# The dimension of planar elliptic measures arising from Lipschitz matrices in
Reifenberg flat domains

Ignasi Guillén-Mola  Ignasi Guillén-Mola, Departament de Matemàtiques,
Universitat Autònoma de Barcelona<EMAIL_ADDRESS>,  Martí Prats
Martí Prats, Departament de Matemàtiques, Universitat Autònoma de Barcelona,
and Centre de Recerca Matemàtica, Barcelona, Catalonia<EMAIL_ADDRESS>and  Xavier Tolsa  Xavier Tolsa, ICREA, Barcelona, Departament de
Matemàtiques, Universitat Autònoma de Barcelona, and Centre de Recerca
Matemàtica, Barcelona, Catalonia<EMAIL_ADDRESS>
###### Abstract.

In this paper we show that, given a planar Reifenberg flat domain with small
constant and a divergence form operator associated to a real (not necessarily
symmetric) uniformly elliptic matrix with Lipschitz coefficients, the
Hausdorff dimension of its elliptic measure is at most $1$. More precisely, we
prove that there exists a subset of the boundary with full elliptic measure
and with $\sigma$-finite one-dimensional Hausdorff measure. For Reifenberg
flat domains, this result extends a previous work of Thomas H. Wolff for the
harmonic measure.

###### Acknowledgements.

Part of this work was carried out while I.G.M. and X.T. were visiting the
Hausdorff Institute for Mathematics in Bonn (Germany) during the trimester
program _Interactions between Geometric measure theory, Singular integrals,
and PDE_.

I.G.M. was supported by the PIF UAB 2020/21 from the Universitat Autònoma de
Barcelona, and by Generalitat de Catalunya’s Agency for Management of
University and Research Grants (AGAUR) (2021 FI_B 00637 and 2023 FI-3 00151).
I.G.M. and M.P. were partially supported by the Spanish State Research Agency
(AEI) project PID2021-125021NAI00 (Spain). M.P. was partially supported by the
projects PID2021-123405NB-I00 (AEI), and 2021-SGR-00087 (AGAUR). M.P. and X.T.
were supported by the AEI, through the Severo Ochoa and María de Maeztu
Program for Centers and Units of Excellence in R&D (CEX2020-001084-M). X.T.
was supported by the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (grant agreement
101018680), and project 2021-SGR-00071 (AGAUR). All the authors were also
supported by PID2020-114167GB-I00. We thank CERCA Programme/Generalitat de
Catalunya for institutional support.

###### Key words and phrases:

Elliptic measure, Reifenberg flat domain.

###### 1991 Mathematics Subject Classification:

28A12, 31A15, 35J25. Secondary: 28A25, 28A78, 31B05, 35J08.

###### Contents

  1. 1 Introduction and Main results
  2. 2 Preliminaries and definitions
    1. 2.1 Notation
    2. 2.2 CDC, NTA and Reifenberg flat domains
    3. 2.3 Partial differential equations
    4. 2.4 The fundamental solution and the Green function
  3. 3 Main Lemma and preliminary reductions
    1. 3.1 The change of variables in Reduction 1
    2. 3.2 Reduction 1: Lipschitz diagonalization of the symmetric part
    3. 3.3 Reduction 2: The dependence on the Lipschitz seminorm
  4. 4 Elliptic measures in CDC domains
  5. 5 Proof of the weak version of the Main Lemma
  6. 6 Proof of Theorem 1.1
    1. 6.1 Reduction to bounded domains
    2. 6.2 Proof for bounded domains
  7. 7 $L\log L(d\sigma)$ type estimate for small densities: Proof of (1.2)
    1. 7.1 Directional derivatives and the dual space
    2. 7.2 Sketch of the proof
    3. 7.3 Second derivatives of the Green function
    4. 7.4 Whitney cubes and proof of Step 9
    5. 7.5 Proof of Steps 1 to 8

##  1\. Introduction and Main results

We study the dimension of planar elliptic measures in Reifenberg flat domains
with small constant, assuming also Lipschitz continuity of the coefficients of
the matrix. In fact, that regularity is only needed near the boundary.

Let $L_{A}u\coloneqq-\operatorname{div}(A\nabla u)$ be a second order operator
in divergence form associated to the $(n+1)\times(n+1)$ real matrix
$A(\cdot)=(a_{ij}(\cdot))_{1\leq i,j\leq n+1}$ such that there exists
$\lambda\geq 1$ with

(1.1a) $\displaystyle\lambda^{-1}|\xi|^{2}\leq\langle A(x)\xi,\xi\rangle$
$\displaystyle\quad\text{for all }\xi\in{\mathbb{R}}^{n+1}\text{ and a.e.\
}x\in{\mathbb{R}}^{n+1},$ (1.1b) $\displaystyle\langle
A(x)\xi,\eta\rangle\leq\lambda|\xi||\eta|$ $\displaystyle\quad\text{for all
}\xi,\eta\in{\mathbb{R}}^{n+1}\text{ and a.e.\ }x\in{\mathbb{R}}^{n+1}.$

We say that a measurable matrix is uniformly elliptic with ellipticity
constant $\lambda\geq 1$ if (1.1a) and (1.1b) are satisfied. The uniform
ellipticity condition implies that the matrix has bounded coefficients.

Let $\Omega\subset{\mathbb{R}}^{2}$ be an open set and fix a point
$p\in\Omega$. Consider the operator

$\begin{array}[]{cccc}T:&C(\partial\Omega)&\to&\mathbb{R}\\\
&f&\mapsto&u_{f}^{L_{A}}(p),\end{array}$

where $u_{f}^{L_{A}}$ is the $L_{A}$-harmonic extension of $f$ in $\Omega$ via
Perron’s method. By the maximum principle for $L_{A}$-harmonic functions with
uniformly elliptic matrices (see [GT01, p. 46] for example), the operator $T$
is linear, bounded and positive. For bounded open sets
$\Omega\subset{\mathbb{R}}^{n+1}$ with $n\geq 2$ we do the same construction.

The elliptic ($L_{A}$-harmonic) measure in $\Omega$ with pole $p\in\Omega$ is
the unique Radon probability measure $\omega_{\Omega,A}^{p}$ (by the Riesz
representation theorem) such that

$u_{f}^{L_{A}}(p)=\int_{\partial\Omega}f(\xi)\,d\omega_{\Omega,A}^{p}(\xi)\quad\text{for
every }f\in C(\partial\Omega).$

Via the Perron method construction, although the characteristic function
$\mathbf{1}_{E}$ of a set $E\subset\partial\Omega$ is not continuous in
$\partial\Omega$, the function $x\mapsto\omega_{\Omega,A}^{x}(E)$ is
$L_{A}$-harmonic in $\Omega$. For a detailed construction of elliptic/harmonic
measures, see [HKM06, Section 11]. If the set $\Omega$, the matrix $A$, or the
pole $p\in\Omega$ is clear from the context, we will omit it in
$\omega_{\Omega,A}^{p}$.

In this work, we study the Hausdorff dimension of elliptic measures arising
from uniformly elliptic matrices. This is defined as

$\dim_{\mathcal{H}}\omega_{\Omega,A}^{p}\coloneqq\inf~\\{\dim_{\mathcal{H}}F:\omega_{\Omega,A}^{p}(F^{c})=0\\}.$

Naturally $\dim_{\mathcal{H}}\omega\leq\dim_{\mathcal{H}}\partial\Omega$. The
Hausdorff dimension of the harmonic measure (i.e., with $\omega=\omega_{Id}$)
has been studied by several authors, both in the plane and in higher
dimensions, showing a different behavior in each case.

In the plane, Carleson proved $\dim_{\mathcal{H}}\omega\leq 1$ for “snowflake
type” sets and $\dim_{\mathcal{H}}\omega<1$ for some self similar “Cantor
type” sets, both results in [Car85]. More precise results were obtained for
simply connected domains by Makarov in [Mak85] by showing
$\dim_{\mathcal{H}}\omega=1$. The upper bound $\dim_{\mathcal{H}}\omega\leq 1$
with no assumptions on the domain was shown by Jones and Wolff in [JW88], and
later it was improved by Wolff in [Wol93] by proving that there is a subset
$F\subset\partial\Omega$ with full harmonic measure $\omega(F)=1$ and
$\sigma$-finite length.

In the same direction for higher dimensions, in ${\mathbb{R}}^{n+1}$ with
$n\geq 2$, Bourgain proved in [Bou87] that there exists a dimensional constant
$b_{n}>0$ such that $\dim_{\mathcal{H}}\omega\leq n+1-b_{n}$. By the results
on the previous paragraph, we can take $b_{1}=1$ and this choice is optimal.
For $n\geq 2$, Wolff constructed in [Wol95] a domain
$\Omega_{n}\subset{\mathbb{R}}^{n+1}$ such that
$\dim_{\mathcal{H}}\omega_{\Omega_{n}}>n$, and in particular the Bourgain
constant can’t equal $1$, i.e., $b_{n}<1$. In a recent work [BG22], Badger and
Genshaw refined the proof in [Bou87] to find estimates on the Bourgain
constant $b_{n}\in(0,1)$.

From the results in the previous paragraphs we have that
$\dim_{\mathcal{H}}\omega<n+1$ (with some precise gap), but possibly
$\dim_{\mathcal{H}}\omega=\dim_{\mathcal{H}}\partial\Omega$. In fact, the
situation $\dim_{\mathcal{H}}\omega<\dim_{\mathcal{H}}\partial\Omega$ holds
for many non-trivial domains. This phenomenon is frequently called the
“dimension drop” for harmonic measure. The dimension drop is closely related
to the results mentioned above. Indeed, similar techniques to the ones in
[Bou87, JW88] are used in some of the following articles. The first work in
this direction is due to Carleson [Car85] for self-similar Cantor sets in the
plane. For this type of domains, see Makarov and Volberg [MV86], Batakis
[Bat96], and also [Vol92, Vol93]. Jones and Wolff showed that the dimension
drop happens for some uniform and disconnected planar domains, see Theorem 2.1
in [GM05, Section IX.2]. For IFS domains (iterated functions systems), see
Urbański and Zdunik [UZ02], and Batakis and Zdunik [BZ15].

For general AD-regular domains of fractional dimension, one may ask if the
dimension drop happens. It holds on uniformly “non-flat” AD-regular domains
with codimension smaller than $1$, as shown by Azzam in [Azz20]. Recently, for
$n\geq 1$ and $s\in[n-\frac{1}{2},n)$, the third author showed in [Tol23] that
the dimension drop occurs for higher codimensional $s$-AD-regular subsets of
$C^{1}$ $n$-dimensional manifolds in ${\mathbb{R}}^{n+1}$. In contrast, David,
Jeznach and Julia proved in [DJJ23] that this last result may fail for $s$
close enough to $n-1$.

The situation is different for uniformly elliptic matrices. In [Swe92] in the
planar case and in [Swe94] in higher dimensions, for every $\varepsilon>0$,
Sweezy constructed a domain $\Omega\subset{\mathbb{R}}^{n+1}$ and an elliptic
operator in divergence form $L_{A}$ whose associated elliptic measure
$\omega_{\Omega,A}$ satisfies $\dim_{\mathcal{H}}\omega_{\Omega,A}\geq
n+1-\varepsilon$. However, as $\varepsilon$ becomes smaller, the ellipticity
constant of the resulting matrix increases. Such planar domains and elliptic
measures are constructed by the push forward under quasiconformal mappings,
and the higher dimensional analog is deduced from the planar case.

Using a new approach in the planar case, David and Mayboroda in [DM21]
constructed an elliptic operator $L=-\operatorname{div}a\nabla$, where $a$ is
a uniformly elliptic and continous scalar function on the complementary of the
four corner Cantor set of dimension $1$, whose elliptic measure $\omega_{aId}$
is proportional to the one-dimensional Hausdorff measure on the Cantor set.
Operators of this form are the so-called “good elliptic operators”. Following
the same strategy, Perstneva in [Per23] constructed “good elliptic operators”
on the complementary of the planar $d$-dimensional Koch-type snowflake with
$1<d<\log 4/\log 3$, whose elliptic measure is proportional to the
$d$-dimension Hausdorff measure on the Koch snowflake.

After Sweezy’s results, it is natural to ask which conditions on the matrix
imply the analogous result of [JW88, Wol93] for the elliptic case. In this
paper we study the metric properties of the elliptic measure when assuming
regularity conditions on the matrix $A$. For other results in this line, see
for example [PPT21], about the rectifiability of the elliptic measure for
Hölder matrices.

Suppose from now on that the domain is $(\delta,r_{0})$-Reifenberg flat and
the coefficients of the matrix are also Lipschitz. Roughly speaking, a domain
$\Omega\subset{\mathbb{R}}^{2}$ is $(\delta,r_{0})$-Reifenberg flat if for
every ball $B$ centered at $\partial\Omega$ and with radius smaller than
$r_{0}$, the $\delta$-neighborhood of a line through the center of $B$
contains $B\cap\partial\Omega$. See Definition 2.7 for the details. In this
situation we show that the dimension of the elliptic measure in Reifenberg
flat domains with small constant is at most $1$. More precisely, for this type
of domains we obtain the analogous result of [Wol93, Theorem 1] in the
following theorem.

######  Theorem 1.1.

Let $\Omega\subset{\mathbb{R}}^{2}$ be a $(\delta,r_{0})$-Reifenberg flat
domain, $p\in\Omega$, and $A$ be a real uniformly elliptic (not necessarily
symmetric) matrix with ellipticity constant $\lambda$. Suppose also that $A$
is Lipschitz. Then there exists $\delta_{0}=\delta_{0}(\lambda)>0$ such that
if $0<\delta\leq\delta_{0}$ then there is a set $F\subset\partial\Omega$
satisfying $\omega_{\Omega,A}(F)=1$ and with $\sigma$-finite one-dimensional
Hausdorff measure. In particular $\dim_{\mathcal{H}}\omega_{\Omega,A}\leq 1$.

Despite we are requiring to work with $\delta$-Reifenberg flat domains with
small enough constant $\delta$, such sets can be constructed with Hausdorff
dimension strictly larger than 1. For example, a suitable variant of the Koch
snowflake can be constructed to be $\delta$-Reifenberg flat.

It is well-known that elliptic measures only depend on how the matrix is near
the boundary and hence it is only necessary to assume the Lipschitz regularity
around the boundary. In fact, in our proof we use the regularity only around
the boundary, and so the theorem is still true assuming Lipschitz continuity
in a small neighbourhood of $\partial\Omega$.

Similarly as in [Wol93, Proof of Theorem 1], Theorem 1.1 follows from a more
quantitative result involving a good covering of a subset of the boundary with
big elliptic measure, see 3.1 for the precise statement.

One might think that this result could be obtained by the application of
quasiconformal mappings. For symmetric matrices with determinant 1, the
principal solution of the associated Beltrami equation, which depends only on
the matrix, can act as a change of variables which inherits the extra
regularity of the coefficients. Then, we obtain that the elliptic measure is
the pushforward of the harmonic measure in the image domain. In the general
case, we can obtain the elliptic measure as a pushforward of a harmonic
measure using a quasiconformal change of variables which depends also on the
Green function of the domain, as long as it satisfies the capacity density
condition. See the forthcoming article [GM] by the first author. In this case,
the extra dependence of the Green function does not allow us to obtain extra
regularity estimates for the change of variables, and the $\sigma$-finiteness
of length can not be attained.

A key point, and the main difficulty in this paper is to obtain the lower
bound

$\int_{\partial\widetilde{\Omega}}\log\frac{d\widetilde{\omega}^{p}}{d\sigma}(\xi)\,d\widetilde{\omega}^{p}(\xi)\geq-\textrm{const}>-\infty,$

with a bound independent of the smoothness, where $\widetilde{\Omega}$ is the
modified domain appearing in the proof of Lemma 3.2 and
$\widetilde{\omega}^{p}$ the elliptic measure in $\widetilde{\Omega}$ with
respect to the matrix $A$. This lower bound is known in the harmonic case for
general smooth domains in the plane. This fact is the key point in the study
of the dimension in the planar case, see for example [Car85], [JW88], [Wol93].
Actually, the behavior of this integral is also crucial in the study of the
dimension of harmonic measures in higher dimensions in [Wol95].

The first occurrence (as far as we know) of the use of this lower bound in the
study of the dimension of the harmonic measure is in [Car85]. For the proof
see [JW88], and for further details see [CTV19].

The previous lower bound is used in the proof of Lemma 3.2 to obtain a subset
with big elliptic measure as it is done in [JW88] and [Wol93], and it can be
deduced from

(1.2) $\int_{\partial\widetilde{\Omega}}\log|S\nabla
g_{p}^{T}(\xi)|^{2}\,d\widetilde{\omega}^{p}(\xi)\geq-\textrm{const}>-\infty,$

where $g_{p}^{T}$ is the Green function in $\widetilde{\Omega}$ with respect
to the matrix $A^{T}$, and $S$ is the square root matrix of the symmetric part
$A_{0}=(A+A^{T})/2$, i.e., $S^{T}S=A_{0}$. In Section 7 we obtain also an
upper bound for the integral above, see Lemma 7.1.

A fundamental tool to obtain the estimate (1.2) is the relation

$|\nabla g_{p}^{T}(y)|\gtrsim\frac{g_{p}^{T}(y)}{{\rm
dist}(y,\partial\widetilde{\Omega})}$

near the boundary. In Reifenberg flat domains this estimate was obtained by
Lewis, Lundström and Nyström in [LLN08], see Lemma 2.13 below. The converse
inequality is well-known for general domains and Hölder matrices and follows
by Schauder estimates. Obtaining an inequality of this type for other domains
may allow to apply the techniques exposed in this paper.

Very similar results to the one in Theorem 1.1 about the dimension are true
for $p$-harmonic measures, i.e., when the associated operator is the
$p$-Laplacian $\operatorname{div}\left(|\nabla u|^{p-2}\nabla u\right)$ for
Reifenberg flat sets with small constant (see [LNV13]), and simply connected
sets in the plane (see [LNPC11]).

##  2\. Preliminaries and definitions

###  2.1. Notation

  * •

We use $c,C\geq 1$ to denote constants that may depend only on the dimension
and the constants appearing in the hypotheses of the results, and whose values
may change at each occurrence.

  * •

We write $a\lesssim b$ if there exists a constant $C\geq 1$ such that $a\leq
Cb$, and $a\approx b$ if $C^{-1}b\leq a\leq Cb$.

  * •

If we want to stress the dependence of the constant on a parameter $\eta$, we
write $a\lesssim_{\eta}b$ or $a\approx_{\eta}b$ meaning that
$C=C(\eta)=C_{\eta}$.

  * •

The ambient space is ${\mathbb{R}}^{2}$. However, some auxiliary results will
be stated in general, i.e., in ${\mathbb{R}}^{n+1}$ for $n\geq 1$.

  * •

The diameter of a set $E\subset{\mathbb{R}}^{n+1}$ is denoted by ${\rm
diam}E\coloneqq\sup_{x,y\in E}|x-y|$.

  * •

We denote by $B_{r}(x)$ or $B(x,r)$ the open ball with center $x$ and radius
$r$, i.e., $B_{r}(x)=B(x,r)=\\{y\in{\mathbb{R}}^{n+1}:|y-x|<r\\}$. We denote
$B_{r}\coloneqq B_{r}(0)$.

  * •

Given a ball $B$, we denote by $r_{B}$ or $r(B)$ its radius, and by $c_{B}$ or
$c(B)$ its center.

  * •

We say that a matrix $A$ is Hölder continuous with exponent $\alpha\in(0,1]$
in a set $U$, or briefly $C^{0,\alpha}(U)$, if its coefficients are Hölder
continuous with exponent $\alpha$. That is, there exists a constant
$C_{\alpha}>0$ (called the Hölder seminorm) such that

$\left|a_{ij}(x)-a_{ij}(y)\right|\leq C_{\alpha}|x-y|^{\alpha}\quad\text{for
all }x,y\in U\text{ and }1\leq i,j\leq n+1.$

For shortness we write $C^{\alpha}$ instead of $C^{0,\alpha}$ if
$\alpha\in(0,1)$, and when $\alpha=1$ we say “Lipschitz continuous”. In this
case we write $C_{L}$ instead of $C_{1}$, i.e.,

$\left|a_{ij}(x)-a_{ij}(y)\right|\leq C_{L}|x-y|\quad\text{for all }x,y\in
U\text{ and }1\leq i,j\leq n+1.$

  * •

We say that a function $f$ is $\kappa$-Lipschitz in $U$ if
$|f(x)-f(y)|\leq\kappa|x-y|$ for all $x,y\in U$.

  * •

We denote the characteristic function of a set $E$ by $\mathbf{1}_{E}$.

  * •

Denote ${\mathcal{D}}({\mathbb{R}}^{n+1})$ the standard dyadic grid. That is,
${\mathcal{D}}({\mathbb{R}}^{n+1})=\bigcup_{k\in{\mathbb{Z}}}{\mathcal{D}}_{k}({\mathbb{R}}^{n+1})$
where ${\mathcal{D}}_{k}({\mathbb{R}}^{n+1})$ is the collection of all cubes
of the form

$\\{x\in{\mathbb{R}}^{n+1}:m_{i}2^{-k}\leq x_{i}<(m_{i}+1)2^{-k}\quad\text{for
}i=1,\ldots,n+1\\},$

where $m_{i}\in{\mathbb{Z}}$.

  * •

Given $t>0$ and a set $E\subset{\mathbb{R}}^{n+1}$, we write
$U_{t}(E)\coloneqq\\{x\in{\mathbb{R}}^{n+1}:{\rm dist}(x,E)<t\\}$ for the
$t$-neighborhood $E$.

###  2.2. CDC, NTA and Reifenberg flat domains

In this subsection we introduce the capacity density condition (CDC), non-
tangentially accessible domains (NTA) and Reifenberg flat domains, the main
object of our study.

######  Definition 2.1.

Let $K$ be a compact subset of $\Omega$. Its capacity is

$\operatorname{Cap}(K,\Omega)=\inf\left\\{\int_{\Omega}|\nabla u|^{2}\,dx:u\in
C^{\infty}_{0}(\Omega)\text{ and }u\geq 1\text{ on }K\right\\}.$

Just from the definition of capacity we obtain the following facts. For more
properties see [HKM06, 2.2. Theorem].

######  Lemma 2.2.

The set function $E\mapsto\operatorname{Cap}(E,\Omega)$, $E\subset\Omega$,
enjoys the following properties:

  1. (1)

If $E_{1}\subset E_{2}$, then
$\operatorname{Cap}(E_{1},\Omega)\leq\operatorname{Cap}(E_{2},\Omega)$.

  2. (2)

If $\Omega_{1}\subset\Omega_{2}$ are open and $E\subset\Omega_{1}$, then
$\operatorname{Cap}(E,\Omega_{2})\leq\operatorname{Cap}(E,\Omega_{1})$.

######  Definition 2.3 (CDC domain. [HKM06, (11.20), (2.13)]).

A domain $\Omega\subset{\mathbb{R}}^{n+1}$, $n\geq 1$, satisfies the capacity
density condition, CDC for short, if there exist constants $c_{0},r_{0}>0$
such that

$\operatorname{Cap}\left(\overline{B(x_{0},r)}\cap\Omega^{c},B(x_{0},2r)\right)\geq
c_{0}r^{n-1},$

for all $x_{0}\in\partial\Omega$, $x_{0}\not=\infty$ and $r\leq r_{0}$.

In order to define NTA domains we need to introduce some concepts. Given a
domain $\Omega\subset{\mathbb{R}}^{n+1}$, $n\geq 1$, and a fixed constant $C$,
we define:

  * •

$C$-Whitney ball: A ball $B(x,r)\subset\Omega$ is a $C$-Whitney ball in
$\Omega$ if $C^{-1}r<{\rm dist}(B(x,r),\partial\Omega)<Cr$.

  * •

$C$-Harnack chain: For $p_{1},p_{2}\in\Omega$, a $C$-Harnack chain from
$p_{1}$ to $p_{2}$ in $\Omega$ is a sequence of $C$-Whitney balls such that
the first ball contains $p_{1}$, the last contains $p_{2}$, and such that
consecutive balls intersect. The number of balls is called the length of the
$C$-Harnack chain.

Consecutive balls in a $C$-Harnack chain must have comparable radius. Given a
positive $L_{A}$-harmonic function in $\Omega$, a $C$-Harnack chain between
two points $p_{1},p_{2}\in\Omega$ allows us (via Harnack’s inequality) to
obtain $u(p_{1})\approx u(p_{2})$, where the constant involved depends on $C$
and the length of the $C$-Harnack chain.

######  Definition 2.4 (NTA domain. [JK82, Section 3]).

A domain $\Omega\subset{\mathbb{R}}^{n+1}$, $n\geq 1$, is called non-
tangentially accessible, NTA for short, if there exists constants $C>1$ and
$r_{0}>0$ such that:

  1. (1)

Interior Corkscrew condition: For any $\xi\in\partial\Omega$, $0<r<r_{0}$,
there exists a point $A_{r}(\xi)\in\Omega$ such that $|A_{r}(\xi)-\xi|<r$ and
${\rm dist}(A_{r}(\xi),\partial\Omega)>C^{-1}r$. The point
$A_{r}(\xi)=A(\xi,r)$ is called the Corkscrew point of the point $x$ at radius
$r$.

  2. (2)

Exterior Corkscrew condition: $\overline{\Omega}^{c}$ satisfies the interior
Corkscrew condition.

  3. (3)

Harnack chain condition: If $\varepsilon>0$ and $p_{1},p_{2}\in\Omega$ satisfy
that $p_{1},p_{2}\in\Omega\cap B(\xi,r_{0}/4)$ for some
$\xi\in\partial\Omega$, ${\rm dist}(p_{j},\partial\Omega)>\varepsilon$, and
$|p_{1}-p_{2}|<2^{k}\varepsilon$, then there exists a Harnack chain from
$p_{1}$ to $p_{2}$ of length $Ck$ and such that the diameter of each ball is
bounded below by $C^{-1}\min_{j=1,2}{\rm dist}(p_{j},\partial\Omega)$.

######  Remark 2.5.

A domain with the exterior Corkscrew condition satisfies the capacity density
condition, see [HKM06, Theorem 6.31]. In particular, NTA domains satisfy the
capacity density condition.

######  Definition 2.6 (Hausdorff distance).

Given sets $A$ and $B$, we denote their Hausdorff distance ${\rm
dist}_{\mathcal{H}}(A,B)$ by

${\rm dist}_{\mathcal{H}}(A,B)=\max\left\\{\sup_{x\in A}{\rm
dist}(x,B),\sup_{y\in B}{\rm dist}(y,A)\right\\}.$

######  Definition 2.7 (Reifenberg flat domain).

Let $\Omega\subset{\mathbb{R}}^{n+1}$, $n\geq 1$, be an open set, and let
$0<\delta<1/2$ and $r_{0}>0$. We say that $\Omega$ is a
$(\delta,r_{0})$-Reifenberg flat domain if it satisfies the following
conditions:

  1. (1)

For every $x\in\partial\Omega$ and every $0<r\leq r_{0}$, there exists a
hyperplane $\mathcal{P}(x,r)$ containing $x$ such that

${\rm dist}_{\mathcal{H}}\left(\partial\Omega\cap B(x,r),\mathcal{P}(x,r)\cap
B(x,r)\right)\leq\delta r.$

  2. (2)

For every $x\in\partial\Omega$ and every $0<r\leq r_{0}$, one of the connected
components of

$B(x,r)\cap\left\\{y\in{\mathbb{R}}^{n+1}:{\rm dist}(y,\mathcal{P}(x,r))\geq
2\delta r\right\\}$

is contained in $\Omega$ and the other is contained in
${\mathbb{R}}^{n+1}\setminus\Omega$.

For small enough $\delta>0$ we have that a $(\delta,r_{0})$-Reifenberg flat
domain is also an NTA domain (see [KT97, Section 3]) and it satisfies the
capacity density condition, see Remark 2.5.

###  2.3. Partial differential equations

We want to study the elliptic equation

(2.1) $L_{A}u(x)\coloneqq-\operatorname{div}(A(\cdot)\nabla u(\cdot))(x)=0,$

which should be understood in the distributional sense. We simply write $L$
instead of $L_{A}$ when the matrix is clear from the context.

######  Definition 2.8.

We say that a function $u\in W_{\operatorname{loc}}^{1,2}(\Omega)$ is a
solution of 2.1, or $L_{A}$-harmonic, in an open set
$\Omega\subset{\mathbb{R}}^{n+1}$, $n\geq 1$, if

$\int\langle A(y)\nabla u(y),\nabla\varphi(y)\rangle\,dy=0,\quad\text{for all
}\varphi\in C_{c}^{\infty}(\Omega).$

By the De Giorgi-Nash-Moser theorem a solution $u\in
W_{\operatorname{loc}}^{1,2}(\Omega)$ of (2.1) is locally Hölder continuous.
If the matrix has Hölder regularity then the solution is locally $C^{1,\beta}$
for some $\beta\in(0,1)$, see [HL11, Theorem 3.13]. Assuming Lipschitz
regularity of the coefficients, $L_{A}$-harmonic functions enjoy more
regularity. More precisely, weak solutions of $-\operatorname{div}A\nabla u=0$
with $A$ Lipschitz are twice weakly differentiable, and there is a
“Caccioppoli type” inequality for second derivatives.

######  Theorem 2.9 (See [GT01, Theorem 8.8]).

Let $u\in W^{1,2}(\Omega)$ be a weak solution of the equation $L_{A}u=0$, see
(2.1), in $\Omega\subset{\mathbb{R}}^{n+1}$, $n\geq 1$, where $A$ is uniformly
elliptic and Lipschitz continuous in $\Omega$. Then for any subdomain
$\Omega^{\prime}\subset\subset\Omega$, we have $u\in W^{2,2}(\Omega^{\prime})$
and

(2.2) $\|u\|_{W^{2,2}(\Omega^{\prime})}\leq C\|u\|_{W^{1,2}(\Omega)},$

where $C$ depends on the dimension, the ellipticity constant and Lipschitz
constant of $A$, and the value ${\rm dist}(\Omega^{\prime},\partial\Omega)$.

If, in addition, the domain has boundary of class $C^{2}$, then
$L_{A}$-harmonic functions are globally in $W^{2,2}$.

######  Theorem 2.10 (See [GT01, Theorem 8.12]).

Assume, in addition to the hypotheses of Theorem 2.9, that $\Omega$ is bounded
with $\partial\Omega$ of class $C^{2}$ and that there exists a function
$\varphi\in W^{2,2}(\Omega)$ for which $u-\varphi\in W^{1,2}_{0}(\Omega)$.
Then we have also $u\in W^{2,2}(\Omega)$ and

$\|u\|_{W^{2,2}(\Omega)}\leq
C\left(\|u\|_{L^{2}(\Omega)}+\|\varphi\|_{W^{2,2}(\Omega)}\right),$

where $C$ depends on the dimension, the ellipticity constant and Lipschitz
constant of $A$, and $\partial\Omega$.

######  Remark 2.11.

Let $U\subset\Omega$ an open subset. Assuming $A\in C^{0,1}(U)$, then any weak
$L_{A}$-harmonic function $u\in W^{1,2}_{\operatorname{loc}}(\Omega)$ (in fact
$u\in W^{2,2}_{\operatorname{loc}}(U)$ by Theorem 2.9) satisfies
$\operatorname{div}A\nabla u=0$ a.e. in $U$. Indeed, since the matrix $A$ is
differentiable a.e. (by Rademacher’s theorem) and $u\in
W^{2,2}_{\operatorname{loc}}(U)$ (by Theorem 2.9), we have
$\operatorname{div}A\nabla u\in L^{1}_{\operatorname{loc}}(U)$. Moreover,
since $u$ is $L_{A}$-harmonic then $\int\operatorname{div}A\nabla
u\cdot\psi=\int A\nabla u\nabla\psi=0$ for any $\psi\in
C^{\infty}_{c}(U)\subset C^{\infty}_{c}(\Omega)$. By the fundamental lemma of
calculus of variations111The fundamental lemma of calculus of variations: If
$U$ open set, $f\in L^{1}_{\operatorname{loc}}(U)$ and $\int f\phi=0$ for any
$\phi\in C^{\infty}_{c}(U)$, then $f=0$ a.e. in $U$. we conclude
$\operatorname{div}A\nabla u=0$ a.e. in $U$.

The following theorem about the Hölder continuity of $L_{A}$-harmonic
functions up to the boundary of regular enough domains will allow us to bound
the elliptic measure on a specific domain (an annulus) by means of studying
the Green function near the boundary.

######  Theorem 2.12 (See [GT01, Corollary 8.36]).

Let $T$ be a (possibly empty) $C^{1,\alpha}$ boundary portion of a bounded
domain $\Omega\subset{\mathbb{R}}^{n+1}$, $n\geq 1$, and suppose $u\in
W^{1,2}(\Omega)$ is a weak solution of (2.1) in $\Omega$, where $A\in
C^{\alpha}$ $(0<\alpha<1)$, such that $u=0$ on $T$ (in the sense of
$W^{1,2}(\Omega)$). Then $u\in C^{1,\alpha}(\Omega\cup T)$, and for any
$\Omega^{\prime}\subset\subset\Omega\cup T$ we have

$\|u\|_{C^{1,\alpha}(\Omega^{\prime})}\coloneqq\|u\|_{1;\Omega^{\prime}}+[u]_{1,\alpha;\Omega^{\prime}}\leq
C\sup_{\Omega}|u|,$

where

$\|u\|_{1;\Omega^{\prime}}\coloneqq\sup_{\Omega^{\prime}}|u|+\sup_{\Omega^{\prime}}|\nabla
u|,\quad[u]_{1,\alpha;\Omega^{\prime}}\coloneqq\sup_{\begin{subarray}{c}x,y\in\Omega^{\prime}\\\
x\not=y\end{subarray}}\frac{|\nabla u(x)-\nabla u(y)|}{|x-y|^{\alpha}},$

for $C$ depending on $n$, the value of ${\rm
dist}(\Omega^{\prime},\partial\Omega\setminus T)$, the set $T$, and the
ellipticity constants and the Hölder norm of the matrix $A$.

Since bounded Lipschitz functions are also Hölder continuous for any exponent
$\alpha\in(0,1)$, the previous theorem remains true for matrices with bounded
Lipschitz coefficients.

####  2.3.1. Non-degeneracy of $|\nabla u|$ in Reifenberg flat domains with
small constant

Despite the following result applies to more general matrices (see [LLN08,
Definition 1.1 and Lemma 3.35]), it is only stated in our setting for our
purposes, i.e., for uniformly elliptic 1.1a-1.1b and Hölder continuous
matrices. In fact, the Hölder continuity is only used near the boundary.

######  Lemma 2.13 ([LLN08, Lemma 3.35]).

Let $\Omega\subset{\mathbb{R}}^{n+1}$, $n\geq 1$, be a
$(\delta,r_{0})$-Reifenberg flat domain, $\xi\in\partial\Omega$, and
$0<r<\min\\{r_{0},1\\}$. Let $0<\alpha<1$ and $A\in C^{\alpha}(\\{x:{\rm
dist}(x,\partial\Omega)<10r_{0}\\})$ be a real uniformly elliptic (not
necessarily symmetric) matrix with ellipticity constant $\lambda$ and Hölder
seminorm $C_{\alpha}$. Suppose that $u$ is a positive $L_{A}$-harmonic
function in $\Omega\cap B(\xi,4r)$, that is continuous in
$\overline{\Omega}\cap B(\xi,4r)$, and that $u=0$ on $\partial\Omega\cap
B(\xi,4r)$. There exist
$\hat{\delta}=\hat{\delta}(n,\lambda,C_{\alpha},\alpha)$,
$\gamma=\gamma(n,\lambda,C_{\alpha},\alpha)$ and
$\hat{c}=\hat{c}(n,\lambda,C_{\alpha},\alpha)$ such that if
$0<\delta\leq\hat{\delta}$, then

$\gamma^{-1}\frac{u(y)}{{\rm dist}(y,\partial\Omega)}\leq\left|\nabla
u(y)\right|\leq\gamma\frac{u(y)}{{\rm
dist}(y,\partial\Omega)}\quad\text{whenever }y\in\Omega\cap B(\xi,r/\hat{c}).$

######  Remark 2.14.

Note that if the matrix has bounded Lipschitz coefficients, i.e., $\alpha=1$,
then the same result holds with constants depending on $n$, the ellipticity
constant $\lambda$ and the value
$C_{L}\|A\|_{L^{\infty}({\mathbb{R}}^{n+1})}$, where $C_{L}$ is the Lipschitz
seminorm of $A$. The value $C_{L}\|A\|_{L^{\infty}({\mathbb{R}}^{n+1})}$ comes
from the fact that bounded Lipschitz functions are Hölder continuous for any
exponent. Indeed, a quick computation shows that the matrix $A$ is Hölder
continuous with exponent $1/2$ with Hölder seminorm
$C_{1/2}\coloneqq\left(2C_{L}\|A\|_{L^{\infty}({\mathbb{R}}^{n+1})}\right)^{1/2}$.

The comparability in Lemma 2.13 will allow to bound the error terms in the
study of the key term in (7.1).

###  2.4. The fundamental solution and the Green function

We denote by ${\mathcal{E}}_{x}^{A}(y)$ the fundamental solution with pole at
$x$ for $L_{A}$ in ${\mathbb{R}}^{n+1}$, $n\geq 1$, so that
$L_{A}{\mathcal{E}}_{x}^{A}(\cdot)=\delta_{x}$ in the distributional sense,
where $\delta_{x}$ is the Dirac mass at the point $x\in{\mathbb{R}}^{n+1}$. We
write ${\mathcal{E}}_{x}(y)$ when the matrix $A$ is clear from the context.
For a construction of the fundamental solution for real and uniformly elliptic
matrices we refer to [HK07] for higher dimensions, ${\mathbb{R}}^{n+1}$ with
$n\geq 2$, and [KN85, Appendix] for the planar case.

In higher dimensions the fundamental solution behaves “in many senses” as in
the harmonic case, see [HK07] for more details, but in the plane the situation
is more delicate due to the change of sign of
${\mathcal{E}}_{0}^{Id}(x)=\log|x|$ in $|x|=1$, the fundamental solution of
the Laplacian in the plane.

Here we collect the formal definition and some properties of the fundamental
solution in the plane.

######  Definition 2.15 ([KR09, Definition 2.5]).

A function ${\mathcal{E}}_{x}:{\mathbb{R}}^{2}\to{\mathbb{R}}$ is called a
fundamental solution for $L_{A}=\operatorname{div}A\nabla\cdot$ with pole at
$x$ if

  1. (1)

${\mathcal{E}}_{x}\in
W_{\operatorname{loc}}^{1,2}({\mathbb{R}}^{2}\setminus\\{x\\})\cap
W_{\operatorname{loc}}^{1,p}({\mathbb{R}}^{2})$ for all $p<2$, and

$\int_{{\mathbb{R}}^{2}}\langle
A(z)\nabla{\mathcal{E}}_{x}(z),\nabla\varphi(z)\rangle\,dz=-\varphi(x),\quad\text{for
all }\varphi\in C_{c}^{\infty}({\mathbb{R}}^{2}),$

  2. (2)

$\left|{\mathcal{E}}_{x}(y)\right|={\mathcal{O}}(\log|x-y|)\text{ as
}|y|\rightarrow\infty$.

The following result controls the fundamental solution far from the pole
similarly as the fundamental solution for the harmonic case.

######  Theorem 2.16 ([KR09, Theorem 2.6]).

For each $x\in{\mathbb{R}}^{2}$ there exists a unique (modulo an additive
constant) fundamental solution ${\mathcal{E}}_{x}$ for $L_{A}$ with pole at
$x$, and positive constants $C_{1}$, $C_{2}$, $R_{1}<1<R_{2}$, which depend
only on $\lambda$, such that

$\begin{aligned} C_{1}\log(1/|x-y|)&\leq{}\\\
C_{1}\log(|x-y|)&\leq{}\end{aligned}\\!\begin{gathered}-{\mathcal{E}}_{x}(y)\\\
{\mathcal{E}}_{x}(y)\end{gathered}\\!\begin{aligned} {}&\leq
C_{2}\log(1/|x-y|)\\\ {}&\leq C_{2}\log(|x-y|)\end{aligned}\begin{aligned}
{}&\quad\text{for }|x-y|<R_{1},\text{ and}\\\ {}&\quad\text{for
}|x-y|>R_{2}.\end{aligned}$

From the previous result and the maximum principle we obtain the following
pointwise bound.

######  Corollary 2.17.

$|{\mathcal{E}}_{x}(y)|\lesssim 1+\left|\log|x-y|\right|$ for all
$x,y\in{\mathbb{R}}^{2}$, where the constant depends on $\lambda$.

###### Proof.

For $|x-y|<R_{1}$ and $|x-y|>R_{2}$, Theorem 2.16 gives
$|{\mathcal{E}}_{x}(y)|\lesssim\left|\log|x-y|\right|\leq
1+\left|\log|x-y|\right|$. In
$\mathcal{A}=\\{y\in{\mathbb{R}}^{2}:R_{1}<|x-y|<R_{2}\\}$ we have
$L{\mathcal{E}}_{x}(y)=0$, and hence by the maximum principle we obtain
$C_{2}\log R_{1}\leq{\mathcal{E}}_{x}(\cdot)\leq C_{2}\log R_{2}$ in the
annulus $\mathcal{A}$. So $|{\mathcal{E}}_{x}(y)|\lesssim 1\leq
1+\left|\log|x-y|\right|$. ∎

We have the following relation between the fundamental solutions of the
operators with matrices $A$ and $A^{T}$. The same holds in higher dimensions,
see [HK07, (3.43)].

######  Lemma 2.18 ([KR09, Lemma 2.7]).

Fix $x,y\in{\mathbb{R}}^{2}$. Let ${\mathcal{E}}_{x}$ be the fundamental
solution for an elliptic operator $L_{A}$ with pole at $x$, and
${\mathcal{E}}_{y}^{T}$ be the fundamental solution to the adjoint operator
$L_{A^{T}}$ with pole at $y$. Then
${\mathcal{E}}_{x}(y)={\mathcal{E}}_{y}^{T}(x)$.

Now we focus on Green’s function. Given a bounded Wiener regular domain
$\Omega\subset{\mathbb{R}}^{n+1}$, $n\geq 1$, and a uniformly elliptic matrix
$A$, let $g_{x}=g_{x}^{\Omega,A}$ denote the Green function, in $\Omega$ with
pole at $x\in\Omega$ with respect to the matrix $A$, constructed in [DK09,
Theorem 2.12] in the planar case, and [GW82, Theorem 1.1] in higher
dimensions. We denote $g_{x}^{T}=g_{x}^{\Omega,A^{T}}$ the Green function with
respect to the matrix $A^{T}$. In particular, $g_{x}$ satisfies

(2.3) $\int_{\Omega}A(z)\nabla
g_{x}(z)\nabla\varphi(z)\,dz=\varphi(x),\quad\text{for all }\varphi\in
C_{c}^{\infty}(\Omega),$

and the following:

  1. (1)

$g_{x}(y)=g_{y}^{T}(x)$ for all $x,y\in\Omega$ and $x\not=y$. See [DK09,
Theorem 2.12 (2.18)] in the plane, and [GW82, Theorem 1.3] in higher
dimensions.

  2. (2)

For each $x\in\Omega$ and any $0<r<{\rm dist}(x,\partial\Omega)$, $g_{x}\in
W^{1,2}(\Omega\setminus B_{r}(x))$. See [DK09, Theorem 2.12 (2.15)] in the
plane, and [GW82, Theorem 1.1 (1.3)] in higher dimensions.

If the domain $\Omega$ has boundary of class $C^{2}$ and the matrix is
Lipschitz continuous in a neighborhood
$U_{2s}(\partial\Omega)=\\{x\in{\mathbb{R}}^{n+1}:{\rm
dist}(x,\partial\Omega)<2s\\}$, then the Green function also satisfies:

  1. (3)

For each $x\in\Omega$ and any $0<r<\min\\{s,{\rm dist}(x,\partial\Omega)\\}$,
$g_{x}\in W^{2,2}(\Omega\cap U_{r}(\partial\Omega))$. This is a consequence of
(2) by Theorems 2.9 and 2.10.

Next we show that $L_{A^{T}}g^{T}_{x}=\omega_{\Omega,A}^{x}-\delta_{x}$ in the
distributional sense. With this identity we can move from integrating on the
boundary to the interior of the set.

######  Lemma 2.19.

Let $\Omega\subset{\mathbb{R}}^{n+1}$, $n\geq 1$, be a bounded Wiener regular
domain and $\varphi\in C(\overline{\Omega})\cap W^{1,2}(\Omega)$. Then

(2.4)
$\int_{\partial\Omega}\varphi(\xi)\,d\omega^{x}_{\Omega,A}(\xi)-\varphi(x)=-\int_{\Omega}A^{T}(z)\nabla
g_{x}^{T}(z)\nabla\varphi(z)\,dz,\quad\text{for a.e.\ }x\in\Omega.$

###### Sketch of proof.

In higher dimensions this is proved in [AGMT22, (2.6)]. Here we detail the
differences in the planar case.

As $\Omega$ is bounded Wiener regular and $\varphi\in C(\overline{\Omega})\cap
W^{1,2}(\Omega$), the $L_{A}$-harmonic function $u$ solving the Dirichlet
problem with boundary data $\varphi_{|\partial\Omega}$ can be taken to be in
$C(\overline{\Omega})\cap W^{1,2}(\Omega)$ and $u-\varphi\in
W^{1,2}_{0}(\Omega)$. Indeed, by the Lax-Milgram theorem in
$W_{0}^{1,2}(\Omega)$ there is a unique function $v\in W_{0}^{1,2}(\Omega)$
with $\int_{\Omega}A\nabla
v\nabla\vartheta=-\int_{\Omega}A\nabla\varphi\nabla\vartheta$ for any
$\vartheta\in W_{0}^{1,2}(\Omega)$. Taking $u=v+\varphi$ it is clear that
$L_{A}u=0$, $u\in W^{1,2}(\Omega)$ and $u-\varphi\in W^{1,2}_{0}(\Omega)$. On
the other hand, since $\Omega$ is bounded Wiener regular, the $L_{A}$-harmonic
extension $u$ of $\varphi_{|\partial\Omega}$ is continuous up to the boundary,
see [HKM06, Theorem 6.27]. Moreover, by the definition of elliptic measure and
the uniqueness of solutions we have

$u(x)=\int_{\partial\Omega}\varphi(\xi)\,d\omega^{x}_{\Omega,A}(\xi).$

Write

$\int_{\Omega}A^{T}\nabla g_{x}^{T}\nabla\varphi=\int_{\Omega}A^{T}\nabla
g_{x}^{T}\nabla u+\int_{\Omega}A^{T}\nabla
g_{x}^{T}\nabla(\varphi-u)\eqqcolon\textup{I}+\textup{II}.$

The same proof of [AGMT22, (2.10) and (2.12)] applies also in the plane to
have that the left-hand side integral is absolutely convergent and
$\textup{II}=\varphi(x)-u(x)$ for a.e. $x\in\Omega$, replacing the use of
(2.8) and (2.9) in [AGMT22] by the fact that the Green function satisfies
$\nabla g_{z}\in L^{p}(\Omega)$ for all $p\in[1,2)$ and $g_{z}\in
W_{\operatorname{loc}}^{1,2}(\Omega\setminus\\{z\\})$, see [DK09, Remark 2.19
and (3.66)] and item 2 in item 2 respectively. Using that
$|g_{x}^{T}(z)|\lesssim\left|\log|z-x|\right|$ when $|x-z|\leq 2\varepsilon$
is small enough, see [DK09, (2.17)], in the planar case the term
$\textup{I}_{\varepsilon}^{2}$ defined in [AGMT22, p. 10855] is controlled by

$|\textup{I}_{\varepsilon}^{2}|\lesssim\frac{\left|\log\varepsilon\right|}{\varepsilon}\int_{B_{2\varepsilon}(x)}|\nabla
u|\lesssim\varepsilon\left|\log\varepsilon\right|\mathcal{M}(\nabla
u\mathbf{1}_{\Omega})(x),$

and as $\varepsilon\left|\log\varepsilon\right|\to 0$ as $\varepsilon\to 0$,
the same proof there implies that $\textup{I}=0$ for a.e. $x\in\Omega$.
Therefore, 2.4 holds also in the planar case. ∎

We will show below that from the equality (2.4) it follows that

(2.5)
$g_{y}(x)=-{\mathcal{E}}_{y}(x)+\int_{\partial\Omega}{\mathcal{E}}_{y}(\xi)\,d\omega_{\Omega,A}^{x}(\xi),\quad\text{for
all }x,y\in\Omega.$

(Probably this is already known but we will show the full details in the plane
for completeness). Recall that
$x\mapsto\int_{\partial\Omega}{\mathcal{E}}_{y}(\xi)\,d\omega_{\Omega,A}^{x}(\xi)$
is the $L_{A}$-harmonic extension of ${\mathcal{E}}_{y}$ inside $\Omega$.
Assuming that $g_{y}(x)=0$ if $y\not\in\Omega$, we have that (2.5) also holds
in this case, since $\Omega$ is Wiener regular and therefore the Green
function is continuous through the boundary. Moreover, by (2.5) and since
$\Omega$ is bounded, the Green function also satisfies:

  1. (4)

For each $x\in\Omega$, $g_{x}(y)\geq 0$ for any $y\in\Omega\setminus\\{x\\}$.

This was already proved in higher dimension in [GW82, Theorem 1.1]. However,
since the situation is more delicate for unbounded planar domains due to the
logarithmic behaviour of the fundamental solution, we only consider bounded
planar Wiener regular domains.

###### Proof of (2.5) in the planar case.

Let $0<\varepsilon\ll\min\\{|x-y|,{\rm dist}(x,\partial\Omega),{\rm
dist}(y,\partial\Omega)\\}$, $\psi^{y}=\psi^{y}_{\varepsilon}\in
C^{\infty}_{c}(B_{2\varepsilon}(y))$ such that $\psi^{y}_{\varepsilon}=1$ in
$B_{\varepsilon}(y)$ and $|\nabla\phi^{y}_{\varepsilon}|\lesssim
1/\varepsilon$, and $\psi^{x}=\psi^{x}_{\varepsilon}$ defined analogously.

Applying (2.4) to $(1-\psi^{y}){\mathcal{E}}_{y}\in C(\overline{\Omega})\cap
W^{1,2}(\Omega)$ and using that $1-\psi^{y}=1$ in $\partial\Omega\cup\\{x\\}$,
we have

$\int_{\partial\Omega}{\mathcal{E}}_{y}(\xi)\,d\omega_{\Omega,A}^{x}(\xi)-{\mathcal{E}}_{y}(x)=-\int_{\Omega}A^{T}(z)\nabla
g_{x}^{T}(z)\nabla((1-\psi^{y}){\mathcal{E}}_{y})(z)\,dz.$

Write the right-hand side term as

$\displaystyle-\int_{\Omega}A^{T}\nabla
g_{x}^{T}\nabla((1-\psi^{y}){\mathcal{E}}_{y})\,dz=$
$\displaystyle\int_{B_{2\varepsilon}(y)\setminus
B_{\varepsilon}(y)}A^{T}\nabla
g_{x}^{T}\nabla\psi^{y}\cdot{\mathcal{E}}_{y}\,dz+\int_{B_{2\varepsilon}(y)}A\nabla{\mathcal{E}}_{y}\nabla
g_{x}^{T}\cdot\psi^{y}\,dz$ $\displaystyle-\int_{B_{2\varepsilon}(x)\setminus
B_{\varepsilon}(x)}A\nabla{\mathcal{E}}_{y}\nabla\psi^{x}\cdot
g_{x}^{T}\,dz-\int_{\Omega}A\nabla{\mathcal{E}}_{y}\nabla((1-\psi^{x})g_{x}^{T})\,dz$
$\displaystyle\eqqcolon$
$\displaystyle\,\textup{I}_{\varepsilon}+\textup{II}_{\varepsilon}+\textup{III}_{\varepsilon}+\textup{IV}_{\varepsilon}.$

Using that $|g_{x}^{T}(z)|\lesssim\left|\log|x-z|\right|$ and
$|{\mathcal{E}}_{y}(z)|\lesssim\left|\log|y-z|\right|$ when $|x-z|\leq
2\varepsilon$ and $|y-z|\leq 2\varepsilon$ for small enough $\varepsilon>0$,
see [DK09, (2.17)] and Theorem 2.16, the terms $\textup{I}_{\varepsilon}$ and
$\textup{III}_{\varepsilon}$ are bounded by

(2.6)
$|\textup{I}_{\varepsilon}|+|\textup{III}_{\varepsilon}|\lesssim\varepsilon\left|\log\varepsilon\right|\left\\{\left(\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{B_{2\varepsilon}(y)}|\nabla
g_{x}^{T}|^{2}\,dz\right)^{1/2}+\left(\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{B_{2\varepsilon}(x)}|\nabla{\mathcal{E}}_{y}|^{2}\,dz\right)^{1/2}\right\\}.$

For the bound of $\textup{II}_{\varepsilon}$, since $g_{x}^{T}$ is
$L_{A^{T}}$-harmonic in $\Omega\setminus\\{x\\}\supset B_{10\varepsilon}(y)$,
there exists $p=p(\lambda)>2$ such that

$\left(\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{B_{2\varepsilon}(y)}|\nabla
g_{x}^{T}|^{p}\,dz\right)^{1/p}\lesssim_{\lambda}\left(\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{B_{4\varepsilon}(y)}|\nabla
g_{x}^{T}|^{2}\,dz\right)^{1/2},$

see [Ken94, Lemma 1.1.12], and let $1\leq q<2$ be its Hölder exponent
conjugate, i.e., $1/p+1/q=1$. By Hölder’s inequality and the choice of $p>2$,
the term $\textup{II}_{\varepsilon}$ is controlled by

$|\textup{II}_{\varepsilon}|=\left|\int_{B_{2\varepsilon}(y)}A\nabla{\mathcal{E}}_{y}\nabla
g_{x}^{T}\cdot\psi^{y}\,dz\right|\lesssim\varepsilon^{2}\left(\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{B_{2\varepsilon}(y)}|\nabla{\mathcal{E}}_{y}|^{q}\,dz\right)^{1/q}\left(\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{B_{4\varepsilon}(y)}|\nabla
g_{x}^{T}|^{2}\,dz\right)^{1/2}.$

By [CL92, Theorem 0.1] we have $\left(\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{B_{2\varepsilon}(y)}|\nabla{\mathcal{E}}_{y}|^{q}\,dz\right)^{1/q}\lesssim_{\lambda}1/\varepsilon$,
and so

(2.7)
$|\textup{II}_{\varepsilon}|\lesssim\varepsilon\left(\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{B_{4\varepsilon}(x)}|\nabla
g_{x}^{T}|^{2}\,dz\right)^{1/2}.$

Since $g_{x}^{T}\in W_{\operatorname{loc}}^{1,2}(\Omega\setminus\\{x\\})$ and
${\mathcal{E}}_{y}\in W_{\operatorname{loc}}^{1,2}(\Omega\setminus\\{y\\})$,
in particular $|\nabla g_{x}^{T}|^{2}\in
L_{\operatorname{loc}}^{1}(\Omega\setminus\\{x\\})$ and
$|\nabla{\mathcal{E}}_{y}|^{2}\in
L_{\operatorname{loc}}^{1}(\Omega\setminus\\{y\\})$, by the Lebesgue
differentiation theorem we have that $\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{B_{4\varepsilon}(y)}|\nabla
g_{x}^{T}|^{2}\to|\nabla g_{x}^{T}(y)|^{2}$ for a.e. $y\in\Omega$ and
$\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{B_{4\varepsilon}(x)}|\nabla{\mathcal{E}}_{y}|^{2}\to|\nabla{\mathcal{E}}_{y}(x)|^{2}$
for a.e. $x\in\Omega$ respectively. That is, by 2.6 and 2.7 we have
$|\textup{I}_{\varepsilon}|+|\textup{II}_{\varepsilon}|+|\textup{III}_{\varepsilon}|=0$
a.e. $x,y\in\Omega$.

On the other hand, from the Dirac delta property of the fundamental solution,
$(1-\psi^{x})g_{x}^{T}\in W^{1,2}_{0}(\Omega)$ and the density of
$C^{\infty}_{c}(\Omega)\subset C^{\infty}_{c}({\mathbb{R}}^{n+1})$ in
$W^{1,2}_{0}(\Omega)$, we obtain
$\textup{IV}_{\varepsilon}=(1-\psi^{x}(y))g_{x}^{T}(y)=g_{y}(x)$, and (2.5) is
proved for a.e. $x,y\in\Omega$. By continuity, it also holds for all
$x,y\in\Omega$. ∎

To end this section, we see how the Green function is related to the density
of the elliptic measure in smooth domains. Assume now $A$ is Lipschitz
continuous in an open neighborhood of $\partial\Omega$, say
$U_{s}(\partial\Omega)=\\{x\in{\mathbb{R}}^{n+1}:{\rm
dist}(x,\partial\Omega)<s\\}$. Under this assumption $A$ is differentiable
a.e. by Rademacher’s theorem, and $L_{A}$-harmonic functions are in $W^{2,2}$
by Theorem 2.10.

######  Lemma 2.20.

Let $\Omega\subset{\mathbb{R}}^{n+1}$, $n\geq 1$, be a bounded domain with
smooth boundary (and hence Wiener regular) and $A\in
C^{0,1}(U_{s}(\partial\Omega))$. The elliptic measure $\omega^{p}_{\Omega,A}$
can be written as

(2.8) $d\omega_{\Omega,A}^{p}=-\langle A^{T}\nabla
g_{p}^{T},\nu\rangle\,d\sigma,\quad\text{for a.e.\ }p\in\Omega\setminus
U_{2s}(\partial\Omega),$

where $\nu$ is the unit outer normal to $\partial\Omega$ and $\sigma$ is the
surface measure on $\partial\Omega$.

###### Proof.

Let $\varphi\in C^{\infty}_{c}(U_{s}(\partial\Omega))$ and set
$\phi_{p}(z)=g_{p}^{T}(z)+{\mathcal{E}}_{p}^{T}(z)$ for
$z\in{\mathbb{R}}^{n+1}$, see (2.5) when $z\in\Omega$. Notice that
$\phi_{p}(z)=-{\mathcal{E}}^{T}_{p}(z)$ in $\Omega^{c}$, and hence $\phi_{p}$
is $L_{A^{T}}$-harmonic in $\Omega$ and $\overline{\Omega}^{c}$.

The claim follows since the right-hand side of (2.4) is

$-\int_{\partial\Omega}\varphi(\xi)\langle A^{T}(\xi)\nabla
g_{p}^{T}(\xi),\nu(\xi)\rangle\,d\sigma(\xi)-\varphi(p).$

Indeed, since $A$ is differentiable a.e. in $\operatorname{supp}\varphi$
(Rademacher’s theorem), $g_{p}^{T}\in W^{2,2}(\operatorname{supp}\varphi)$
(Theorem 2.10) and $\varphi\in C^{\infty}_{c}(U_{s}(\partial\Omega))$, in
particular $A^{T}\nabla g_{p}^{T}\cdot\varphi\in
W^{1,2}(\operatorname{supp}\varphi)$. As the integration by parts formula
holds for $W^{1,2}$ functions, then

$\displaystyle-\int_{\Omega}A^{T}\nabla g_{p}^{T}\nabla\varphi=$
$\displaystyle\int_{\Omega}\varphi\operatorname{div}\left(A^{T}\nabla
g_{p}^{T}\right)-\int_{\Omega}\operatorname{div}\left(A^{T}\nabla
g_{p}^{T}\cdot\varphi\right)$ $\displaystyle=$
$\displaystyle\int_{\Omega}\varphi\operatorname{div}\left(A^{T}\nabla
g_{p}^{T}\right)-\int_{\partial\Omega}\varphi\langle A^{T}\nabla
g_{p}^{T},\nu\rangle\,d\sigma$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}^{n+1}}\varphi\operatorname{div}\left(A^{T}\nabla\phi_{p}\right)-\int_{{\mathbb{R}}^{n+1}}\varphi\operatorname{div}\left(A^{T}\nabla{\mathcal{E}}^{T}_{p}\right)-\int_{\partial\Omega}\varphi\langle
A^{T}\nabla g_{p}^{T},\nu\rangle\,d\sigma$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}^{n+1}}\varphi\operatorname{div}\left(A^{T}\nabla\phi_{p}\right)-\varphi(p)-\int_{\partial\Omega}\varphi\langle
A^{T}\nabla g_{p}^{T},\nu\rangle\,d\sigma,$

where in the last equality we used that $\varphi$ has compact support and
$\int\varphi\operatorname{div}\left(A^{T}\nabla{\mathcal{E}}_{p}^{T}\right)=\varphi(p)$
by the definition of the fundamental solution. Since $\phi_{p}$ is
$L_{A^{T}}$-harmonic in ${\mathbb{R}}^{n+1}\setminus\partial\Omega$,
${\mathcal{H}}^{n+1}(\partial\Omega)=0$ and $A\in
C^{0,1}(\operatorname{supp}\varphi)$, we have that
$\operatorname{div}\left(A^{T}\nabla\phi_{p}\right)=0$ a.e. in
$\operatorname{supp}\varphi$ by Remark 2.11, and the claim follows. ∎

##  3\. Main Lemma and preliminary reductions

As in [Wol93], Theorem 1.1 will follow from the following more quantitative
result.

######  Main Lemma 3.1.

Let $\Omega\subset{\mathbb{R}}^{2}$ be a bounded $(\delta,r_{0})$-Reifenberg
flat domain, a point $p\in\Omega$ with ${\rm dist}(p,\partial\Omega)>r_{0}$,
and $A$ be a real uniformly elliptic (not necessarily symmetric) matrix with
ellipticity constant $\lambda$, and suppose also that $A$ is
$\kappa$-Lipschitz in
$U_{r_{0}}(\partial\Omega)\coloneqq\\{x\in{\mathbb{R}}^{2}:{\rm
dist}(x,\partial\Omega)<r_{0}\\}$. For a given $0<r\leq 1$ satisfying
$r\kappa\|A\|_{L^{\infty}({\mathbb{R}}^{2})}\leq 1$, there exists
$\delta_{0}=\delta_{0}(\lambda)>0$ such that for every
$0<\delta\leq\delta_{0}$ we have the following:

For any $0<\tau<1$, sufficiently large $M$, and $\rho\in\left(0,r/M\right)$
there is a set $F\subset\partial\Omega$ such that
$\omega_{\Omega,A}^{p}(F)\geq C^{-1}\tau$ and a countable covering
$F\subset\bigcup_{i}B(z_{i},r_{i})$ where

  1. (1)

$\sum_{i}r_{i}\leq CM^{\tau}$,

  2. (2)

$\sum_{\\{i\,:\,r_{i}>\rho\\}}r_{i}\leq CM^{-1}$,

with universal constant $C$.

By means of a linear deformation of the plane (see Section 3.1 below) and a
rescaling, we see that it suffices to prove the following weaker lemma to
obtain 3.1.

######  Lemma 3.2 (Weak form of 3.1).

Let $\Omega\subset{\mathbb{R}}^{2}$, $p\in\Omega$ and $A$ as in 3.1. Suppose
also that $A_{0}=\frac{A+A^{T}}{2}$ is of the form $A_{0}=R^{T}BR$ with $R\in
C^{0,1}(U_{r_{0}}(\partial\Omega))$ a rotation, and $B\in
C^{0,1}(U_{r_{0}}(\partial\Omega))$ diagonal. Then there exists
$\delta_{0}=\delta_{0}(\lambda,\kappa\|A\|_{L^{\infty}({\mathbb{R}}^{2})})>0$
such that for every $0<\delta\leq\delta_{0}$ we have the following:

For any $0<\tau<1$, sufficiently large $M$ (how large depends on $\tau$ and on
the constants in the hypothesis), and $\rho\in(0,1/M)$ there is a set
$F\subset\partial\Omega$ such that $\omega_{\Omega,A}^{p}(F)\geq C^{-1}\tau$
and a countable covering $F\subset\bigcup_{i}B(z_{i},r_{i})$ with

  1. (1enumi)

$\sum_{i}r_{i}\leq CM^{\tau}$,

  2. (2enumi)

$\sum_{\\{i\,:\,r_{i}>\rho\\}}r_{i}\leq CM^{-1}$,

with universal constant $C$.

We remark that this weaker form replaces the assumption on the parameter $r$
by the additional assumption $A_{0}=R^{T}BR$, and allows $\delta_{0}$ to
depend also on $\kappa\|A\|_{L^{\infty}}$, and $\rho$ to be in $(0,1/M)$. For
the proof of Lemma 3.2, see Section 5.

###  3.1. The change of variables in Reduction 1

In this subsection we first collect some auxiliary results about changes of
variables that will be useful to prove some technical lemmas, secondly we
construct the precise linear deformation in the plane that allows the
reduction from 3.1 to Lemma 3.2, and finally we see how it distorts planar
Reifenberg flat domains.

####  3.1.1. Linear changes of variables

We will see how $L_{A}$-harmonic functions behave under linear changes of
variables. See [AM19, Lemmas 3.8 and 3.9] for a detailed proof of the
following two results.

######  Lemma 3.3.

Let $D\in{\mathbb{R}}^{(n+1)\times(n+1)}$ be a constant matrix with $\det
D\not=0$, $n\geq 0$. A function $f$ is $L_{A}$-harmonic in $\Omega$ if and
only if $\widetilde{f}=f\circ D$ is $L_{\widetilde{A}}$-harmonic in
$D^{-1}(\Omega)$, where $\widetilde{A}(\cdot)=D^{-1}A(D\cdot)(D^{-1})^{T}$ and
$D^{-1}(\Omega)=\\{D^{-1}x:x\in\Omega\\}$.

By the definition of elliptic measure, the previous lemma implies the
following relation of elliptic measures under a linear change of variables.

######  Corollary 3.4.

Let $D\in{\mathbb{R}}^{(n+1)\times(n+1)}$ be a constant matrix such that $\det
D\not=0$, $n\geq 0$, and let $\Omega$ be a Wiener regular domain. Let
$\omega=\omega_{\Omega,A}$ be the elliptic measure in $\Omega$ with matrix
$A$, and $\widetilde{\omega}=\omega_{D^{-1}(\Omega),\widetilde{A}}$ where
$\widetilde{A}(\cdot)=D^{-1}A(D\cdot)\left(D^{-1}\right)^{T}$. Then
$\omega^{x}(E)=\widetilde{\omega}^{D^{-1}x}(D^{-1}(E))$ for every $x\in\Omega$
and $E\subset\partial\Omega$.

####  3.1.2. Lipschitz diagonalization of symmetric matrices in the plane

In the study of the integral (7.1) in Section 7 we will use that after a
suitable linear change of variables $D$, the symmetric part of the matrix
$\widetilde{A}$ in 3.4 diagonalizes in the form $R^{T}BR$, where $R$ is a
Lipschitz rotation and $B$ is Lipschitz diagonal. In this subsection we see
that we can always reduce to this case.

We need to follow this strategy because in general it is not true that
Lipschitz elliptic symmetric matrices diagonalize in the aforementioned form,
as we can see in the following example.

######  Example 3.5.

Let $A_{1},A_{2}$ be two constant symmetric matrices diagonalizing with
different eigenvectors, and $f:{\mathbb{R}}^{n+1}\to{\mathbb{R}}$ be a
Lipschitz function with $f\mathbf{1}_{\\{|x|<1\\}}<0$,
$f\mathbf{1}_{\\{|x|>1\\}}>0$ and $\|f\|_{\infty}\leq\varepsilon$ for small
enough fixed constant $\varepsilon>0$.

Set $A(x)\coloneqq
Id+f(x)\mathbf{1}_{\\{|x|<1\\}}(x)A_{1}+f(x)\mathbf{1}_{\\{|x|>1\\}}(x)A_{2}$
and take $\varepsilon>0$ small enough to ensure the ellipticity condition on
the matrix $A$. Moreover, with this choice of the function $f$ we have that
the matrix $A$ has Lipschitz coefficients.

Let $v_{1}$ be an eigenvector of $A_{1}$ with eigenvalue $\mu_{1}$, i.e.,
$A_{1}v_{1}=\mu_{1}u_{1}$, and let $v_{2}$ be an eigenvector of $A_{2}$ with
eigenvalue $\mu_{2}$, i.e., $A_{2}v_{2}=\mu_{2}v_{2}$. Then, for $|x|<1$ the
vector $u$ is an eigenvector of $A$,

$A(x)v_{1}=(Id+f(x)A_{1})v_{1}=(1+f(x)\mu_{1})v_{1},$

and for $|x|>1$ the vector $v$ is an eigenvector of $A$,

$A(x)v_{2}=(Id+f(x)A_{2})v_{2}=(1+f(x)\mu_{2})v_{2}.$

From this we get that the matrix $A$ diagonalizes with the same basis as
$A_{1}$ if $|x|<1$, and with the same basis as $A_{2}$ if $|x|>1$, whence we
obtain that the basis is not continuous.

In the following lemma we see that we can avoid the situation seen in the
previous example by using a linear change of variables.

######  Lemma 3.6.

Let $U\subset{\mathbb{R}}^{2}$ be a set. Let $A\in C^{0,1}(U)$ be a uniformly
elliptic and symmetric $2\times 2$ matrix with ellipticity constant $\lambda$,
and let $D=\begin{pmatrix}1/K&0\\\ 0&1\end{pmatrix}$ with
$K^{2}\geq\lambda^{2}+\lambda$. Then the matrix
$\widetilde{A}(\cdot)=D^{-1}A(D\cdot)D^{-1}$ is of the form
$\widetilde{A}=R^{T}BR\in C^{0,1}(D^{-1}(U))$, with $B\in C^{0,1}(D^{-1}(U))$
diagonal and $R\in C^{0,1}(D^{-1}(U))$ a rotation.

###### Proof.

Denote the matrix $A(x)=\begin{pmatrix}a(x)&b(x)\\\ b(x)&d(x)\end{pmatrix}$,
and let

$\widetilde{A}=D^{-1}(A\circ
D)D^{-1}=\begin{pmatrix}K^{2}\widetilde{a}&K\widetilde{b}\\\
K\widetilde{b}&\widetilde{d}\end{pmatrix},$

where we write $\widetilde{a}(x)=a(Dx)$ and the analogous expressions for the
other elements of the matrix. We want to see that when
$K^{2}\geq\lambda^{2}+\lambda$ we can write $\widetilde{A}=R^{T}BR$ where $B$
is Lipschitz and diagonal, and $R$ is a Lipschitz rotation matrix.

The eigenvalues $\lambda_{\pm}=\lambda_{\pm}(\cdot)$ of $\widetilde{A}$ are

(3.1)
$\lambda_{\pm}=\frac{K^{2}\widetilde{a}+\widetilde{d}\pm\sqrt{\left(K^{2}\widetilde{a}-\widetilde{d}\right)^{2}+4K^{2}\widetilde{b}^{2}}}{2},$

and we want to see that they are Lipschitz if $K^{2}\geq\lambda^{2}+\lambda$.
Note that

(3.2) $\lambda_{+}+\lambda_{-}=K^{2}\widetilde{a}+\widetilde{d}.$

For shortness, let
$f=\left(K^{2}\widetilde{a}-\widetilde{d}\right)^{2}+4K^{2}\widetilde{b}^{2}$
be an auxiliar function. Note that since $a,b,d\in C^{0,1}(U)\cap
L^{\infty}(U)$, and so $\widetilde{a},\widetilde{b},\widetilde{d}\in
C^{0,1}(D^{-1}U)\cap L^{\infty}(D^{-1}U)$, we have $f\in C^{0,1}(D^{-1}U)\cap
L^{\infty}(D^{-1}U)$.

For $x,y\in D^{-1}U$, i.e., $Dx,Dy\in U$,

(3.3) $\displaystyle 2|\lambda_{\pm}(x)-\lambda_{\pm}(y)|$ $\displaystyle\leq
K^{2}|\widetilde{a}(x)-\widetilde{a}(y)|+|\widetilde{d}(x)-\widetilde{d}(y)|+|\sqrt{f(x)}-\sqrt{f(y)}|$
$\displaystyle\leq
C|x-y|+|\sqrt{f(x)}-\sqrt{f(y)}|=C|x-y|+\frac{|f(x)-f(y)|}{|\sqrt{f(x)}+\sqrt{f(y)}|}.$

Since $a\geq\lambda^{-1}$ and $d\leq\lambda$ by ellipticity (indeed
$a,d\approx_{\lambda}1$), and so $\widetilde{a}\geq\lambda^{-1}$ and
$\widetilde{d}\leq\lambda$, in particular
$K^{2}\widetilde{a}-\widetilde{d}\geq K^{2}\lambda^{-1}-\lambda$. Since
$K^{2}\geq\lambda^{2}+\lambda$, we obtain that

(3.4) $K^{2}\widetilde{a}-\widetilde{d}\geq K^{2}\lambda^{-1}-\lambda\geq 1,$

and with this we have

(3.5)
$\sqrt{f}=\sqrt{\left(K^{2}\widetilde{a}-\widetilde{d}\right)^{2}+4K^{2}\widetilde{b}^{2}}\geq
K^{2}\widetilde{a}-\widetilde{d}\geq 1.$

Combining the estimates 3.3 and 3.5 we get

$|\lambda_{\pm}(x)-\lambda_{\pm}(y)|\lesssim|x-y|+|f(x)-f(y)|\lesssim|x-y|,$

i.e., $\lambda_{\pm}$ are Lipschitz.

It remains to see that the matrix diagonalizes in the form $R^{T}BR$ and that
the eigenvectors are also Lipschitz. Let $u^{\pm}=(u_{1}^{\pm},u_{2}^{\pm})$
be the eigenvectors of the eigenvalues $\lambda_{\pm}$. Hence,
$\left(\widetilde{A}-\lambda_{\pm}Id\right)u^{\pm}=0$, i.e.,

(3.6)
$\begin{cases}(K^{2}\widetilde{a}-\lambda_{\pm})u_{1}^{\pm}+K\widetilde{b}u_{2}^{\pm}=0,\\\
K\widetilde{b}u_{1}^{\pm}+(\widetilde{d}-\lambda_{\pm})u_{2}^{\pm}=0.\end{cases}$

Consider the vectors
$v^{+}=\left(\lambda_{+}-\widetilde{d},K\widetilde{b}\right)$ and
$v^{-}=\left(K\widetilde{b},\lambda_{-}-K^{2}\widetilde{a}\right)$, which are
clearly Lipschitz by the preceding discussion. We claim that $v^{+}$ and
$v^{-}$ satisfy (3.6). Indeed, $v^{+}$ satisfy the second equality in (3.6) by
the definition of $v^{+}$, and the first equality follows from the definition
of the eigenvalues $\lambda_{\pm}$ and the equality
$\lambda_{+}-\widetilde{d}=K^{2}\widetilde{a}-\lambda_{-}$, see (3.2). The
vector $v^{-}$ satisfy (3.6) by the same reason.

By (3.2), we can write
$v^{+}=\left(K^{2}\widetilde{a}-\lambda_{-},K\widetilde{b}\right)$ (and so
$v^{-}$ and $v^{+}$ are orthogonal), and hence $\|v^{+}\|=\|v^{-}\|$.
Moreover,

$\|v^{\pm}\|^{2}=\left(K^{2}\widetilde{a}-\lambda_{-}\right)^{2}+K^{2}\widetilde{b}^{2}\geq\left(K^{2}\widetilde{a}-\lambda_{-}\right)^{2}=\left(\frac{K^{2}\widetilde{a}-\widetilde{d}+\sqrt{f}}{2}\right)^{2}\geq
1,$

by 3.4 and 3.5. To conclude, set the unitary vectors
$u^{\pm}\coloneqq\frac{v^{\pm}}{\|v^{\pm}\|}$. They are orthonormal and hence
we conclude that $\widetilde{A}=R^{T}BR$ with

$B=\begin{pmatrix}\lambda_{-}&0\\\ 0&\lambda_{+}\end{pmatrix},\quad
R=\begin{pmatrix}u_{1}^{-}&u_{1}^{+}\\\
u_{2}^{-}&u_{2}^{+}\end{pmatrix}=\begin{pmatrix}u_{2}^{+}&u_{1}^{+}\\\
-u_{1}^{+}&u_{2}^{+}\end{pmatrix},$

and $u^{\pm}$ are Lipschitz since $v^{\pm}$ are Lipschitz and $\|v^{\pm}\|\geq
1$. ∎

We also need to control how Reifenberg flat sets change under the linear
planar deformation in the previous lemma.

######  Lemma 3.7.

Let $K\geq 1$, $D=\begin{pmatrix}1/K&0\\\ 0&1\end{pmatrix}$ and $\Omega$ be a
$(\delta,r_{0})$-Reifenberg flat domain. If $\delta<\sqrt{15}/(16K)$ then
$D^{-1}(\Omega)$ is a
$\left(\frac{8\sqrt{15}}{15}K\delta,r_{0}\right)$-Reifenberg flat domain.

###### Proof.

Let $0<r\leq r_{0}$, $\xi\in\partial\Omega$ and
$\mathcal{P}\coloneqq\mathcal{P}(\xi,r)\ni\xi$. Denote
$\Omega^{\prime}=D^{-1}(\Omega)$, $\xi^{\prime}=D^{-1}\xi$ and
$\mathcal{P}^{\prime}=D^{-1}(\mathcal{P})$. We want to check the conditions in
Definition 2.7 with the point $\xi^{\prime}$, radius $r$ and the hyperplane
$\mathcal{P}^{\prime}$. See Figure 1.

$\xi\in\mathcal{P}\cap\partial\Omega$$\mathcal{P}$$\delta r$$2\delta
r$$\Omega$${\mathbb{R}}^{2}\setminus\overline{\Omega}$ (a) $\Omega$, $0<r\leq
r_{0}$ and $\xi\in\mathcal{P}\cap\partial\Omega$.

$\mathcal{P}^{\prime}$$2K\delta
r$$\Omega^{\prime}$${\mathbb{R}}^{2}\setminus\overline{\Omega^{\prime}}$$\xi^{\prime}\in\mathcal{P}^{\prime}\cap\partial\Omega^{\prime}$
(b) $D^{-1}$ of fig. 1(a) and $\\{x:{\rm
dist}(x,\mathcal{P}^{\prime})=2K\delta r\\}$ in red.

Figure 1. Almost the worst situation: $\mathcal{P}\perp\\{y=0\\}$.

######  Claim 3.8.

For any $x,y\in{\mathbb{R}}^{2}$, $K^{-1}{\rm dist}(x,y)\leq{\rm
dist}(Dx,Dy)\leq{\rm dist}(x,y)$.

###### Proof.

Indeed, since the minimum (resp. maximum) eigenvalue of $D$ is $K^{-1}$ (resp.
1), we have

$\frac{{\rm dist}(Dx,Dy)}{{\rm dist}(x,y)}=\frac{\|Dx-
Dy\|}{\|x-y\|}\in[K^{-1},1].$

∎

######  Claim 3.9.

One component of

(3.7) $B_{r}(\xi^{\prime})\cap\\{x\in{\mathbb{R}}^{2}:{\rm
dist}(x,\mathcal{P}^{\prime})\geq 2K\delta r\\}$

is contained in $\Omega^{\prime}$ and the other is contained in
${\mathbb{R}}^{2}\setminus\Omega^{\prime}$.

###### Proof.

By the previous claim we get

(3.8) $D^{-1}\left(\\{x\in{\mathbb{R}}^{2}:{\rm dist}(x,\mathcal{P})\geq
2\delta r\\}\right)\supset\\{x\in{\mathbb{R}}^{2}:{\rm
dist}(x,\mathcal{P}^{\prime})\geq 2K\delta r\\},$

and from the definition of $D^{-1}$ we have $B_{r}(\xi^{\prime})\subset
D^{-1}\left(B_{r}(\xi)\right)$. By (3.8), and since
$2K\delta<2\sqrt{15}/16<1$, in particular $B_{r}(\xi^{\prime})\cap
D^{-1}\left(\\{x:{\rm dist}(x,\mathcal{P})\geq 2\delta
r\\}\right)\not=\emptyset$, and the claim follows from the Reifenberg flat
condition of $\Omega$. ∎

First we check

(3.9) $\sup_{y\in\partial\Omega^{\prime}\cap B_{r}(\xi^{\prime})}{\rm
dist}(y,\mathcal{P}^{\prime}\cap B_{r}(\xi^{\prime}))\leq K\delta r.$

From the Reifenberg flatness of $\Omega$, see Definition 2.7(1), we have

$\sup_{x\in\partial\Omega\cap B_{r}(\xi)}{\rm dist}(x,\mathcal{P}\cap
B(\xi,r))\leq\delta r.$

Given $y\in{\partial\Omega}^{\prime}\cap B_{r}(\xi^{\prime})$, take
$z\in\mathcal{P}\cap B_{r}(\xi)$ with ${\rm dist}(Dy,z)\leq\delta r$. Since
$\mathcal{P}^{\prime}$ is a line through $\xi$, the distance ${\rm
dist}(y,\mathcal{P}^{\prime})$ is attained at $B_{r}(\xi^{\prime})$, that is,

${\rm dist}(y,\mathcal{P}^{\prime}\cap B_{r}(\xi^{\prime}))={\rm
dist}(y,\mathcal{P}^{\prime})\leq{\rm dist}(y,D^{-1}z).$

By Claim 3.8,

${\rm dist}(y,\mathcal{P}^{\prime}\cap B_{r}(\xi^{\prime}))\leq K{\rm
dist}(Dy,z)\leq K\delta r,$

and (3.9) follows.

Now we turn to prove

(3.10) $\sup_{y\in\mathcal{P}^{\prime}\cap B_{r}(\xi^{\prime})}{\rm
dist}(y,\partial\Omega^{\prime}\cap
B_{r}(\xi^{\prime}))\leq\frac{2\sqrt{2}K\delta}{\sqrt{1+\sqrt{1-4K^{2}\delta^{2}}}}r.$

$\frac{2\sqrt{2}K\delta}{\sqrt{1+\sqrt{1-4K^{2}\delta^{2}}}}r$$r(1-\sqrt{1-4K^{2}\delta^{2}})$$y$$B_{r}(\xi^{\prime})$${\mathbb{R}}^{2}\setminus\overline{\Omega}$$\Omega$$\ell$$\xi^{\prime}\in\mathcal{P}^{\prime}\cap\partial\Omega^{\prime}$$\leq
r$$y$$p\in\partial\Omega^{\prime}$$\leq 2K\delta
r$$\mathcal{P}^{\prime}$$2K\delta r$ Figure 2. Setting of (3.10).

By Claim 3.9 we have that for each line $\ell$ orthogonal to
$\mathcal{P}^{\prime}$ such that

$\ell\cap B_{r}(\xi^{\prime})\cap\\{x\in{\mathbb{R}}^{2}:{\rm
dist}(x,\mathcal{P}^{\prime})\geq 2K\delta r\\}\not=\emptyset,$

there is a point $p\in\ell\cap\partial\Omega^{\prime}\cap\\{x\in
B_{r}(\xi^{\prime}):{\rm dist}(x,\mathcal{P}^{\prime})<2K\delta r\\}$, since
$\partial\Omega^{\prime}$ must separate each component in (3.7). Using this
fact and $\xi^{\prime}\in\mathcal{P}^{\prime}\cap\partial\Omega^{\prime}$ (the
center of the ball), for every $y\in\mathcal{P}^{\prime}\cap
B_{r}(\xi^{\prime})$ we obtain

${\rm dist}(y,\partial\Omega^{\prime}\cap
B_{r}(\xi^{\prime}))\leq\min\left\\{r,{\rm
dist}\left(y,B_{r}(\xi^{\prime})\cap\\{x:{\rm
dist}(x,\mathcal{P}^{\prime})\geq 2K\delta r\\}\right)\right\\}.$

Using basic trigonometric computations, see Figure 2, we have that for every
$y\in\mathcal{P}^{\prime}\cap B_{r}(\xi^{\prime})$,

${\rm dist}\left(y,B_{r}(\xi^{\prime})\cap\\{x:{\rm
dist}(x,\mathcal{P}^{\prime})\geq 2K\delta
r\\}\right)\leq\frac{2\sqrt{2}K\delta}{\sqrt{1+\sqrt{1-4K^{2}\delta^{2}}}}r.$

Since $K\delta<\sqrt{15}/16<\sqrt{3}/4$, the previous value is less than $r$,
and hence (3.10) is proved.

Notice that $K\delta r<2K\delta
r<\frac{2\sqrt{2}K\delta}{\sqrt{1+\sqrt{1-4K^{2}\delta^{2}}}}r$. From this,
3.9 and 3.10 we get

${\rm dist}_{\mathcal{H}}(\partial\Omega^{\prime}\cap
B_{r}(\xi^{\prime}),\mathcal{P}^{\prime}\cap
B_{r}(\xi^{\prime}))\leq\frac{2\sqrt{2}K\delta}{\sqrt{1+\sqrt{1-4K^{2}\delta^{2}}}}r,$

and Definition 2.7(1) is verified.

Since the last term is strictly larger than $2K\delta r$, and
$2\frac{2\sqrt{2}K\delta}{\sqrt{1+\sqrt{1-4K^{2}\delta^{2}}}}<1$ when
$\delta<\frac{\sqrt{15}}{16K}$, we get

$B_{r}(\xi^{\prime})\cap\\{x:{\rm dist}(x,\mathcal{P}^{\prime})\geq 2K\delta
r\\}\supset B_{r}(\xi^{\prime})\cap\left\\{x:{\rm
dist}(x,\mathcal{P}^{\prime})\geq
2\frac{2\sqrt{2}K\delta}{\sqrt{1+\sqrt{1-4K^{2}\delta^{2}}}}r\right\\}\not=\emptyset,$

and the second condition of Reifenberg flat, Definition 2.7(2), is achieved by
Claim 3.9.

In conclusion, if $\delta<\sqrt{15}/(16K)$ then
$\Omega^{\prime}=D^{-1}(\Omega)$ is
$\left(\frac{2\sqrt{2}K\delta}{\sqrt{1+\sqrt{1-4K^{2}\delta^{2}}}},r_{0}\right)$-Reifenberg
flat. In particular $\Omega^{\prime}=D^{-1}(\Omega)$ is
$\left(\frac{8\sqrt{15}}{15}K\delta,r_{0}\right)$-Reifenberg flat. ∎

###  3.2. Reduction 1: Lipschitz diagonalization of the symmetric part

Let us see how Lemma 3.2 implies 3.1. We will do this in two steps. First we
show how to get rid of the assumption on the decomposition of $A_{0}$, using
the linear transformation in Lemma 3.6.

######  Claim 3.10.

In Lemma 3.2, the hypothesis $A_{0}=R^{T}BR$ is unnecessary.

###### Proof.

For every square matrix $X$ we define its symmetric part
$X_{0}\coloneqq\frac{X+X^{T}}{2}$.

By Lemma 3.6 there exists a constant diagonal matrix
$D=\begin{pmatrix}1/K&0\\\ 0&1\end{pmatrix}$ with $K=K(\lambda)\geq 1$ such
that $\widetilde{(A_{0})}(\cdot)=D^{-1}A_{0}(D\cdot)D^{-1}$ can be written as
$\widetilde{(A_{0})}=R^{T}BR\in C^{0,1}(D^{-1}(U_{r_{0}}(\partial\Omega)))$
with $R\in C^{0,1}(D^{-1}(U_{r_{0}}(\partial\Omega)))$ a rotation, and $B\in
C^{0,1}(D^{-1}(U_{r_{0}}(\partial\Omega)))$ diagonal.

Setting $\widetilde{A}(\cdot)\coloneqq D^{-1}A(D\cdot)D^{-1}$, we have that
the symmetric part of the matrix $\widetilde{A}$ is

$\widetilde{A}_{0}\coloneqq\frac{D^{-1}A(D\cdot)D^{-1}+\left(D^{-1}A(D\cdot)D^{-1}\right)^{T}}{2}=\frac{D^{-1}A(D\cdot)D^{-1}+D^{-1}A^{T}(D\cdot)D^{-1}}{2}=\widetilde{(A_{0})},$

and hence $\widetilde{A}_{0}=R^{T}BR$ as before.

Note that

$U_{r_{0}}(\partial D^{-1}(\Omega))\coloneqq\\{x:{\rm dist}(x,\partial
D^{-1}\Omega)<r_{0}\\}\subset D^{-1}(U_{r_{0}}(\partial\Omega)),$

and so these matrices are Lipschitz in $U_{r_{0}}(\partial D^{-1}(\Omega))$.

Denoting $\widetilde{\omega}\coloneqq\omega_{D^{-1}\Omega,\widetilde{A}}$ the
elliptic measure in $D^{-1}\Omega$ with matrix $\widetilde{A}$, by 3.4 we have
$\omega^{x}(\cdot)=\widetilde{\omega}^{D^{-1}x}(D^{-1}\cdot)$ for any
$x\in\Omega$. By Lemma 3.7 we have that $D^{-1}\Omega$ is
$(8\sqrt{15}K\delta/15,r_{0})$-Reifenberg flat. Set $\widetilde{p}\coloneqq
D^{-1}p\in D^{-1}(\Omega)$. Since ${\rm dist}(p,\partial\Omega)>r_{0}$, ${\rm
dist}(\widetilde{p},\partial D^{-1}\Omega)>r_{0}$ and we are in position to
apply Lemma 3.2 with this pole $\widetilde{p}$.

First, we need to compute the ellipticity constant, the Lipschitz seminorm,
and the $L^{\infty}$ norm of the matrix $\widetilde{A}$. Recall $\lambda\geq
1$ is the ellipticity constant of $A$. For $\xi,\eta\in{\mathbb{R}}^{2}$,

$\displaystyle\langle\widetilde{A}\xi,\eta\rangle$ $\displaystyle=\langle
A(D\cdot)D^{-1}\xi,D^{-1}\eta\rangle\leq\lambda|D^{-1}\xi||D^{-1}\eta|\leq\lambda
K^{2}|\xi||\eta|,$ $\displaystyle\langle\widetilde{A}\xi,\xi\rangle$
$\displaystyle=\langle
A(D\cdot)D^{-1}\xi,D^{-1}\xi\rangle\geq\lambda^{-1}|D^{-1}\xi|^{2}\geq\lambda^{-1}|\xi|^{2}\geq\left(\lambda
K^{2}\right)^{-1}|\xi|^{2},$

i.e., the ellipticity constant of $\widetilde{A}$ is $\lambda K^{2}$. If one
seeks optimal constants, choosing $\widetilde{A}/K$ the ellipticity constant
becomes $\lambda K$. The Lipschitz seminorm of $\widetilde{A}$ in
$D^{-1}(U_{r_{0}}(\partial\Omega))$ is at most
$K^{2}[A]_{C^{0,1}(U_{r_{0}}(\partial\Omega))}$. Indeed, for any two points
$x,y\in D^{-1}(U_{r_{0}}(\partial\Omega))$ with $x\not=y$,

$\displaystyle\frac{|D^{-1}A(Dx)D^{-1}-D^{-1}A(Dy)D^{-1}|}{|x-y|}$
$\displaystyle\leq K^{2}\frac{|A(Dx)-A(Dy)|}{|x-y|}$ $\displaystyle\leq
K^{2}[A]_{C^{0,1}(U_{r_{0}}(\partial\Omega))}\frac{|Dx-Dy|}{|x-y|}$
$\displaystyle\leq K^{2}[A]_{C^{0,1}(U_{r_{0}}(\partial\Omega))}.$

The $L^{\infty}$ norm is $\|\widetilde{A}\|_{L^{\infty}({\mathbb{R}}^{2})}\leq
K^{2}\|A\|_{L^{\infty}({\mathbb{R}}^{2})}$.

By Lemma 3.2 there exists

$\delta_{0}=\delta_{0}(K(\lambda)^{2}\lambda,K(\lambda)^{2}[A]_{C^{0,1}(U_{r_{0}}(\partial\Omega))}\cdot
K(\lambda)^{2}\|A\|_{L^{\infty}({\mathbb{R}}^{2})})>0$

such that if $0<8\sqrt{15}K(\lambda)\delta/15\leq\delta_{0}$, i.e.,
$0<\delta\leq 15\delta_{0}/(8\sqrt{15}K(\lambda))$, and taking $M$ big enough
such that Lemma 3.2 holds, then there is a set $\widetilde{F}\subset\partial
D^{-1}(\Omega)$ such that
$\widetilde{\omega}^{\widetilde{p}}(\widetilde{F})\geq C^{-1}\tau$ and with a
covering $\widetilde{F}\subset\bigcup_{i}B(\widetilde{z}_{i},r_{i})$ where

  1. (1enumi)

$\sum_{i}r_{i}\leq CM^{\tau}$,

  2. (2enumi)

$\sum_{\\{i\,:\,r_{i}>\rho\\}}r_{i}\leq CM^{-1}$,

with universal constant $C$.

Defining $F\coloneqq D(\widetilde{F})$ we have
$\omega^{p}(F)=\widetilde{\omega}^{\widetilde{p}}(\widetilde{F})\geq
C^{-1}\tau$, and $F\subset\bigcup_{i}D(B(\widetilde{z}_{i},r_{i}))$. Finally,
as $D(B(\widetilde{z}_{i},r_{i}))\subset B(D\widetilde{z}_{i},r_{i})$, then
$\\{B(D\widetilde{z}_{i},r_{i})\\}_{i}$ is a covering of $F$ satisfying the
same properties. ∎

###  3.3. Reduction 2: The dependence on the Lipschitz seminorm

This is the second step to show that Lemma 3.2 implies 3.1. Note that the
flatness constant of the Reifenberg flat domain on Lemma 3.2 (hence also on
Claim 3.10) depends also on the Lipschitz seminorm $\kappa$ of the matrix.
Below, we see that in fact these results imply 3.1 by a rescaling argument.
Here the flatness constant is determined solely by the ellipticity of the
matrix.

###### Proof of 3.1 assuming Lemma 3.2.

Fix $r\in(0,1]$ be such that $r\kappa\|A\|_{L^{\infty}}\leq 1$, and let
$\widetilde{A}(\cdot)=A(r\cdot)$, $\widetilde{\Omega}=\Omega/r$ and
$\widetilde{\omega}\coloneqq\omega_{\widetilde{\Omega},\widetilde{A}}$ be the
elliptic measure with respect to the matrix $\widetilde{A}$ in
$\widetilde{\Omega}$.

The matrix $\widetilde{A}$ is Lipschitz in $\\{x/r:{\rm
dist}(x,\partial\Omega)<r_{0}\\}=\\{x:{\rm
dist}(x,\partial\widetilde{\Omega})<r_{0}/r\\}$, and $\widetilde{\Omega}$ is
$(\delta,r_{0}/r)$-Reifenberg flat since $\Omega$ is
$(\delta,r_{0})$-Reifenberg flat. By the uniqueness of the elliptic measure we
have $\widetilde{\omega}^{z/r}(\cdot/r)=\omega^{z}(\cdot)$ for any
$z\in\Omega$.

With this “zoom” the ellipticity constant of $\widetilde{A}$ becomes the same
as the one of $A$, $\|\widetilde{A}\|_{L^{\infty}}=\|A\|_{L^{\infty}}$ and
$[\widetilde{A}]_{C^{0,1}}=r[A]_{C^{0,1}}$, which implies
$[\widetilde{A}]_{C^{0,1}}\cdot\|\widetilde{A}\|_{L^{\infty}}=r\kappa\|A\|_{L^{\infty}}\leq
1$.

This allows us to invoke Claim 3.10 for the elliptic measure
$\widetilde{\omega}^{p/r}$ since ${\rm dist}(p/r,\Omega/r)>r_{0}/r$, as ${\rm
dist}(p,\Omega)>r_{0}$. Note that now we don’t have the dependence on the
Lipschitz seminorm and $L^{\infty}$ norm of the matrix since we are in the
case $[\widetilde{A}]_{C^{0,1}}\|\widetilde{A}\|_{L^{\infty}}\leq 1$. Hence,
there exists $\delta_{0}=\delta_{0}(\lambda)>0$ such that if
$0<\delta\leq\delta_{0}$, then for $M$ big enough (to satisfy Claim 3.10) and
setting $\rho$ such that $0<\rho/r<1/M$, we can find a set
$F^{\prime}\subset\partial\Omega/r$ such that
$\widetilde{\omega}^{p/r}(F^{\prime})\geq C^{-1}\tau$ and with a covering
$F^{\prime}\subset\bigcup_{i}B(z^{\prime}_{i},r_{i})$ such that

  1. (1enumi)

$\sum_{i}r_{i}\leq CM^{\tau}$,

  2. (2enumi)

$\sum_{\\{i\,:\,r_{i}>\rho/r\\}}r_{i}\leq CM^{-1}$.

Set $F=rF^{\prime}$. Then
$\omega^{p}(F)=\widetilde{\omega}^{p/r}(F^{\prime})\geq C^{-1}\tau$, and
$F=rF^{\prime}\subset\bigcup_{i}B(z_{i},rr_{i})$, which implies

  1. (1)

$\sum_{i}rr_{i}\leq rCM^{\tau}\leq CM^{\tau}$,

  2. (2)

$\sum_{\\{i\,:\,rr_{i}>\rho\\}}rr_{i}=\sum_{\\{i\,:\,r_{i}>\rho/r\\}}rr_{i}\leq
rCM^{-1}\leq CM^{-1}$,

as claimed. ∎

##  4\. Elliptic measures in CDC domains

In this section we collect the key properties of elliptic measures in CDC
domains for the proof of 3.1. The first one, frequently called Bourgain’s
lemma (see [Bou87, Lemma 1] for the harmonic case), is the following lemma.

######  Lemma 4.1 ([HKM06, Lemma 11.21]).

Let $\Omega\subset{\mathbb{R}}^{n+1}$, $n\geq 1$, be a bounded CDC domain with
constant $c_{0}$ and radius $s_{0}$, and let $A$ be a real uniformly elliptic
(not necessarily symmetric) matrix. Then there exists a constant
$\tau\in(0,1)$, depending only on $n$, $c_{0}$ and the ellipticity constant of
the matrix $A$, such that for $E\subset\partial\Omega$,
$x_{0}\in\partial\Omega$ and $0<r\leq s_{0}$, we have the following:

  1. (1)

if $B(x_{0},2r)\cap E=\emptyset$, then $\omega^{p}_{\Omega,A}(E)\leq
1-\tau<1$, and

  2. (2)

if $B(x_{0},2r)\cap\partial\Omega\subset E$, then
$\omega^{p}_{\Omega,A}(E)\geq\tau>0$,

for any point $p\in B(x_{0},r)\cap\Omega$.

The proof of 3.1 is based on a modification of the domain, without losing the
initial information. In the following lemma we obtain the first step in that
modification. This is the analogue of [Wol93, Lemma 1.1].

######  Lemma 4.2.

Let $\Omega\subset{\mathbb{R}}^{n+1}$, $n\geq 1$, be a bounded CDC domain with
constants $c_{0}$ and radius $s_{0}$, let $x\in\partial\Omega$ and $0<r\leq
s_{0}$, and let $A$ be a real uniformly elliptic (not necessarily symmetric)
matrix. Then for any $k>2$, there exists a constant $C$ depending only on $n$,
$c_{0}$ and the ellipticity constant of the matrix $A$ such that

$\omega_{\Omega\setminus\overline{B(x,r)},A}^{p}\left(\overline{B(x,r)}\right)\leq
C\omega_{\Omega,A}^{p}(B(x,kr)),\quad\text{for all
}p\in\Omega\setminus\overline{B(x,r)}.$

###### Proof.

The proof follows the same argument as [Wol93, Lemma 1.1].

Since $k>2$, by Lemma 4.1(2) there exists $C>1$, depending on $n$, $c_{0}$ and
the ellipticity constant of the matrix, such that
$\omega^{p}_{\Omega,A}(B(x,kr))\geq\omega^{p}_{\Omega,A}(\overline{B(x,2r)})\geq
C^{-1}$ for any $p\in\overline{B(x,r)}\cap\Omega$. The lemma follows by the
maximum principle in $\Omega\setminus\overline{B(x,r)}$ by standard
techniques. ∎

Later on we will need to have some control on the Radon-Nikodym derivative of
the elliptic measure of the modified domain, see Lemma 4.5 below. First we
compute the CDC constants of an annulus, which will be used later to control
this density in a modified domain.

######  Lemma 4.3.

Let $k>3$. The annulus
$\mathcal{A}_{k}=B(0,k^{2}r)\setminus\overline{B(0,r)}\subset{\mathbb{R}}^{n+1}$,
$n\geq 1$, satisfies the CDC with constant $c_{0}=c_{0}(k)$ and radius
$s_{0}\coloneqq\left(\frac{k+1}{2}\right)^{2}r$, and moreover

  1. (1)

$\overline{B(0,kr)}\subset B(x_{0},s_{0})$ and

  2. (2)

$\overline{B(x_{0},2s_{0})}\subset B(0,k^{2}r)$,

for any $x_{0}\in\partial B(0,r)$, i.e., the inner circle.

###### Proof.

In the following computations we find the radius $s_{0}$ to have the CDC on
the annulus $\mathcal{A}_{k}$ with the conditions (1) and (2). From the first
condition we get $s_{0}>(k+1)r$, and from the second $2s_{0}+r<k^{2}r$, i.e.,
$2s_{0}<(k^{2}-1)r$. In order to have existence in $s_{0}$ we need
$2(k+1)r<(k^{2}-1)r$, whence we need $k>3$. Let $s_{0}$ be the middle point in
$\left((k+1)r,\frac{k^{2}-1}{2}r\right)$,

$s_{0}\coloneqq\frac{(k+1)r+\frac{k^{2}-1}{2}r}{2}=\frac{k^{2}+2k+1}{4}r\eqqcolon
C(k)\cdot r.$

Now we want to see that for $k>3$, the annulus $\mathcal{A}_{k}$ satisfies the
capacity density condition with $s_{0}=\frac{k^{2}+2k+1}{4}r=C(k)\cdot r$. By
definition of $C(k)$, given a point $x_{0}\in\partial\mathcal{A}_{k}$, the
ball $B(x_{0},C(k)\cdot r)$ does not intersect the other component of
$\partial\mathcal{A}_{k}$, by condition (2).

We want to see that there exists $c_{0}=c_{0}(k)$ such that

$\operatorname{Cap}\left(\overline{B(x_{0},s)}\cap\mathcal{A}_{k}^{c},B(x_{0},2s)\right)\geq
c_{0}(k)\cdot s^{n-1},$

for all $x_{0}\in\partial\mathcal{A}_{k}$ and $0<s\leq s_{0}$.

Case 1. Suppose $x_{0}\in\partial B(0,r)\subset\partial\mathcal{A}_{k}$. Let
$0<s\leq s_{0}=C(k)\cdot r$. By the choice of $C(k)$ we have
$B(x_{0},s)\cap\mathcal{A}_{k}^{c}=B(x_{0},s)\cap\overline{B(0,r)}$.

Set $\xi=x_{0}-x_{0}\frac{s}{2s_{0}}$. So $|\xi-
x_{0}|=\left|x_{0}\frac{s}{2s_{0}}\right|=\frac{rs}{2s_{0}}=\frac{s}{2C(k)}$,
and note that $|\xi-x_{0}|\leq\frac{r}{2}$. In particular
$B\left(\xi,\frac{s}{2C(k)}\right)\subset B(x_{0},s)\cap\mathcal{A}_{k}^{c}$.
Also, $B(x_{0},2s)\subset
B\left(\xi,2s+|x_{0}-\xi|\right)=B\left(\xi,2s+\frac{s}{2C(k)}\right)$. From
these two inclusions, the monotonicity of the capacity and [HKM06, (2.13)], we
have

$\operatorname{Cap}\left(\overline{B(x_{0},s)}\cap\mathcal{A}_{k}^{c},B(x_{0},2s)\right)\geq\operatorname{Cap}\left(\overline{B\left(\xi,\frac{s}{2C(k)}\right)},B\left(\xi,2s+\frac{s}{2C(k)}\right)\right)\approx_{k}s^{n-1}.$

Case 2. Suppose $x_{0}\in\partial B(0,k^{2}r)\subset\partial\mathcal{A}_{k}$.
Let $0<s\leq s_{0}=C(k)\cdot r$. Define
$\xi=x_{0}+\frac{x_{0}}{|x_{0}|}\frac{s}{2}$. Hence
$B\left(\xi,\frac{s}{2}\right)\subset B(x_{0},s)\cap\mathcal{A}_{k}^{c}$ and
$B(x_{0},2s)\subset B\left(\xi,\frac{5}{2}s\right)$. Arguing as before we get

$\operatorname{Cap}\left(\overline{B(x_{0},s)}\cap\mathcal{A}_{k}^{c},B(x_{0},2s)\right)\gtrsim
s^{n-1},$

as claimed. ∎

Now we study the density of the elliptic measure in an annulus. For a Hölder
matrix $A$, here we use that $L_{A}$-harmonic functions are Hölder continuous
up to the boundary, see Theorem 2.12.

######  Lemma 4.4.

Let $k>3$, $0<r\leq 1$ and
$\mathcal{A}_{k}=B(x,k^{2}r)\setminus\overline{B(x,r)}\subset{\mathbb{R}}^{n+1}$,
$n\geq 1$. Let $A$ be a real uniformly elliptic (not necessarily symmetric)
matrix. Suppose also that $A\in C^{\alpha}\left(B(x,2k^{2}r)\right)$ with
$0<\alpha<1$. Then the elliptic measure in the annulus $\mathcal{A}_{k}$
(arising from the matrix $A$) satisfies

$\omega_{\mathcal{A}_{k},A}^{z}(Y)\lesssim\frac{\sigma(Y)}{r},\quad\text{for
any }z\in\partial B(x,kr)\text{ and any }Y\subset\partial B(x,r),$

with constant depending only on $k$, $[A]_{C^{\alpha}}$ and the ellipticity of
$A$.

###### Proof.

Suppose without loss of generality that the annulus is centered at the origin
and denote $B_{t}\coloneqq B(0,t)$. We can also assume that $r=1$. Indeed,
denote $\omega\coloneqq\omega_{\mathcal{A}_{k},A}$ and
$\widetilde{\omega}\coloneqq\omega_{\widetilde{\mathcal{A}_{k}},\widetilde{A}}$
the elliptic measure associated to the matrix $\widetilde{A}(\cdot)\coloneqq
A(r\cdot)$, where the rescaled annulus is
$\widetilde{\mathcal{A}_{k}}=B_{k^{2}}\setminus\overline{B_{1}}$. After
rescaling and by the uniqueness of the elliptic measure,

$\omega^{z}(Y)=\widetilde{\omega}^{z^{\prime}}(Y^{\prime})\quad\text{where
}z^{\prime}=z/r\text{ and }Y^{\prime}=Y/r,$

see 3.4. The matrix $\widetilde{A}$ has the same ellipticity constant as $A$,
and the Hölder seminorm is improved because
$[\widetilde{A}]_{C^{\alpha}}=[A]_{C^{\alpha}}\cdot r^{\alpha}$ whenever
$0<r<1$. If the lemma were true with $r=1$ then writing $p=z/r$ we would get

$\omega^{z}(Y)=\widetilde{\omega}^{p}(Y^{\prime})\leq
C_{k}\sigma(Y^{\prime})=C_{k}\frac{\sigma(Y)}{r},$

as claimed.

Let $p\in\partial B_{k}$ and let $g^{T}_{p}$ the Green function of the annulus
$\mathcal{A}_{k}$ with pole at $p$. Then, using (2.8),

(4.1) $\omega^{p}(Y)=-\int_{Y}\langle{A}^{T}(\xi)\nabla
g^{T}_{p}(\xi),\nu(\xi)\rangle\,d\sigma(\xi)\lesssim\int_{Y}|\nabla
g^{T}_{p}(\xi)|\,d\sigma(\xi).$

We would be done if we can bound $|\nabla g^{T}_{p}|\leq C_{k}$. To obtain
this we apply Theorem 2.12. In the next paragraphs we check its hypothesis.

The function $g^{T}_{p}$ is $L_{A^{T}}$-harmonic in $B_{2}\setminus B_{1}$
since we are in the case $k>3$. Moreover $g^{T}_{p}\equiv 0$ in $\partial
B_{1}$. We need to verify that $g^{T}_{p}\in W^{1,2}\left(B_{2}\setminus
B_{1}\right)$. Recall that the Green function is constructed in (2.5) as
$g^{T}_{p}=-{\mathcal{E}}^{T}_{p}+h$ where $h$ is a
$L_{\widetilde{A}^{T}}$-harmonic function with $h={\mathcal{E}}^{T}_{p}$ on
$\partial{\mathcal{A}_{k}}$, and ${\mathcal{E}}^{T}_{p}$ is the fundamental
solution with pole at $p$. Hence,

(4.2) $\displaystyle\|g^{T}_{p}\|_{L^{2}\left(B_{2}\setminus
B_{1}\right)}\leq$ $\displaystyle\|g^{T}_{p}\|_{L^{2}\left(B_{5/2}\setminus
B_{1}\right)}\leq\max_{y\in\partial B_{5/2}}g^{T}_{p}(y)\leq\max_{y\in\partial
B_{5/2}}|{\mathcal{E}}^{T}_{p}(y)|+\max_{y\in\partial B_{5/2}}|h(y)|$
$\displaystyle\leq$ $\displaystyle\max_{y\in\partial
B_{5/2}}|{\mathcal{E}}^{T}_{p}(y)|+\max_{y\in\partial B_{k^{2}}\cup\partial
B_{1}}|{\mathcal{E}}^{T}_{p}(y)|.$

In the planar case, we have $|{\mathcal{E}}^{T}_{p}(y)|\lesssim
1+\left|\log|y-p|\right|$ by 2.17, and in particular, from (4.2) we obtain

$\|g^{T}_{p}\|_{L^{2}\left(B_{2}\setminus
B_{1}\right)}\lesssim\max_{y\in\partial
B_{5/2}}\left[1+\left|\log|y-p|\right|\right]+\max_{y\in\partial
B_{k^{2}}\cup\partial B_{1}}\left[1+\left|\log|y-p|\right|\right]\leq C_{k}.$

In higher dimensions, $n\geq 2$, the fundamental solution is bounded by
$|{\mathcal{E}}^{T}_{p}(y)|\lesssim|p-y|^{1-n}$ (see [HK07, Theorem 3.1
(3.55)]). From this bound and (4.2) we get

$\|g^{T}_{p}\|_{L^{2}\left(B_{2}\setminus
B_{1}\right)}\lesssim\max_{y\in\partial B_{5/2}}|y-p|^{n-1}+\max_{y\in\partial
B_{k^{2}}\cup\partial B_{1}}|y-p|^{n-1}\leq C_{k}.$

In fact, we have obtained $\|g^{T}_{p}\|_{L^{2}\left(B_{5/2}\setminus
B_{1}\right)}\lesssim C_{k}$, and hence by Caccioppoli’s inequality in the
annulus $B_{2}\setminus B_{1}$ we also obtain the upper bound for the
gradient,

$\|\nabla g^{T}_{p}\|_{L^{2}\left(B_{2}\setminus
B_{1}\right)}\lesssim\|g^{T}_{p}\|_{L^{2}\left(B_{5/2}\setminus
B_{1}\right)}\lesssim C_{k},$

implying $g^{T}_{p}\in W^{1,2}(B(0,2)\setminus B(0,1))$ depending only on $k$
and the ellipticity constant.

Consider $\Omega=B_{2}\setminus\overline{B}_{1}$, $T=\partial B_{1}$ and
$\Omega^{\prime}=B_{3/2}\setminus\overline{B}_{1}$ in Theorem 2.12. Note that
$\Omega^{\prime}\subset\Omega$, $T\subset\partial\Omega^{\prime}$ and ${\rm
dist}\left(\left(B_{3/2}\setminus\overline{B}_{1}\right)\cup\partial
B_{1},\partial B_{2}\right)=1/2$. Then we get $g^{T}_{p}\in
C^{1,\alpha}\left(\left(B_{3/2}\setminus\overline{B}_{1}\right)\cup\partial
B_{1}\right)$ with

$\max_{y\in\partial B_{1}}|\nabla
g^{T}_{p}(y)|\leq\|g^{T}_{p}\|_{1;\Omega^{\prime}}\overset{\text{Thm~{}\ref{C^a
to the boundary}}}{\lesssim}\max_{y\in
B_{2}\setminus\overline{B}_{1}}g^{T}_{p}(y)\leq\max_{y\in\partial
B_{5/2}}g^{T}_{p}(y)\leq C_{k},$

and the lemma follows. ∎

Lemma 4.2 relates the elliptic measure on the initial domain with the elliptic
measure on the domain minus a fixed ball. Next in Lemma 4.5, which is the
analogue of [Wol93, Lemma 1.2], we study the elliptic measure on this last
setting. Combining Lemmas 4.2 and 4.5 we will obtain density properties of the
elliptic measure on a modified domain.

######  Lemma 4.5.

Set $k>3$ and $0<r\leq 1$. Let $\Omega\subset{\mathbb{R}}^{n+1}$, $n\geq 1$,
be a bounded Wiener regular domain, $x\in\partial\Omega$ and let $A$ be a real
uniformly elliptic (not necessarily symmetric) matrix. Suppose also that $A\in
C^{\alpha}(\\{y\in{\mathbb{R}}^{n+1}:{\rm dist}(y,\partial\Omega)<2k^{2}r\\})$
with $0<\alpha<1$.

Set $\widetilde{\Omega}=\Omega\setminus\overline{B}$ where $B=B(x,r)$. Then
$\omega_{\widetilde{\Omega},A}^{p}\big{|}_{\partial B}$ is absolutely
continuous with respect to $\sigma$ for any $p\in\Omega\setminus
k\overline{B}$, and for $z\in\partial B$,

$\frac{d\omega_{\widetilde{\Omega},A}^{p}}{d\sigma}(z)\leq\frac{C}{r}\omega_{\Omega\setminus
k\overline{B},A}^{p}\left(k\overline{B}\right),\quad\text{for any
}p\in\Omega\setminus k\overline{B},$

with constant $C$ depending only on $n$, $k$, $[A]_{C^{\alpha}}$ and the
ellipticity of $A$.

Following the scheme of the proof of [Wol93, Lemma 1.2], to obtain Lemma 4.5
we study the elliptic measure of the annulus $\mathcal{A}_{k}\coloneqq
B(x,k^{2}r)\setminus\overline{B(x,r)}$ when $k>3$ (in order to apply Lemma
4.3) and $0<r\leq 1$ (to have a control on the Hölder seminorm of the matrix
$A\in C^{\alpha}$). However, some technicalities are needed due to the
variability of the coefficients of the matrix.

###### Proof of Lemma 4.5.

During the proof we write $\omega_{\cdot}$ instead of $\omega_{\cdot,A}$.

To obtain the result it suffices to prove
$\omega^{p}_{\widetilde{\Omega}}(Y)\lesssim\frac{\sigma(Y)}{r}\omega^{p}_{\Omega\setminus
k\overline{B}}(k\overline{B})$ for all $p\in\Omega\setminus k\overline{B}$ and
every $Y\subset\partial B$, and in fact, it is enough to assume that $Y$ is
open. Indeed, fixed $p\in\Omega\setminus k\overline{B}$, for $\varepsilon>0$
let $U\supset Y$ be an open set (relative to $\partial B$) such that
$\sigma(U)\leq\sigma(Y)+\varepsilon$. If the lemma were true for open sets,
then
$\omega_{\widetilde{\Omega}}^{p}(Y)\leq\omega_{\widetilde{\Omega}}^{p}(U)\lesssim\frac{\sigma(U)}{r}\omega^{p}_{\Omega\setminus
k\overline{B}}(k\overline{B})=\frac{\sigma(Y)+\varepsilon}{r}\omega^{p}_{\Omega\setminus
k\overline{B}}(k\overline{B})$ and the general case would follow taking
$\varepsilon\to 0$.

Let us assume $Y$ is an open set relative to $\partial B$, and fix
$p\in\Omega\setminus k\overline{B}$. Again, for $\varepsilon>0$ let $V\supset
k\overline{B}$ be an open set such that $\omega_{\Omega\setminus
k\overline{B}}^{p}(V)\leq\omega_{\Omega\setminus
k\overline{B}}^{p}(k\overline{B})+\varepsilon$, $\psi\in C_{c}(V)$ such that
$\mathbf{1}_{k\overline{B}}\leq\psi\leq\mathbf{1}_{V}$, and let
$v_{\Omega\setminus k\overline{B}}$ denote the $L_{A}$-harmonic extension of
$\psi$ in $\Omega\setminus k\overline{B}$.

Let $N\coloneqq\max_{x\in\partial kB}\omega^{x}_{\widetilde{\Omega}}(Y)$ and
let $x_{0}\in\partial kB$ such that
$\omega^{x_{0}}_{\widetilde{\Omega}}(Y)=N$. Define the annulus
$\mathcal{A}_{k}\coloneqq k^{2}B\setminus\overline{B}$. By Lemma 4.6 below we
obtain

(4.3)
$N-\omega^{x_{0}}_{\widetilde{\Omega}\cap\mathcal{A}_{k}}(Y)=\omega^{x_{0}}_{\widetilde{\Omega}}(Y)-\omega^{x_{0}}_{\widetilde{\Omega}\cap\mathcal{A}_{k}}(Y)=\int_{\Omega\cap\partial
k^{2}B}\omega^{\xi}_{\widetilde{\Omega}}(Y)\,d\omega_{\widetilde{\Omega}\cap\mathcal{A}_{k}}^{x_{0}}(\xi).$

By the maximum principle in $\widetilde{\Omega}\setminus k\overline{B}$ we
have that
$\omega^{\xi}_{\widetilde{\Omega}}(Y)\leq\omega^{x_{0}}_{\widetilde{\Omega}}(Y)=N$
for $\xi\in\partial k^{2}B$, and hence

(4.4) $\int_{\Omega\cap\partial
k^{2}B}\omega^{\xi}_{\widetilde{\Omega}}(Y)\,d\omega_{\widetilde{\Omega}\cap\mathcal{A}_{k}}^{x_{0}}(\xi)\leq
N\omega_{\widetilde{\Omega}\cap\mathcal{A}_{k}}^{x_{0}}(\Omega\cap\partial
k^{2}B).$

All in all, from 4.3 and 4.4 then

(4.5) $N-\omega^{x_{0}}_{\widetilde{\Omega}\cap\mathcal{A}_{k}}(Y)\leq
N\omega_{\widetilde{\Omega}\cap\mathcal{A}_{k}}^{x_{0}}(\Omega\cap\partial
k^{2}B).$

Also, by the maximum principle and the fact that the annulus
$\mathcal{A}_{k}=k^{2}B\setminus\overline{B}$ when $k>3$ satisfies the CDC
with the precise conditions in Lemma 4.3, we have that the right-hand side of
(4.5) is controlled by

(4.6)
$\omega_{\widetilde{\Omega}\cap\mathcal{A}_{k}}^{x_{0}}(\Omega\cap\partial
k^{2}B)\leq\omega_{\mathcal{A}_{k}}^{x_{0}}(\partial k^{2}B)\leq 1-\tau<1,$

for $\tau\in(0,1)$, depending also on $k$. Indeed, this last step follows by
applying Lemma 4.1(1) to any $y\in\partial B$ with the choice of $s_{0}$ in
Lemma 4.3, because in that case $x_{0}\in\partial kB\subset B(y,s_{0})$ and
$B(y,2s_{0})\cap\partial k^{2}B=\emptyset$.

From 4.5 and 4.6 we obtain
$N-\omega^{x_{0}}_{\widetilde{\Omega}\cap\mathcal{A}_{k}}(Y)\leq
N\omega_{\widetilde{\Omega}\cap\mathcal{A}_{k}}^{p_{0}}(\partial
k^{2}B)\leq(1-\tau)N$, equivalently
$\omega^{x_{0}}_{\widetilde{\Omega}\cap\mathcal{A}_{k}}(Y)\geq\tau
N=\tau\omega^{x_{0}}_{\widetilde{\Omega}}(Y)$. From this and the maximum
principle,

$\tau\omega^{x_{0}}_{\widetilde{\Omega}}(Y)\leq\omega^{x_{0}}_{\widetilde{\Omega}\cap\mathcal{A}_{k}}(Y)\leq\omega^{x_{0}}_{\mathcal{A}_{k}}(Y).$

We have reduced to the elliptic measure in the annulus $\mathcal{A}_{k}$. By
Lemma 4.4 we have $\omega_{\mathcal{A}_{k}}^{x_{0}}(Y)\lesssim\sigma(Y)/r$,
and so
$\omega^{x_{0}}_{\widetilde{\Omega}}(Y)\lesssim\omega_{\mathcal{A}_{k}}^{x_{0}}(Y)\lesssim\frac{\sigma(Y)}{r}$.
Since $x_{0}\in\partial kB$ was chosen to achieve the maximum of
$\omega^{\cdot}_{\widetilde{\Omega}}(Y)$ in $\partial kB$, we obtain that for
any $x\in\widetilde{\Omega}\cap\partial kB$,

$\omega^{x}_{\widetilde{\Omega}}(Y)\leq\omega^{x_{0}}_{\widetilde{\Omega}}(Y)\lesssim\frac{\sigma(Y)}{r}=\frac{\sigma(Y)}{r}\cdot
v_{\Omega\setminus k\overline{B}}(x),$

where the last equality is just because $v_{\Omega\setminus
k\overline{B}}(\xi)=\psi=1$ for $\xi\in\widetilde{\Omega}\cap\partial kB$.
Moreover, $\omega^{x}_{\widetilde{\Omega}}(Y)=0$ if
$x\in\partial\widetilde{\Omega}\setminus kB$. The same inequality follows for
$x\in\widetilde{\Omega}\setminus k\overline{B}$ by the maximum principle in
$\widetilde{\Omega}\setminus k\overline{B}$. Evaluating at the fixed pole
$p\in\Omega\setminus k\overline{B}$, by the choice of $V$ we have

$\omega_{\widetilde{\Omega}}^{p}(Y)\lesssim\frac{\sigma(Y)}{r}\cdot
v_{\Omega\setminus
k\overline{B}}(p)\leq\frac{\sigma(Y)}{r}\cdot\omega_{\Omega\setminus
k\overline{B}}^{p}(V)\leq\frac{\sigma(Y)}{r}\cdot(\omega_{\Omega\setminus
k\overline{B}}^{p}(k\overline{B})+\varepsilon),$

and the lemma follows by taking $\varepsilon\to 0$. ∎

For the sake of completeness here we provide a proof of (4.3).

######  Lemma 4.6.

Let $\Omega\subset{\mathbb{R}}^{n+1}$, $n\geq 1$, be a Wiener regular domain
and $A$ be a real uniformly elliptic (not necessarily symmetric) matrix. Let
$\widetilde{\Omega}\subset\Omega$ be a Wiener regular domain. For any Borel
set $E\subset\partial\Omega\cap\partial\widetilde{\Omega}$ and
$p\in\widetilde{\Omega}$, there holds

$\omega_{\Omega,A}^{p}(E)-\omega_{\widetilde{\Omega},A}^{p}(E)=\int_{\partial\widetilde{\Omega}\setminus\partial\Omega}\omega_{\Omega,A}^{\xi}(E)\,d\omega_{\widetilde{\Omega},A}^{p}(\xi).$

###### Proof.

During the proof we write $\omega=\omega_{\Omega,A}$ and
$\widetilde{\omega}=\omega_{\widetilde{\Omega},A}$.

Fixed $p\in\widetilde{\Omega}$, for $m\geq 1$, by the inner (just for Borel
sets) and outer regularity of Radon measures, let $K_{m}\subset E$ be a
compact set and $U_{m}\supset E$ be an open set such that

(4.7) $\displaystyle\omega^{p}(U_{m})-1/m$
$\displaystyle\leq\omega^{p}(E)\leq\omega^{p}(K_{m})+1/m,\text{ and}$
$\displaystyle\widetilde{\omega}^{p}(U_{m})-1/m$
$\displaystyle\leq\widetilde{\omega}^{p}(E)\leq\widetilde{\omega}^{p}(K_{m})+1/m.$

Moreover, we take $K_{m}\subset K_{m+1}$ and $U_{m}\supset U_{m+1}$ by
redefining the sequences suitably. Finally, take $K\coloneqq\bigcup_{m\geq
1}K_{m}$ and $U\coloneqq\bigcap_{m\geq 1}U_{m}$.

Let $\varphi=\varphi_{m}\in C_{c}(U_{m})$ be such that
$\mathbf{1}_{K_{m}}\leq\varphi\leq\mathbf{1}_{U_{m}}$, and let $u=u_{m}$ and
$\widetilde{u}=\widetilde{u}_{m}$ denote the $L_{A}$-harmonic extension of
$\varphi$ in $\Omega$ and $\widetilde{\Omega}$ respectively. By the
monotonicity of the integral,

(4.8a) $\displaystyle\omega^{\xi}(K_{m})\leq
u(\xi)\leq\omega^{\xi}(U_{m}),\text{ for all }\xi\in\Omega\text{, and}$ (4.8b)
$\displaystyle\widetilde{\omega}^{\xi}(K_{m})\leq\widetilde{u}(\xi)\leq\widetilde{\omega}^{\xi}(U_{m}),\text{
for all }\xi\in\widetilde{\Omega}.$

Using (4.7), we get

(4.9) $|\omega^{p}(E)-\widetilde{\omega}^{p}(E)-(u(p)-\widetilde{u}(p))|\leq
2/m.$

On the other hand, note that
$u(\xi)-\widetilde{u}(\xi)=(u(\xi)-\varphi(\xi))\cdot\mathbf{1}_{\partial\widetilde{\Omega}\setminus\partial\Omega}(\xi)$
if $\xi\in\partial\widetilde{\Omega}$, and in particular, writing
$u-\widetilde{u}$ as the $L_{A}$-harmonic extension in $\widetilde{\Omega}$ of
its boundary values in $\partial\widetilde{\Omega}$ and evaluating at the
point $p\in\Omega$ we get

$u(p)-\widetilde{u}(p)=\int_{\partial\widetilde{\Omega}\setminus\partial\Omega}(u(\xi)-\varphi(\xi))\,d\widetilde{\omega}^{p}(\xi).$

From this, (4.8a) and
$0\leq\int_{\partial\widetilde{\Omega}\setminus\partial\Omega}\varphi(\xi)\,d\widetilde{\omega}^{p}(\xi)\leq\widetilde{\omega}^{p}(U_{m}\setminus
K_{m})\leq 2/m$, we therefore obtain

$\int_{\partial\widetilde{\Omega}\setminus\partial\Omega}\omega^{\xi}(K_{m})\,d\widetilde{\omega}^{p}(\xi)-\frac{2}{m}\leq
u(p)-\widetilde{u}(p)\leq\int_{\partial\widetilde{\Omega}\setminus\partial\Omega}\omega^{\xi}(U_{m})\,d\widetilde{\omega}^{p}(\xi),$

which together with (4.9) gives

$\int_{\partial\widetilde{\Omega}\setminus\partial\Omega}\omega^{\xi}(K_{m})\,d\widetilde{\omega}^{p}(\xi)-\frac{4}{m}\leq\omega^{p}(E)-\widetilde{\omega}^{p}(E)\leq\int_{\partial\widetilde{\Omega}\setminus\partial\Omega}\omega^{\xi}(U_{m})\,d\widetilde{\omega}^{p}(\xi)+\frac{2}{m}.$

Note that the set $K=\bigcup_{m\geq 1}K_{m}\subset E$ satisfies
$\omega^{p}(E)=\omega^{p}(K)$, and since $K$ is measurable, then
$\omega^{p}(E\setminus K)=0$. Since elliptic measures are mutually absolutely
continuous for any two different poles, this implies that
$\omega^{\xi}(E\setminus K)=0$ for any $\xi\in\Omega$. Again, since $K$ is
measurable we obtain that $\omega^{\xi}(E)=\omega^{\xi}(K)$ for any
$\xi\in\Omega$. By the same argument, using now that $E$ is Borel and so
measurable for every $\omega^{\xi}$ with $\xi\in\Omega$, the set
$U=\bigcap_{m\geq 1}U_{m}$ satisfies $\omega^{\xi}(E)=\omega^{\xi}(U)$ for any
$\xi\in\Omega$.

Taking $m\to\infty$, by the monotone convergence theorem, the equation above
becomes

$\int_{\partial\widetilde{\Omega}\setminus\partial\Omega}\lim_{m\to\infty}\omega^{\xi}(K_{m})\,d\widetilde{\omega}^{p}(\xi)\leq\omega^{p}(E)-\widetilde{\omega}^{p}(E)\leq\int_{\partial\widetilde{\Omega}\setminus\partial\Omega}\lim_{m\to\infty}\omega^{\xi}(U_{m})\,d\widetilde{\omega}^{p}(\xi),$

and the lemma follows since
$\lim_{m\to\infty}\omega^{\xi}(K_{m})=\omega^{\xi}(K)=\omega^{\xi}(E)=\omega^{\xi}(U)=\lim_{m\to\infty}\omega^{\xi}(U_{m})$.
∎

##  5\. Proof of the weak version of the Main Lemma

According to the previous reductions in Section 3, to obtain the 3.1 it
suffices to prove Lemma 3.2, which we intend to do in this section modulo the
proof of (1.2) which is deferred to Section 7.

In this section we work with bounded $(\delta,r_{0})$-Reifenberg flat domains
$\Omega\subset{\mathbb{R}}^{2}$. Recall that for $\delta>0$ small enough we
have that $\Omega$ is an NTA domain (see [KT97, Section 3]), and hence it
satisfies the capacity density condition. See Remark 2.5.

###### Proof of Lemma 3.2.

Let $M>0$ be big enough and $0<\rho<1/M$. Denote
$\omega\coloneqq\omega_{\Omega,A}^{p}$.

For $x\in\partial\Omega$ define the ‘high density value’ as

(5.1) $h(x)\coloneqq\sup\\{r\geq\rho:\omega(B(x,r))\geq Mr\\},$

and $h(x)=\rho$ if the supremum runs over an empty set. Note that $\rho\leq
h(x)\leq 1/M$ for every $x\in\partial\Omega$.

######  Definition 5.1 (Good balls).

For $x\in\partial\Omega$, we say that the ball $B(x,r)$ is good,
$B(x,r)\in{\mathsf{Good}}$, if $r>h(x)$, i.e., for any $s\geq r$ we have
$\omega(B(x,s))<Ms$.

For $x\in{\mathbb{R}}^{2}$ define

$d(x)\coloneqq\inf_{B\in{\mathsf{Good}}}\left[|x-c(B)|+r(B)\right],$

where $c(B)$ is the center of the ball and $r(B)$ its radius. For any
$B\in{\mathsf{Good}}$ we have that $r(B)\geq h(c(B))\geq\rho$, which implies
$d(x)\geq\rho$.

Note that for every ball $B(\xi,r)\in{\mathsf{Good}}$ we have $r\geq h(\xi)$.
Therefore

$d(x)=\inf_{\xi\in\partial\Omega}\left[|x-\xi|+h(\xi)\right].$

This function is $1$-Lipschitz as it is the infimum of $1$-Lipschitz
functions.

######  Remark 5.2.

For $x\in\partial\Omega$ it follows $\rho\leq d(x)\leq h(x)$, and hence
$h(x)=\rho$ implies $d(x)=h(x)$. Moreover, if $d(x)=\rho$ then
$x\in\partial\Omega$.

Let $0<\varepsilon<1$ be small enough (to be fixed in (5.8) below) depending
on the ellipticity constant $\lambda$ and the product
$\kappa\|A\|_{L^{\infty}({\mathbb{R}}^{2})}$. Let
$\mathcal{I}=\mathcal{I}_{\varepsilon^{-2}}$ be the family of maximal dyadic
cubes $Q\in\mathcal{D}({\mathbb{R}}^{2})$ such that
$Q\cap\partial\Omega\not=\emptyset$ and $\ell(Q)\leq\varepsilon^{2}d(x)$ for
all $x\in Q$.

######  Lemma 5.3.

Let $\mathcal{I}$ be the family defined above. Then:

  1. (1)

Every $Q\in\mathcal{I}$ satisfy $\frac{\varepsilon^{2}d(x)}{3}<\ell(Q)\leq
2\varepsilon^{2}d(x)$ for all $x\in\frac{\varepsilon^{-2}}{4}Q$.

  2. (2)

If $Q_{1},Q_{2}\in\mathcal{I}$ and
$\frac{\varepsilon^{-2}}{4}Q_{1}\cap\frac{\varepsilon^{-2}}{4}Q_{2}\not=\emptyset$,
then $\frac{\ell(Q_{1})}{6}<\ell(Q_{2})<6\ell(Q_{1})$.

  3. (3)

$\\{\frac{\varepsilon^{-2}}{4}Q\\}_{Q\in\mathcal{I}}$ has finite
superposition, with superposition number $N=N_{\varepsilon}$ depending on
$\varepsilon$ only.

###### Proof.

Let $x\in\frac{\varepsilon^{-2}}{4}Q$. We start by proving $\ell(Q)\leq
2\varepsilon^{2}d(x)$ in (1). Take any $y\in Q$. By the election of $y$ and
since $d$ is $1$-Lipschitz,

$d(x)=d(y)+d(x)-d(y)\geq\varepsilon^{-2}\ell(Q)-|x-y|\geq\varepsilon^{-2}\ell(Q)-{\rm
diam}\frac{\varepsilon^{-2}}{4}Q\geq\frac{\ell(Q)}{2\varepsilon^{2}}.$

For the other inequality in (1), let $\widehat{Q}$ be the dyadic father of
$Q$, i.e., the unique $\widehat{Q}\in{\mathcal{D}}({\mathbb{R}}^{2})$ such
that $Q\subset\widehat{Q}$ and $\ell(\widehat{Q})=2\ell(Q)$. Since $Q$ is
maximal, there exists $y\in\widehat{Q}$ such that
$2\ell(Q)=\ell(\widehat{Q})>\varepsilon^{2}d(y)$, and hence

$d(x)=d(x)-d(y)+d(y)<|x-y|+\frac{2\ell(Q)}{\varepsilon^{2}}\leq{\rm
diam}\frac{\varepsilon^{-2}}{4}Q+\frac{2\ell(Q)}{\varepsilon^{2}}\leq\frac{3\ell(Q)}{\varepsilon^{2}},$

and with this we conclude the proof of (1).

Let $Q_{1},Q_{2}\in\mathcal{I}$ such that
$\frac{\varepsilon^{-2}}{4}Q_{1}\cap\frac{\varepsilon^{-2}}{4}Q_{2}\not=\emptyset$.
Take $x\in\frac{\varepsilon^{-2}}{4}Q_{1}\cap\frac{\varepsilon^{-2}}{4}Q_{2}$,
and then (2) follows from (1) by

$\ell(Q_{1})\leq 2\varepsilon^{2}d(x)<6\ell(Q_{2}).$

Given $Q\in\mathcal{I}$, there is only a finite number $N=N_{\varepsilon}$ of
cubes $P\in{\mathcal{D}}({\mathbb{R}}^{2})$ such that
$\ell(Q)/6<\ell(P)<6\ell(Q)$ and
$\frac{\varepsilon^{-2}}{4}Q\cap\frac{\varepsilon^{-2}}{4}P\not=\emptyset$,
which gives (3). ∎

######  Lemma 5.4.

There exists $\gamma\geq 1$ such that for every $Q\in\mathcal{I}$ there exists
a ball $G_{Q}\in{\mathsf{Good}}$ with
$r(G_{Q})\approx\varepsilon^{-2}\ell(Q)$, satisfying the inclusions
$\varepsilon^{-2}Q\subset\gamma G_{Q}$ and
$G_{Q}\subset\gamma\varepsilon^{-2}Q$.

###### Proof.

Given $Q\in\mathcal{I}$, fix any $x\in Q\cap\partial\Omega$, and take
$\xi\in\partial\Omega$ such that

(5.2) $d(x)\leq|x-\xi|+h(\xi)\leq 1.1d(x).$

Define the ball $G_{Q}\coloneqq B\left(\xi,2(|x-\xi|+h(\xi))\right)$, and
hence $G_{Q}\in{\mathsf{Good}}$, since $r(G_{Q})\geq 2h(\xi)\geq
h(\xi)+\rho>h(\xi)$.

We claim that $r(G_{Q})\approx\varepsilon^{-2}\ell(Q)$. Indeed, from (5.2) and
(1) in Lemma 5.3,

$\frac{1}{2}r(G_{Q})=|x-\xi|+h(\xi)\overset{\text{{(\ref{comparability d and
x+h})}}}{\approx}d(x)\overset{\text{{(\ref{prop1 family
I})}}}{\approx}\varepsilon^{-2}\ell(Q).$

Also, using the previous comparability, the distance between $x$ and $\xi$ is
controlled above by

$|x-\xi|\leq|x-\xi|+h(\xi)=\frac{1}{2}r(G_{Q})\approx\varepsilon^{-2}\ell(Q),$

which implies that there exists a universal constant $\gamma\geq 1$ such that
$\varepsilon^{-2}Q\subset\gamma G_{Q}$ and
$G_{Q}\subset\gamma\varepsilon^{-2}Q$. ∎

For each cube $Q\in\mathcal{I}$, fix a point $z_{Q}\in Q\cap\partial\Omega$
and define $B_{Q}\coloneqq B(z_{Q},r_{Q})$ with $r_{Q}\coloneqq\varepsilon
d(z_{Q})$. Next, we modify the domain as in [Wol93], but using the family
$\\{B_{Q}\\}_{Q\in\mathcal{I}}$. To do so, define

$\widetilde{\Omega}\coloneqq\Omega\setminus\bigcup_{Q\in\mathcal{I}}B_{Q},$

and denote $\widetilde{\omega}\coloneqq\omega_{\widetilde{\Omega},A}^{p}$ its
elliptic measure with pole $p$. From (1) in Lemma 5.3 and $d(\cdot)\geq\rho$
on $\partial\Omega$ we have that $\mathcal{I}$ is finite. In particular, the
family $\\{B_{Q}\\}_{Q\in\mathcal{I}}$ is finite, and
$\partial\widetilde{\Omega}$ is smooth except at finitely many points.

Since the function $d(\cdot)$ is $1$-Lipschitz, and so $\varepsilon d(\cdot)$
is $\varepsilon$-Lipschitz, the same proof as in [AMT16, Lemma 2.2] applies to
obtain the following lemma.

######  Lemma 5.5.

Let $r_{0}\in(0,\infty]$ and let $\varepsilon>0$ be small enough. Then:

  1. (1)

(Analogue of [AMT16, Lemma 2.2]) There exists
$\delta_{0}=\delta_{0}(\varepsilon)>0$ such that if
$\Omega\subset{\mathbb{R}}^{2}$ is $(\delta,r_{0})$-Reifenberg flat with
$0<\delta<\delta_{0}$, then the modified domain $\widetilde{\Omega}$ as above
is $(c\varepsilon^{1/2}$, $r_{0}/2)$-Reifenberg flat.

  2. (2)

(See [AMT16, Lemma 2.3(c)]) For every $Q\in\mathcal{I}$, there exists a
Lipschitz function $f_{Q}:\mathcal{P}(z_{Q},30r(B_{Q}))\cap
10B_{Q}\to\mathcal{P}(z_{Q},30r(B_{Q}))^{\perp}$ with Lipschitz constant at
most $c\varepsilon^{1/2}$.

For any $Q\in\mathcal{I}$, by the maximum principle and Lemma 4.2 (with
$k=10$) respectively,

(5.3)
$\widetilde{\omega}\left(\overline{B_{Q}}\right)\leq\omega_{\Omega\setminus\overline{B_{Q}}}\left(\overline{B_{Q}}\right)\lesssim\omega(10B_{Q}).$

Moreover, for all $z\in\partial\widetilde{\Omega}\cap\partial B_{Q}$, by Lemma
4.5 (with $k=\sqrt{10}$), the maximum principle and Lemma 4.2 (with
$k=\sqrt{10}$) respectively,

(5.4)
$\frac{d\widetilde{\omega}}{d\sigma}(z)\lesssim\frac{\omega_{\widetilde{\Omega}\setminus\overline{\sqrt{10}B_{Q}}}\left(\overline{\sqrt{10}B_{Q}}\right)}{r(B_{Q})}\leq\frac{\omega_{\Omega\setminus\overline{\sqrt{10}B_{Q}}}\left(\overline{\sqrt{10}B_{Q}}\right)}{r(B_{Q})}\lesssim\frac{\omega\left(10B_{Q}\right)}{r(B_{Q})}.$

By the existence of a good ball $G_{Q}\in{\mathsf{Good}}$ with
$r(G_{Q})\approx\varepsilon^{-2}\ell(Q)$ and $\varepsilon^{-2}Q\subset\gamma
G_{Q}$ (see Lemma 5.4), the ball $10B_{Q}$ has bounded density with respect to
the initial elliptic measure:

$\omega\left(10B_{Q}\right)=\omega\left(B(z_{Q},10\varepsilon
d(z_{Q}))\right)\overset{\text{L.\ref{properties family I}{(\ref{prop1 family
I})}}}{\leq}\omega(3\cdot 3\cdot 10\varepsilon^{-1}Q)\overset{\varepsilon\ll
1}{\leq}\omega(\varepsilon^{-2}Q)\leq\omega(\gamma G_{Q})\\\ \overset{\gamma
G_{Q}\in{\mathsf{Good}}}{\leq}\gamma
r(G_{Q})M\approx\gamma\varepsilon^{-2}\ell(Q)M\overset{\text{L.\ref{properties
family I}{(\ref{prop1 family I})}}}{\approx}\gamma
d(z_{Q})M=\gamma\varepsilon^{-1}r(B_{Q})M.$

In particular, combined with (5.4) this implies

(5.5) $\frac{d\widetilde{\omega}}{d\sigma}(z)\lesssim M\quad\text{for all
}z\in\partial\widetilde{\Omega},$

where the involved constant depends on $\gamma$ and $\varepsilon$.

Let us smooth out the domain $\widetilde{\Omega}$. Recall that
$\widetilde{\Omega}$ is smooth except at finitely many points
$\\{\xi_{j}\\}_{j\in J}$, with $\\#J<\infty$ depending on $M$ and $\rho$. Let
$0<\zeta<\min_{Q\in\mathcal{I}}r(B_{Q})/1000$ be a small enough parameter to
be fixed later. For each point $\xi_{j}$, let
$B_{1},B_{2}\in\\{B_{Q}:Q\in\mathcal{I}\\}$ be the two intersecting balls such
that $\xi_{j}\in\partial B_{1}\cap\partial B_{2}$, and let $c_{1}$, $c_{2}$ be
their centers and $r_{1}$, $r_{2}$ be their radii respectively. Take the
unique point $q\in\widetilde{\Omega}\cap\partial
B(c_{1},r_{1}+\zeta)\cap\partial B(c_{2},r_{2}+\zeta)$, and denote
$B_{q}\coloneqq B(q,\zeta)$. The ball $B_{q}$ is tangent to $B_{1}$ and
$B_{2}$, and define $T_{j}$ to be the bounded open region enclosed between
$B_{1}$, $B_{2}$ and $B_{q}$. We define the new smooth domain

${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}={\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}_{\zeta}\coloneqq\widetilde{\Omega}\setminus\bigcup_{j\in
J}\overline{T_{j}},$

taking small enough $\zeta$ such that

  1. (1)

for each $Q\in\mathcal{I}$, $\sigma(\partial
B_{Q}\cap\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}})\geq
0.9\sigma(\partial B_{Q}\cap\partial\widetilde{\Omega})$, and

  2. (2)

${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}$
is a Lipschitz domain with the same Lipschitz character as
$\widetilde{\Omega}$.

Note that
$\sigma(\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}\setminus\partial\widetilde{\Omega})\leq\\#J\cdot
2\pi\zeta$, and so we can take this value to be as small as needed.

Denote
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\omega}$}}}}\coloneqq\omega_{{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}},A}^{p}$.
By the maximum principle,
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\omega}$}}}}(E)\leq\widetilde{\omega}(E)$
for any
$E\subset\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}\cap\partial\widetilde{\Omega}$.
Consequently,

(5.6)
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\omega}$}}}}(\overline{B_{Q}})\leq\widetilde{\omega}(\overline{B_{Q}})\overset{\text{{(\ref{new
old elliptic measure})}}}{\lesssim}\omega(10B_{Q})\quad\text{for any
}Q\in\mathcal{I},$

and

(5.7)
$\frac{d{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\omega}$}}}}}{d\sigma}(z)\leq\frac{d\widetilde{\omega}}{d\sigma}(z)\overset{\text{{(\ref{not
excessive density})}}}{\lesssim}M\quad\text{for all
}z\in\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set<EMAIL_ADDRESS>
Let
$K(\cdot)\coloneqq\frac{d{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\omega}$}}}}}{d\sigma}(\cdot)$
be the Radon-Nikodym derivative. By (2.8) we have

$K(\xi)=-\langle A^{T}(\xi)\nabla g^{T}(\xi),\nu(\xi)\rangle\quad\text{for
}\xi\in\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}},$

where $g^{T}$ is the Green function in
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}$
with respect to the matrix $A^{T}$. By (1.2) (proved in Section 7), if
$\varepsilon>0$ is small enough depending on the ellipticity constant
$\lambda$ and the product $\kappa\|A\|_{L^{\infty}({\mathbb{R}}^{2})}$, then
there is a constant $\textrm{const}_{\Omega,A}>0$ such that

(5.8)
$-\infty<-\textrm{const}_{\Omega,A}\leq\int_{\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}}\log|S(\xi)\nabla
g^{T}(\xi)|\,d{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\omega}$}}}}(\xi)=\int_{\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}}K(\xi)\log|S(\xi)\nabla
g^{T}(\xi)|\,d\sigma(\xi),$

where $A_{0}=\frac{A+A^{T}}{2}$ and $S=A_{0}^{1/2}$, i.e., $S^{T}S=A_{0}$.

For every
$\xi\in\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}$
we can write

$\langle A^{T}(\xi)\nabla g^{T}(\xi),\nu(\xi)\rangle=\langle\nabla
g^{T}(\xi),A(\xi)\nu(\xi)\rangle=\langle\nabla
g^{T}(\xi),c_{1}(\xi)\nu(\xi)\rangle+\langle\nabla
g^{T}(\xi),c_{2}(\xi)t(\xi)\rangle,$

where $\nu$ (resp. $t$) is the outward normal (resp. tangential) vector of
$\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}$,
and $c_{1}(\xi)$ (resp. $c_{2}(\xi)$) is the projection of $A(\xi)\nu(\xi)$
into $\nu(\xi)$ (resp. $t(\xi)$). In particular $c_{1}(\xi)=\langle
A(\xi)\nu(\xi),\nu(\xi)\rangle\approx 1$, and since
$\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}$
is smooth, we have that $\langle\nabla
g^{T}(\xi),c_{2}(\xi)t(\xi)\rangle=c_{2}(\xi)\partial_{t}g^{T}(\xi)=0$ in
$\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}$
as $g^{T}_{|\partial\Omega}=0$. Hence $-\langle A^{T}(\xi)\nabla
g^{T}(\xi),\nu(\xi)\rangle\approx|\nabla g^{T}(\xi)|$, and since $|\nabla
g^{T}(\xi)|^{2}\approx\langle A_{0}\nabla g^{T}(\xi),\nabla
g^{T}(x)\rangle=|S(\xi)\nabla g^{T}(\xi)|^{2}$, we obtain

(5.9) $|S(\xi)\nabla g^{T}(\xi)|\leq-C\langle A^{T}(\xi)\nabla
g^{T}(\xi),\nu(\xi)\rangle=CK(\xi).$

From 5.8 and 5.9,

$\displaystyle-\infty<-\textrm{const}_{\Omega,A}$
$\displaystyle\leq\int_{\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}}K(\xi)\log|S(\xi)\nabla
g^{T}(\xi)|\,d\sigma(\xi)\leq\int_{\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}}K(\xi)\log
CK(\xi)\,d\sigma(\xi)$
$\displaystyle=\int_{\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}}K(\xi)\log
C\,d\sigma(\xi)+\int_{\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}}K(\xi)\log
K(\xi)\,d\sigma(\xi)$ $\displaystyle=\log
C+\int_{\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}}K(\xi)\log
K(\xi)\,d\sigma(\xi),$

whence we obtain

(5.10)
$-\infty<-C_{\Omega,A}\leq\int_{\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}}K(\xi)\log
K(\xi)\,d\sigma(\xi).$

In view of (5.10) and the fact that $K(\cdot)\lesssim M$ on
$\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}\cap\partial\widetilde{\Omega}$
by (5.7), if $M$ is big enough (provided $\zeta$ is small enough depending on
$M$ and $\rho$), then the set of points with very small density has indeed
small elliptic measure. More precisely, we will obtain by the same
computations as [Wol93, p. 170] that

(5.11)
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\omega}$}}}}\left(\left\\{\xi\in\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}\cap\partial\widetilde{\Omega}:K(\xi)\geq
M^{-\tau}\right\\}\right)=\int_{\left\\{\xi\in\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}\cap\partial\widetilde{\Omega}:K(\xi)\geq
M^{-\tau}\right\\}}K(\xi)\,d\sigma(\xi)\geq\frac{\tau}{4}$

whenever $M$ is big enough depending on $\tau$ and the constant
$\textrm{const}_{\Omega,A}$, taking small enough $\zeta$ depending on $M$ and
$\rho$.

###### Proof of (5.11).

First of all we note that, in our setting, $K$ satisfies a reverse Hölder
inequality. As the matrix $A$ is Lipschitz in a neighborhood of
$\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}$
and
${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}$
is a Lipschitz domain with the same Lispchitz character as
$\widetilde{\Omega}$, by [DP22, Theorem 2.3] and the connection between the
solvability of Dirichlet problem and the reverse Hölder inequality, see [DP22,
p. 12], there exists $\beta>0$ such that $K\in\mathrm{RH}_{1+\beta}$. That is,
for all
$\xi\in\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}$
and $0<r\leq 2{\rm
diam}{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}$,

$\left(\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{B(z,r)\cap\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}}K^{1+\beta}\,d\sigma\right)^{1/(1+\beta)}\leq
C_{1+\beta}\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.57222pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.70335pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.59167pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.10834pt}}\\!\int_{B(z,r)\cap\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}}K\,d\sigma.$

We remark that the exponent $\beta$ and the constant $C_{1+\beta}$ depend on
$M$.

Now, we show that it suffices to see

(5.12)
$\int_{\left\\{\xi\in\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}:K(\xi)\geq
M^{-\tau}\right\\}}K\,d\sigma\geq\frac{\tau}{2}.$

Indeed, (5.11) follows from (5.12) as

$\int_{\left\\{\xi\in\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}:K(\xi)\geq
M^{-\tau}\right\\}}K\,d\sigma=\int_{\left\\{\xi\in\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}\cap\partial\widetilde{\Omega}:K(\xi)\geq
M^{-\tau}\right\\}}K\,d\sigma+\int_{\left\\{\xi\in\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}\setminus\partial\widetilde{\Omega}:K(\xi)\geq
M^{-\tau}\right\\}}K\,d\sigma$

and the last term is controlled by

(5.13)
$\displaystyle\int_{\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}\setminus\partial\widetilde{\Omega}}K\,d\sigma$
$\displaystyle\leq\left(\int_{\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}}K^{1+\beta}\,d\sigma\right)^{1/(1+\beta)}\sigma(\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}\setminus\partial\widetilde{\Omega})^{1-1/(1+\beta)}$
$\displaystyle\overset{(\mathrm{RH}_{1+\beta})}{}$
$\displaystyle{\leq}C_{1+\beta}{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\omega}$}}}}(\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}})^{1/(1+\beta)}\sigma(\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}\setminus\partial\widetilde{\Omega})^{1-1/(1+\beta)}\leq
C_{1+\beta}\sigma(\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}\setminus\partial\widetilde{\Omega})^{1-1/(1+\beta)}\leq\frac{\tau}{4},$

where we took $\zeta$ small enough in the last step.

Let us prove (5.12). Let $\log^{+}(\cdot)\coloneqq\max\\{0,\log(\cdot)\\}$ and
$\log^{-}(\cdot)\coloneqq-\min\\{0,\log(\cdot)\\}$. First we bound

(5.14)
$\int_{\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}}K\log^{+}K=\int_{\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}\cap\partial\widetilde{\Omega}}K\log^{+}K+\int_{\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set<EMAIL_ADDRESS>
Writing $K(\cdot)\leq e^{C}M$ on
$\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}\cap\partial\widetilde{\Omega}$,
see (5.7),

(5.15)
$\int_{\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}\cap\partial\widetilde{\Omega}}K\log^{+}K\leq(\log
M+C)\int_{\left\\{\xi\in\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}:K(\xi)\geq
1\right\\}}K\,d\sigma\leq\log
M\int_{\left\\{\xi\in\partial{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\Omega}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\Omega}$}}}}:K(\xi)\geq
1\right\\}}K\,d\sigma+C.$