+Cr^{-\frac{\omega_{0}}{(d+4)}}]r\
\hbox{ in }\ B_{\eta r}(0)$

for some $\nu^{\prime}$ with

$|\nu^{\prime}-\nu|\leqslant
C\left[\delta+\sigma^{\frac{1}{d+4}}+r^{-\frac{\omega_{0}}{(d+4)}}\right].$

From here we will call $\omega=\frac{\omega_{0}}{d+4}$. Then we carry the
flatness condition back to $u$ using (6.2)

$|u(x)-\tfrac{1}{(\nu^{\prime}\cdot\overline{a}\nu^{\prime})^{1/2}}x\cdot\nu^{\prime}|\leqslant|\overline{u}(x)-\tfrac{1}{(\nu^{\prime}\cdot\overline{a}\nu^{\prime})^{1/2}}(x\cdot\nu^{\prime})_{+}|+C\eta^{-1}r^{-1/2}\eta
r\ \hbox{ in }\ B_{\eta r}(0)$

and the new error term can also be absorbed into the $Cr^{-\omega}$ term.

∎

###### Proof of Lemma 6.4.

Iterate Lemma 6.3 taking $r_{0}=R$, $\delta_{0}=\delta$,

$r_{k}=\eta r_{k-1},\ \sigma_{k}=C(r_{k}/R_{0})^{\beta},$

and

$\delta_{k}=\eta^{\alpha}(\delta_{k-1}+C\sigma_{k-1}^{\frac{1}{d+4}}+Cr_{k-1}^{-\omega}).$

As long as

$\delta_{k}\leqslant\bar{\delta}$

and $\sigma_{0}\leqslant\bar{\sigma}$ then Lemma 6.4 applies. This holds, for
example, if

(6.3) $\delta\leqslant\bar{\delta}/3,\
C(R/R_{0})^{\frac{\beta}{d+4}}\leqslant\min\\{\bar{\delta}/3,\bar{\sigma}\\},\
\hbox{ and }\ r_{k}\geqslant C\bar{\delta}^{-6(d+4)}$

with $\bar{\delta}$ from Lemma 6.4. Note that this requires that $R\leqslant
cR_{0}$ for a small enough constant $c$ depending on $\beta$ and
$\bar{\delta}$. Call $\bar{k}$ to be the largest integer $k$ so that the third
requirement of (6.3) holds, note that $r_{\bar{k}}\leqslant C(d,\Lambda,L)$.

We compute the iteration of $\delta_{k}$

$\displaystyle\delta_{k}$
$\displaystyle=\eta^{\alpha}(\delta_{k-1}+C\sigma_{k-1}^{\frac{1}{d+4}}+Cr_{k-1}^{-\omega})$
$\displaystyle=\eta^{k\alpha}\delta_{0}+C\sum_{j=0}^{k-1}\eta^{(k-j)\alpha}[\sigma_{j}^{\frac{1}{d+4}}+r_{j}^{-\omega}]$
$\displaystyle\leqslant\eta^{k\alpha}\delta_{0}+C\sum_{j=0}^{k-1}\eta^{(k-j)\alpha}[(r_{j}/R_{0})^{\frac{\beta}{d+4}}+r_{j}^{-\omega}]$
$\displaystyle\leqslant\eta^{k\alpha}\delta_{0}+C\sum_{j=0}^{k-1}\eta^{(k-j)\alpha}[\eta^{\frac{\beta}{d+4}j}(R/R_{0})^{\frac{\beta}{d+4}}+\eta^{-{\omega}j}R^{-{\omega}}]$
$\displaystyle\leqslant\eta^{k\alpha}\delta_{0}+C\eta^{k\alpha}(R/R_{0})^{\frac{\beta}{d+4}}+C\eta^{-k{\omega}}R^{-{\omega}}$
$\displaystyle\leqslant\left[(r_{k}/R)^{\alpha}(\delta_{0}+C(R/R_{0})^{\frac{\beta}{d+4}})+r_{k}^{-{\omega}}\right]$

using for the second to last line that we can take
$0<\alpha<\frac{\beta}{d+4}$. We also obtain the following estimate of the
variation of the normal directions

$|\nu_{k}-\nu_{0}|\leqslant\sum_{j=1}^{k}|\nu_{j}-\nu_{j-1}|\leqslant
C\sum_{j=0}^{k-1}(\delta_{j}+\sigma_{j}^{\frac{1}{d+4}}+Cr_{j}^{-{\omega}})\leqslant
C[\delta_{0}+(R/R_{0})^{\frac{\beta}{d+4}}+r_{k}^{-{\omega}}].$

We are using several times here that we are summing geometric series and the
finite sum is bounded by a constant times the largest term, of course the
constant depends on $\eta$, which depends on $d,\Lambda,L$, and on $\beta$.

Finally the intermediate radii $r_{k}<r<r_{k-1}$ or $1\leqslant r\leqslant
r_{\bar{k}}$ are estimated by using the flatness on $B_{r_{k-1}}$ or
$B_{r_{\bar{k}}}$ respectively. ∎

######  Remark 6.6.

The almost minimality hypothesis in Lemma 6.4 could be replaced by a more
general one: for all $1\leqslant r\leqslant R$ and any $v\in
u+H^{1}_{0}(B_{r}(0))$

(6.4) $J(u,B_{r}(0))\leqslant J(v,B_{r}(0))+\omega(r/R_{0})|B_{r}|,$

as long as $\omega^{\frac{1}{d+4}}$ is a Dini modulus with $\omega(1)\leqslant
1$. In the proof the parameter $\sigma_{k}$ would change to
$\sigma_{k}=\omega(r_{k}/R_{0})=\omega(\eta^{k}R/R_{0})$. This would then
affect the estimate of the term

$\sum_{j=0}^{k-1}\eta^{(k-j)\alpha}\omega(\eta^{j}R/R_{0})^{\frac{1}{d+4}}\leqslant\eta^{k\alpha/2}+\sum_{k/2}^{k-1}\omega(\eta^{j}R/R_{0})^{\frac{1}{d+4}}\leqslant\eta^{k\alpha/2}+C(\eta)\int_{0}^{\eta^{k/2}}\omega(t)^{\frac{1}{d+4}}\frac{dt}{t}.$

This results in a decay of oscillations with an altered modulus

$\tilde{\omega}(\frac{r}{R})=\int_{0}^{(r/R)^{1/2}}\omega(t)^{\frac{1}{d+4}}\frac{dt}{t}$

assuming that this is not better than the $(r/R)^{\alpha/2}$ modulus.

###  6.1. Liouville property of flat energy minimizers on $\varmathbb{R}^{d}$

We conclude the section with an interesting corollary about the plane-like
property of large scale flat absolute minimizers on the whole space (global).

A function $u\in H^{1}_{loc}(\varmathbb{R}^{d})$ is called an absolute
minimizer if it is minimizing with respect to compact perturbations, i.e. if

$J(u,B)\leqslant J(v,B)\ \hbox{ for all }\ v\in u+H^{1}_{0}(B)$

where $B$ is _any_ finite radius ball in $\varmathbb{R}^{d}$. This property is
also referred to as class-A minimizer in the literature [12].

It is known that there are global plane-like absolute minimizers [28], i.e.
the minimizers are sandwiched between two translations of a planar solution of
the effective problem $(\nu\cdot\overline{a}\nu)^{-1}(\nu\cdot x)_{+}$. In
general it seems to be a very difficult problem to classify all the global
absolute minimizers. Even for homogeneous media there can be non planar global
absolute minimizers in higher dimensions [16], which is why the flatness
assumption we have been using is essential. The following result says that all
large scale flat global absolute minimizers grow sublinearly away from a
planar solution of the effective problem. It is not clear whether such
solutions are a _constant_ distance from a planar solution of the effective
problem, this seems like a very difficult question.

######  Theorem 6.7.

Suppose that $u$ is an absolute minimizer of $J$ on $\varmathbb{R}^{d}$ with

$L=\limsup_{r\to\infty}\|\nabla u\|_{\underline{L}^{2}(B_{r})}<+\infty$

and

$\frac{1}{r}\inf_{\nu\in
S^{d-1}}\sup_{B_{r}}|u(x)-\tfrac{1}{(\nu\cdot\bar{a}\nu)^{1/2}}(x\cdot\nu)_{+}|\leqslant\delta$

for all $r$ sufficiently large.

There is $\bar{\delta}$ depending on universal constants and $L$ so that if
$0<\delta\leqslant\bar{\delta}$ then there is a slope $\nu_{*}\in S^{d-1}$
with $|\nu_{*}-\nu|\leqslant C\delta$ so that for all $r\geqslant 1$

$\frac{1}{r}\sup_{B_{r}}|u(x)-\tfrac{1}{(\nu_{*}\cdot\bar{a}\nu_{*})^{1/2}}(x\cdot\nu_{*})_{+}|\leqslant
Cr^{-{\omega}}.$

######  Remark 6.8.

This kind of Liouville property is a standard corollary of the $C^{1,\alpha}$
estimate, see [4, 5, 8, 25] for related results. In the case of standard
elliptic homogenization the decay $Cr^{-\delta}$ can be upgraded to $Cr^{-1}$,
the key point is subtracting off an appropriate corrector and then re-applying
the estimates. This relies on invariance of the equations with respect to
vertical translations $u\mapsto u+c$, our PDE is more analogous to the
minimization of integral functionals $\int a(x,u)\nabla u\cdot\nabla u\ dx$
where $a$ is periodic in $(x,u)$ and this invariance is missing. See Moser
[24] and Moser and Struwe [25] for some discussion of related Liouville
questions for those models.

###### Proof.

By Theorem 1.1 $u$ is Lipschitz in $\varmathbb{R}^{n}$ with constant $CL$ with
$C$ universal. Let

$\bar{\delta}(d,\Lambda,\|\nabla u\|_{L^{\infty}})>0$

from Lemma 6.4. By Lemma 6.4, there is $\alpha<1$ so that for all $R$
sufficiently large (so that the $\bar{\delta}$ flatness assumption holds) and
for $1\leqslant r\leqslant R$

$\frac{1}{r}\inf_{\nu\in
S^{d-1}}\sup_{B_{r}}|u(x)-\frac{1}{(\nu\cdot\bar{a}\nu)^{1/2}}(x\cdot\nu)_{+}|\leqslant
C\left((\frac{r}{R})^{\alpha}\left[\frac{1}{R}\inf_{\nu\in
S^{d-1}}\sup_{B_{R}}|u(x)-\frac{1}{(\nu\cdot\bar{a}\nu)^{1/2}}(x\cdot\nu)_{+}|\right]+\frac{1}{r^{\omega}}\right)$

First of all, since the term in square brackets on the right is smaller than
$\delta$ for all $R$ sufficiently large, sending $R\to\infty$ we find,

$\frac{1}{r}\inf_{\nu\in
S^{d-1}}\sup_{B_{r}}|u(x)-\frac{1}{(\nu\cdot\bar{a}\nu)^{1/2}}(x\cdot\nu)_{+}|\leqslant
Cr^{-\omega}.$

For each $r\geqslant 1$ let $\nu(r)\in S^{d-1}$ achieve the previous infimum.
Then this estimate shows that for all $r\leqslant s\leqslant 2r$

$\sup_{B_{r}}|\frac{1}{(\nu(s)\cdot\bar{a}\nu(s))^{1/2}}(x\cdot\nu(s))_{+}-\frac{1}{(\nu(r)\cdot\bar{a}\nu(r))^{1/2}}(x\cdot\nu(r))_{+}|\leqslant
Cr^{1-\omega}$

applying this with
$x=\frac{1}{\Lambda}r(\nu(s)\cdot\bar{a}\nu(s))^{1/2}\nu(r)$ and with
$x=\frac{1}{\Lambda}r(\nu(r)\cdot\bar{a}\nu(r))^{1/2}\nu(s)$ gives

$|\nu(r)\cdot\nu(s)-\frac{(\nu(s)\cdot\bar{a}\nu(s))^{1/2}}{(\nu(r)\cdot\bar{a}\nu(r))^{1/2}}|+|\nu(r)\cdot\nu(s)-\frac{(\nu(r)\cdot\bar{a}\nu(r))^{1/2}}{(\nu(s)\cdot\bar{a}\nu(s))^{1/2}}|\leqslant
Cr^{-\omega}$

since $\nu(r)\cdot\nu(s)$ is close to two numbers which are reciprocals of one
another

$\frac{1}{2}|\nu(r)-\nu(s)|^{2}=1-\nu(r)\cdot\nu(s)\leqslant Cr^{-\omega}.$

Thus the sequence $\nu(2^{k}r_{0})$ converges as $k\to\infty$, call $\nu_{*}$
to be the limit. Now for any $r\geqslant r_{0}$ choose $k$ so that
$2^{k-1}r_{0}<r\leqslant 2^{k}r$ and then, using the previous estimate again,

$|\nu(r)-\nu_{*}|\leqslant|\nu(r)-\nu(2^{k}r_{0})|+\sum_{j=k}^{\infty}|\nu(2^{j+1}r_{0})-\nu(2^{j}r_{0})|\leqslant
Cr^{-\omega}.$

∎

##  Appendix A Proofs of technical results

In this section we provide the detailed proofs of several of the technical
results appearing in the paper.

###  A.1. Proof of Lemma 4.2

The result is a standard Hopf Lemma barrier argument, using that $a$ is
Lipschitz to write the equation in non-divergence form. We just need to
establish that the Green’s function $G(x,y)$ for the operator
$-\nabla\cdot(a(rx)\nabla\cdot)$ in $B_{1}$ has

$\inf_{y\in B_{\eta}}G(0,y)\geqslant c$

for $\eta$ and $c$ depending on universal constants and $r_{0}$.

For this take $G_{0}(x,y)$ to be the Green’s function for
$-\nabla\cdot(a(0)\nabla\cdot)$ in $B_{1}$ and $G(x,y)$ to be the Green’s
function for $-\nabla\cdot(a(rx)\nabla\cdot)$ in $B_{1}$. Call
$w(x)=G(x,0)-G_{0}(x,0)$ so that

$-\nabla\cdot(a(0)\nabla w)=\nabla\cdot((a(rx)-a(0))\nabla G(x,0))\ \hbox{ and
}\ w=0\ \hbox{ on }\ \partial B_{1}.$

Then

$w(x)=\int_{B_{r}}\nabla G_{0}(x,y)(a(y)-a(0))\nabla G(y,0)\ dy.$

Applying Green’s function bounds [21][Lemma 2.5],

$|\nabla G(x,y)|\leqslant C|x-y|^{1-n}$

which hold uniformly in $r>0$, and the Hölder continuity of $a$

$\displaystyle|w(x)|\leqslant
C\int_{B_{1}}|x-y|^{1-n}r^{\gamma}|y|^{\gamma}|y|^{1-n}\ dy.$

We estimate in a standard way splitting the integral

$\displaystyle|w(x)|$ $\displaystyle\leqslant
Cr^{\gamma}\bigg{[}\int_{1\geqslant|y|\geqslant 2|x|}|y|^{\gamma-2(n-1)}\
dy+\int_{|y|\leqslant|x|/2}|x|^{1-n}|y|^{\gamma+1-n}\ dy\cdots$
$\displaystyle\quad\quad\quad\quad\cdots+\int_{|x|/2\leqslant|y|\leqslant
2|x|}|x-y|^{1-n}|x|^{\gamma+1-n}\ dy\bigg{]}$

this results in

$|w(x)|\leqslant Cr^{\gamma}|x|^{2-n+\gamma}.$

Because $G_{0}$ solves a constant coefficient equation we have
$G_{0}(x,0)\geqslant c|x|^{2-n}$ so

$G(x,0)\geqslant G_{0}(x,0)-w(x)\geqslant(c-Cr^{\gamma}|x|^{\gamma})|x|^{2-n}$

so we can guarantee a universal positive lower bound on $|x|\leqslant
cr_{0}^{-1}$. ∎

###  A.2. Proof of Lemma 5.6

Without loss, by a linear transformation, we can assume that
$\overline{a}=id$. This comes at the cost of changing the coefficient of ${\bf
1}_{\\{u>0\\}}$ to $\textup{det}(\overline{a})$, since this is just another
constant depending on ellipticity we will just pretend it is $1$ to reduce the
number of constants below.

Step 1 (Hölder continuity): We consider continuity at a boundary point
$y\in\partial B_{1}$. Call $L=\max\\{1,\|\nabla g\|_{\infty}\\}$. First we use
a barrier argument to show that $u$ is positive in a ball of radius
$r=c(d)L^{-1}g(y)$ centered at $y$. Then $u$ is harmonic in $B_{1}\cap
B_{r}(y)$.

Since $u$ is subharmonic in the entire $B_{1}$, see [2], we can do a standard
barrier argument from above to find, for any $0<\alpha<1$,

$\max_{B_{1}\cap B_{\rho}(y)}u\leqslant g(y)+C(d,\alpha)L\rho^{\alpha}.$

This argument also works from below if $r\geqslant 2$ since then $u$ is
harmonic in the entire ball $B_{1}$.

In the case $r\leqslant 2$ we can apply a similar barrier argument from below
in $B_{r}(y)\cap B_{1}$ to find for all $0<\alpha<1$ and
$0\leqslant\rho\leqslant r$

$\min_{B_{1}\cap B_{\rho}(y)}u\geqslant
g(y)-C(d,\alpha)g(y)(\rho/r)^{\alpha}=g(y)-CL^{\alpha}g(y)^{1-\alpha}\rho^{\alpha}.$

Note that since $r\leqslant 2$ also $L^{\alpha}g(y)^{1-\alpha}\leqslant CL$ so
the estimate from below is of the same form as the one from above.

We just need to check the positivity in $B_{r}(y)$ for $r=c_{0}L^{-1}g(y)$ and
$c_{0}>0$ chosen sufficiently small dimensional. We can assume to start that
$c_{0}>0$ small enough so that

$\inf_{B_{2r}(y)}g\geqslant g(y)-2Lr\geqslant\frac{1}{2}g(y).$

Let $\phi$ be the harmonic function in $B_{2}\setminus B_{1}$ which is equal
to $1$ on $\partial B_{1}$ and $0$ on $\partial B_{2}$, extend $\phi$ by $0$
outside of $B_{2}$. Make a family of barriers with $r=c_{0}L^{-1}g(y)$

$\psi_{t}(x)=\frac{1}{2}g(y)\phi(r^{-1}(x-z_{t}))$

where $(z_{t})_{t\in[0,1]}$ is

$z_{t}=(1+\frac{2r}{(1+t)})y$

so that $B_{2r}(z_{0})$ is exterior tangent to $B_{1}$ at $y$ and
$B_{r}(z_{1})$ is exterior tangent to $B_{1}$ at $y$. Note that

$\psi_{t}\leqslant\frac{1}{2}g(y)\leqslant g(x)\ \hbox{ for $x$ in
$B_{2r}(z_{t})\cap B_{1}\subset B_{2r}(y)\cap B_{1}$}$

and $\psi_{t}=0\leqslant g$ outside of $B_{2r}(z_{t})$.

The plan is to show that $\psi_{t}$ are subsolutions of the Euler-Lagrange
equation and below $g$ on $\partial B_{1}$ and then slide inwards from $t=0$
to $t=1$ to find $\psi_{1}\leqslant u$. Note that $\psi_{0}\leqslant u$ in
$B_{1}$ because $\psi_{0}\equiv 0$ in $B_{1}$. We have also established above
that $\psi_{t}\leqslant g$ on $\partial B_{1}$ for all $t\in[0,1]$.

Note that on $\partial B_{2r}(z_{t})$

$|\nabla\psi_{t}|=c(d)c_{0}^{-1}L\geqslant 1$

where the last inequality comes from $L\geqslant 1$ and choosing $c_{0}$ large
enough dimensional. In particular $\psi_{t}$ is a smooth strict subsolution of
the Euler-Lagrange equation for $J_{0}$ outside of $B_{r}(z_{t})$.

Technically we are using that $u$ being a $J_{0}$ minimizer in $B_{1}$ is a
viscosity solution of the Euler-Lagrange equation

$-\Delta u=0\ \hbox{ in }\ \\{u>0\\}\cap B_{1}\ \hbox{ and }\ |\nabla u|=1\
\hbox{ on }\ \partial\\{u>0\\}\cap B_{1}.$

Then we are applying a sliding maximum principle, for reference see
[10][Theorem 2.2].

Step 2 (Energy comparison): Define an energy competitor

$v=\phi_{\delta}g+u(1-\phi_{\delta})$

where $\phi_{\delta}$ is a Lipschitz function with
$0\leqslant\phi_{\delta}\leqslant 1$, $|\nabla\phi_{\delta}|\leqslant C(\delta
r)^{-1}$, and

$\phi_{\delta}(x)=\begin{cases}1&|x|\leqslant(1-2\delta)r\\\
0&|x|\geqslant(1-\delta)r.\end{cases}$

Since $\phi_{\delta}\equiv 0$ in a neighborhood of $\partial B_{r}$ we do have
$v\in H^{1}_{0}(U)+g$.

Energy comparison gives

$J(u,B_{r})\leqslant J(v,B_{r})=J(u,B_{(1-2\delta)r})+J(v,B_{r}\setminus
B_{(1-2\delta)r})$

so

$\displaystyle J(u,B_{r}\setminus B_{(1-\delta)r})$ $\displaystyle\leqslant
J(v,B_{r}\setminus B_{(1-2\delta)r})-J(u,B_{(1-\delta)r}\setminus
B_{(1-2\delta)r})$ $\displaystyle\leqslant\int_{B_{r}\setminus
B_{(1-2\delta)r}}|\phi_{\delta}\nabla g+(1-\phi_{\delta})\nabla
u+(g-u)\nabla\phi_{\delta}|^{2}\ dx$
$\displaystyle\quad\quad\cdots-\int_{B_{(1-\delta)r}\setminus
B_{(1-2\delta)r}}|\nabla u|^{2}dx+\Lambda|B_{r}\setminus B_{(1-2\delta)r}|$
$\displaystyle\leqslant\int_{B_{r}\setminus
B_{(1-2\delta)r}}(1+\eta)|\phi_{\delta}\nabla g+(1-\phi_{\delta})\nabla
u|^{2}+(1+\frac{1}{\eta})|(g-u)\nabla\phi_{\delta}|^{2}\ dx$
$\displaystyle\quad\quad\cdots-\int_{B_{(1-\delta)r}\setminus
B_{(1-2\delta)r}}|\nabla u|^{2}dx+\Lambda|B_{r}\setminus B_{(1-2\delta)r}|$
$\displaystyle\leqslant\int_{B_{r}\setminus
B_{(1-2\delta)r}}(1+\eta)\phi_{\delta}|\nabla
g|^{2}+(1+\eta)(1-\phi_{\delta})|\nabla
u|^{2}+(1+\frac{1}{\eta})|(g-u)|^{2}|\nabla\phi_{\delta}|^{2}\ dx$
$\displaystyle\quad\quad\cdots-\int_{B_{(1-\delta)r}\setminus
B_{(1-2\delta)r}}|\nabla u|^{2}dx+\Lambda|B_{r}\setminus B_{(1-2\delta)r}|$
$\displaystyle\leqslant(\Lambda+2L^{2})|B_{r}\setminus
B_{(1-2\delta)r}|+\int_{B_{r}\setminus B_{(1-2\delta)r}}\eta|\nabla
u|^{2}+(1+\frac{1}{\eta})|(g-u)|^{2}|\nabla\phi_{\delta}|^{2}\ dx$
$\displaystyle\leqslant\eta\|\nabla
u\|^{2}_{L^{2}(B_{r})}+C(1+\eta^{-1}\delta^{2(\alpha-1)}r^{-2})|B_{r}\setminus
B_{(1-2\delta)r}|.$

We used Young’s inequality in the form $2ab\leqslant\eta
a^{2}+\frac{1}{\eta}b^{2}$ for the third inequality, and we used convexity of
$a\mapsto|a|^{2}$ for the fourth inequality. Note that

$\|\nabla u\|^{2}_{L^{2}(B_{r})}\leqslant J(u,B_{r})\leqslant
J(g,B_{r})\leqslant C(L)|B_{r}|$

and $|B_{r}\setminus B_{(1-2\delta)r}|\leqslant C\delta|B_{r}|$ so

$J(u,B_{r}\setminus B_{(1-\delta)r})\leqslant
C(\eta+\delta+\eta^{-1}\delta^{2(\alpha-1)+1}r^{-2})|B_{r}|$

choosing $\eta=\delta^{\frac{1}{2}+(\alpha-1)}r^{-1}$ to match the first and
third terms

$J(u,B_{r}\setminus B_{(1-\delta)r})\leqslant
C(\delta+r^{-1}\delta^{\alpha-\frac{1}{2}})|B_{r}|.$

###  A.3. Proof of Corollary 6.5

We can take the flatness direction in the assumption to be $\nu=e_{d}$. By the
assumption of the corollary then

$|u(x)-\tfrac{1}{e_{d}\cdot\overline{a}e_{d}}(x_{d})_{+}|\leqslant\delta r.$

Let $y=(y^{\prime},y_{d})$ and $z=(z^{\prime},z_{d})$ be two points in
$\partial\\{u>0\\}\cap B_{r/8}$. So, in particular,

$z\in B_{r/4}(y)\subset B_{r/2}(y)\subset B_{5r/8}(0)$

Now we aim to apply Lemma 6.4 to $u$ in $B_{r/2}(y)$. By the original flatness
assumption $u$ is $2\delta r$-flat in $B_{r/2}(y)$ so as long as
$4\delta\leqslant\bar{\delta}$ from Lemma 6.4 we can apply the result at the
scale $s=|y-z|+s_{0}$ to obtain

$|u(x)-\tfrac{1}{(\nu\cdot\overline{a}\nu)^{1/2}}(x\cdot\nu)_{+}|\leqslant Ar\
\hbox{ with }\ A=C(\bar{\delta}+s^{-\omega})$

for a direction $\nu$ with

$|\nu-e_{d}|\leqslant A$

where the scale $s_{0}$ is chosen depending on $L$ and universal constants so
that we can have $A\leqslant\frac{1}{4}$, also choosing $\bar{\delta}$ smaller
if necessary.

In particular

$|(z-y)\cdot\nu|\leqslant As=A(|z-y|+s_{0})$

and

$|(z-y)\cdot\nu|\geqslant|(z-y)\cdot
e_{d}|-|(z-y)\cdot(e_{d}-\nu)|\geqslant|x_{d}-z_{d}|-A|z-y|.$

So

$|z_{d}-y_{d}|\leqslant
2A(|z-y|+s_{0})\leqslant\frac{1}{2}(|z_{d}-y_{d}|+|z^{\prime}-y^{\prime}|+s_{0})$

and we can move the first term over to the left and get

(A.1) $|z_{d}-y_{d}|\leqslant 2|z^{\prime}-y^{\prime}|+2s_{0}.$

The final conclusion follows then from taking the graph $\psi$ to be the
supremum of cones $a-2|x^{\prime}-x_{0}^{\prime}|$ whose subgraph lies inside
$\\{u>0\\}$ in $B_{r/8}$. Then (A.1) implies that $\partial\\{u>0\\}$ is
contained between the graphs of $\psi$ and $\psi+2s_{0}$.

∎

###  A.4. Density estimates

In this appendix we show interior and exterior density estimates.

The main result of the section is

######  Proposition A.1.

Suppose that $u$ is non-negative $H^{1}$ function in a ball $B_{r}(0)$ with

  1. (a)

$u$ is $L$-Lipschitz

  2. (b)

$u$ is $\ell$-non-degenerate at its free boundary

  3. (c)

There is $\sigma>0$ so that for any ball $B_{\rho}\subset B_{r}(0)$

$J(u,B_{\rho})\leqslant J(v,B_{\rho})+\sigma|B_{\rho}|$

where $v\in u+H^{1}_{0}(B_{\rho})$ is the $a$-harmonic replacement.

There is $\sigma_{0}>0$ and $c>0$, $C\geqslant 1$ depending on $L$, $\ell$ and
universal parameters so that if $\sigma\leqslant\sigma_{0}$ then

(A.2) $c\leqslant\frac{|\\{u>0\\}\cap B_{\rho}(x)|}{|B_{\rho}|}\leqslant 1-c\
\hbox{ for any }\ B_{\rho}\subset B_{r}(0).$

The interior density only requires the Lipschitz and non-degenerate hypotheses
so the conclusion is standard, the exterior density estimate does require the
almost minimality property and some inputs from homogenization.

###### Proof of Proposition A.1.

The inner density follows directly from non-degeneracy and the Lipschitz
estimate, see for example [10].

Now we consider the exterior density. We follow a standard line of argument,
see [29, Lemma 5.1], but we need to additionally use homogenized problem
replacement at large scales and continuity of $a$ at small scales to apply the
constant coefficient argument (Poisson kernel lower bound).

Let $h$ be the $a$-harmonic replacement of $u$ in $B_{r}$. Then

$\displaystyle\sigma|B_{r}|+\Lambda|\\{u=0\\}\cap B_{r}|$
$\displaystyle\geqslant\int_{B_{r}}a(x)\nabla u\cdot\nabla u-a(x)\nabla
h\cdot\nabla h\ dx$
$\displaystyle=\int_{B_{r}}a(x)\nabla(u-h)\cdot\nabla(u-h)\ dx$
$\displaystyle\geqslant\Lambda^{-1}\int_{B_{r}}|\nabla(u-h)|^{2}\ dx.$

We will first argue for $r\geqslant r_{0}\geqslant 1$ large enough but bounded
by universal constants, then we will argue for $r\leqslant r_{0}$ at the end.
Let $\overline{h}$ be the $\overline{a}$-harmonic replacement of $h$ in
$B_{r}$ then by Theorem 2.1

$r^{-2}\|h-\overline{h}\|^{2}_{L^{2}(B_{r})}\leqslant Cr^{-\alpha}|B_{r}|.$

By Poincaré inequality in $B_{r}$

$\displaystyle\int_{B_{r}}|\nabla(u-h)|^{2}\ dx$
$\displaystyle\geqslant\frac{\Lambda^{-1}c_{d}}{r^{2}}\int_{B_{r}}(h-u)^{2}\
dx$ $\displaystyle\geqslant\frac{c}{r^{2}}\int_{B_{r}}(\overline{h}-u)^{2}\
dx-\frac{c}{r^{2}}\int_{B_{r}}(\overline{h}-h)^{2}\ dx$
$\displaystyle\geqslant\frac{c}{|B_{r}|}\left(\frac{1}{r}\int_{B_{r}}h-u\
dx\right)^{2}-Cr^{-\alpha}|B_{r}|.$

By the non-degeneracy assumption of $u$ we know $\max_{\partial
B_{r}}u\geqslant\ell r$. By Lipschitz continuity $u\geqslant\frac{1}{2}\ell r$
in a $\frac{1}{2}\frac{\ell}{L}r$-neighborhood of the point where the maximum
is achieved so

$\int_{\partial B_{r}}u(x)dS\geqslant\frac{1}{2}\ell r\int_{\partial
B_{r}}{\bf 1}_{\\{u(x)\geqslant\frac{1}{2}\ell r\\}}dS\geqslant
c\tfrac{\ell^{2}}{L}r|\partial B_{r}|$

By the Poisson kernel formula for $\overline{h}$, now dropping the $\ell$ and
$L$ dependence of the constant,

$\overline{h}(0)\geqslant cr.$

and by Harnack inequality (recall $\overline{h}$ is harmonic up to a linear
transformation)

$\overline{h}(x)\geqslant c\overline{h}(0)\geqslant cr\ \hbox{ in }\ x\in
B_{r/2}.$

The Lipschitz estimate of $u$ gives

$u(x)\leqslant L|x|.$

Thus

$u(x)-\overline{h}(x)\geqslant\frac{1}{2}c\ell r\ \hbox{ in }|x|\leqslant
c\ell r/2L.$

Combining this with previous

$\displaystyle\frac{1}{|B_{r}|}\left(\frac{1}{r}\int_{B_{r}}\overline{h}-u\
dx\right)^{2}\geqslant\frac{1}{|B_{r}|}\left(\frac{1}{r}\int_{B_{c\ell
r/2L}}\frac{1}{2}c\ell r\ dx\right)^{2}\geqslant 3c_{0}|B_{r}|.$

For $r\geqslant r_{0}$ (universal) the term
$C\mathcal{E}(\frac{1}{r})\leqslant c_{0}$ so

$\int_{B_{r}}|\nabla(u-h)|^{2}\ dx\geqslant 2c_{0}|B_{r}|$

and for $\sigma\leqslant c_{0}$ as well we get finally

$\Lambda|\\{u=0\\}\cap B_{r}|\geqslant c_{0}|B_{r}|.$

Next we argue for $r\leqslant r_{0}$. Actually it is more convenient to argue
for $r\leqslant r_{1}\ll 1$ again a universal constant. For intermediate
values $r_{1}\leqslant r\leqslant r_{0}$ we will just use a lower bound

$|\\{u=0\\}\cap B_{r}|\geqslant|\\{u=0\\}\cap B_{r_{1}}|\geqslant
c_{1}|B_{r_{1}}|\geqslant c_{1}(\frac{r_{1}}{r_{0}})^{d}|B_{r}|.$

The small $r_{1}$ case follows a very similar line of argument to the previous
using the $a(0)$-harmonic replacement $\tilde{h}$ in $B_{r}$ (instead of
$\overline{a}$-harmonic replacement) along with the estimate

$\|h-\tilde{h}\|_{\underline{L}^{2}(B_{r})}^{2}\leqslant
Cr^{2}\|\nabla(h-\tilde{h})\|_{\underline{L}^{2}(B_{r})}^{2}\leqslant
Cr^{2}r^{2\gamma}|B_{r}|$

which follows from the standard energy estimate

$\|\nabla(h-\tilde{h})\|_{L^{2}(B_{r})}\leqslant
C\|a-a(0)\|_{L^{\infty}(B_{r})}\|\nabla\tilde{h}\|_{L^{2}(B_{r})}\leqslant
Cr^{\gamma}(\int_{\partial B_{r}}|u||\nabla u|)^{1/2}\leqslant
Cr^{\gamma}L|B_{r}|^{1/2}$

using that $a$ is $C^{0,\gamma}$. ∎

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