# Nodal sets of Dirichlet eigenfunctions in quasiconvex Lipschitz domains

Jiuyi Zhu  Department of Mathematics
Louisiana State University
Baton Rouge, LA 70803, USA
Email<EMAIL_ADDRESS>and  Jinping Zhuge  Morningside Center of Mathematics
Academy of Mathematics and Systems Science
Chinese Academy of Sciences
Beijing, China
Email<EMAIL_ADDRESS>
###### Abstract.

We introduce the class of quasiconvex Lipschitz domains, which covers both
$C^{1}$ and convex domains, to the study of boundary unique continuation for
elliptic operators. In particular, we prove the upper bound of the size of
nodal sets for Dirichlet eigenfunctions of general elliptic equations in
bounded quasiconvex Lipschitz domains. Our result is new even for Laplace
operator in convex domains.

###### 2010 Mathematics Subject Classification:

35A02, 35P05, 35J15.

##  1\. Introduction

###  1.1. Motivation

In the study of boundary unique continuation for elliptic equations, there are
two important questions. The first question originated from a conjecture of L.
Bers. Consider a harmonic function $u\in C^{1}(\overline{\mathbb{R}^{d}_{+}})$
in the upper half space $\mathbb{R}^{d}_{+}$. Does $u=\partial_{d}u=0$ on a
subset $U\subset\partial\mathbb{R}^{d}_{+}$ with positive surface measure
imply that $u$ is identically zero in $\mathbb{R}^{d}_{+}$? This conjecture is
true in two dimensions by tools from complex analysis. However, for $d\geq 3$
Bourgain-Wolff [4] constructed a striking counterexample in the class of
$C^{1,\alpha}(\overline{\mathbb{R}^{d}_{+}})$ (with $U$ containing no open
subsets). Given this failure, a variant of the conjecture was asked by Lin
[18]:

Let $u$ be a bounded harmonic function in a Lipschitz domain
$\Omega\subset\mathbb{R}^{d}$. Suppose that $u$ vanishes on a relatively open
set $U\subset\partial\Omega$ and $\nabla u$ vanishes on a subset of $U$ with
positive surface measure. Then $u=0$ in $\Omega$.

The above full conjecture is still open for $d\geq 3$ in general Lipschitz
domains. But many efforts have been made on reducing the geometric condition
of $\partial\Omega$ and obtaining quantitative estimate of the singular sets
on $\partial\Omega$. In [18], Lin proved the conjecture for $C^{1,1}$ domains
by showing that the singular sets have Hausdorff dimension at most $d-2$. This
result was extended to $C^{1,\alpha}$ domains by Adolfsson-Escauriaza [1], and
to $C^{1}$-Dini domains by Kukavica-Nyström [16]. More recently, Tolsa in [24]
proved the above conjecture for $C^{1}$ domains (or Lipschitz domains with
small constant). More quantitative estimate of the $(d-2)$-dimensional
Hausdorff measure of the singular sets was obtained by Kenig-Zhao in [14] for
$C^{1}$-Dini domains. They also showed that the $C^{1}$-Dini type condition is
optimal for singular sets having finite $(d-2)$-dimensional Hausdorff measure
[15]. Along another line, Adolfsson-Escauriaza-Kenig [2] confirmed the
conjecture for arbitrary convex domains. Recently, Mccurdy [22] obtained more
quantitative result in arbitrary convex domains.

The second question is the quantitative estimate of the nodal sets near the
boundary with less regularity. This is particularly directed to Yau’s
conjecture on manifolds, which states the following:

Let $(M,g)$ be a $d$-dimensional smooth compact Riemannian manifold and
$\varphi_{\lambda}$ be the eigenfunction of $-\Delta_{g}$ corresponding to the
eigenvalue $\lambda>0$, i.e.,
$-\Delta_{g}\varphi_{\lambda}=\lambda\varphi_{\lambda}$. Let
$Z(\varphi_{\lambda})=\\{x\in M|\varphi_{\lambda}(x)=0\\}$ be the nodal set of
$\varphi_{\lambda}$. Then
$c\sqrt{\lambda}\leq\mathcal{H}^{d-1}(Z(\varphi_{\lambda}))\leq
C\sqrt{\lambda}$, where $\mathcal{H}^{d-1}$ denotes the $(d-1)$-dimensional
Hausdorff measure.

In Yau’s conjecture, if $M$ is a compact Riemannian manifold with boundary,
then both the regularity of the metric (or the coefficients of the elliptic
equation) and the boundary will play essential roles. Donnelly-Fefferman first
proved Yau’s conjecture for real analytic metric [6] and real analytic
boundary [7]. See also results on the sharp upper bounds of nodal sets for
various eigenfunctions on analytic domains by Lin and the first author [17].
For smooth manifolds without boundaries, Logunov remarkably proved the sharp
lower bound [20] (thus the lower bound has been completely solved even with
boundaries) and a polynomial upper bound [19]. The polynomial upper bound can
be obtained for Laplace eigenfunctions with various boundary conditions in
bounded smooth domains in $\mathbb{R}^{d}$ as mentioned by the first author in
[25]. Recently, Logunov-Malinnikava-Nadirashvili-Nazarov [21] obtained the
sharp upper bound for Laplace eigenfunctions with Dirichlet boundary condition
in $C^{1}$ domains (or Lipschitz domains with small constant). Then Gallegos
[9] extended the result to general second order elliptic operators with
Lipschitz coefficients in $C^{1}$ domains.

Comparing the above two types of problems, we see that the boundary uniqueness
conjecture of Lin holds for both $C^{1}$ and convex domains, while the
quantitative estimates of nodal sets have only been established for $C^{1}$
domains. Therefore, it is natural to ask if the quantitative estimate of nodal
sets can be obtained in arbitrary convex domains. The main contribution of
this paper is to bring quasiconvex Lipschitz domains to the study of boundary
unique continuation. The quasiconvex Lipschitz domain is a natural class of
Lipschitz domains that contains both $C^{1}$ and convex domains, first
introduced by Jia-Li-Wang [10] for studying the global Carldrón-Zygmund
esitmates. It has been used by the second author of this paper to study the
weak maximum priniciple for biharmonic equations [26]. We emphasize that
quasiconvex Lipschitz domains are not necessarily convex or $C^{1}$. In fact,
we can construct a quasiconvex Lipschitz curve that is nowhere convex or
$C^{1}$; see Appendix A.1. In this paper, we obtain quantitative estimates of
the nodal sets in the new setting of quasiconvex Lipschitz domains for general
second order elliptic equations, which particularly include the previous
results of the second question in $C^{1}$ domains [21, 9] as a special case.
Moreover, our result is new (and sharp) even for Laplace operator in convex
domains.

###  1.2. Assumptions and main results

Consider the second order elliptic operator in the form of
$\mathcal{L}=-\nabla\cdot A\nabla$. A function $u\in H^{1}(D)$ will be called
$A$-harmonic in $D$ if $\mathcal{L}(u)=0$ in $D$. Throughout this paper, we
assume that the coefficient matrix $A$ satisfies the following standard
assmptions:

  * •

Ellipticity condition: there exists $\Lambda\geq 1$ such that
$\Lambda^{-1}I\leq A(x)\leq\Lambda I$ for any $x\in\Omega$.

  * •

Symmetry: $A^{T}=A$.

  * •

Lipschitz continuity: there exists $\gamma\geq 0$ such that
$|A(x)-A(y)|\leq\gamma|x-y|$ for every $x,y\in\Omega$.

If one would like to have the sharp upper bound estimate of nodal sets, then
we need to assume additionally that the coefficients are real analytic; see
Remark 6.2 for a detailed explanation.

Next, we introduce the assumption on the domain $\Omega\subset\mathbb{R}^{d}$.
We say a domain $\Omega$ is Lipschitz, if there exists $r_{0}>0$ such that for
every $x_{0}\in\partial\Omega$, the boundary patch $\partial\Omega\cap
B_{r_{0}}(x_{0})$, after a rigid transformation, can be expressed as a
Lipschitz graph $x_{d}=\phi(x^{\prime})$ such that
$B_{r_{0}}(x_{0})\cap\Omega=B_{r}(x_{0})\cap\\{x=(x^{\prime},x_{d}):x_{d}>\phi(x^{\prime})\\}$.
Moreover, the Lipschitz constants of these graphs are uniformly bounded by
some constant $L$. It should be pointed out that the optimal Lipschitz
constant $L$ may depend on $r_{0}$. For example, for $C^{1}$ domains, the
Lipschitz constant can be arbitrarily small if we take $r_{0}\to 0$. However,
for general Lipschitz domains (including convex domains), the Lipschitz
constant may remain large independent of $r_{0}$.

In this paper, we will call $\omega(\rho):(0,\infty)\to[0,\infty)$ a
quasiconvexity modulus, which is a continuous nondecreasing function such that
$\lim_{\rho\to 0}\omega(\rho)=0$.

######  Definition 1.1.

A Lipschitz domain $\Omega$ is called quasiconvex if it satisfies the
following property: there exist a quasiconvexity modulus $\omega(\rho)$ and
$r_{0}>0$ such that for each point $x_{0}\in\partial\Omega$, one can translate
and rotate the domain so that $x_{0}$ is translated to $0$ and the local graph
of $\partial\Omega$, $x_{d}=\phi(x^{\prime})$, satisfies $\phi(0)=0$ and

$\phi(x^{\prime})\geq-|x^{\prime}|\omega(|x^{\prime}|),\quad\text{for all
}x^{\prime}\text{ with }|x^{\prime}|<r_{0}.$ (1.1)

An alternative equivalent definition of quasiconvex domains is as follows: for
each $x_{0}\in\partial\Omega$ and $r<r_{0}$, there exists a unit vector $n$
(coinciding with the outerward normal at $x_{0}$ if exists) such that

$\Omega\cap B_{r}(x_{0})\subset\\{y:(y-x_{0})\cdot n\leq r\omega(r)\\}.$

It is shown in Lemma 2.3 that a quasiconvex Lipschitz domain is locally almost
convex. In particular, if $\Omega$ is convex, then $\omega(r)=0$. If $\Omega$
is a $C^{1}$ domain, then (1.1) should be strengthened to a two-sided
condition

$|x^{\prime}|\omega(|x^{\prime}|)\geq\phi(x^{\prime})\geq-|x^{\prime}|\omega(|x^{\prime}|),$

which clearly implies the local flatness. On the other hand, if
$\omega(\rho)\leq C\rho$, then the quasiconvex Lipschitz domains defined in
Definition 1.1 satisfy the uniform exterior ball condition (which are called
semiconvex domains in some literature, e.g., [23]).

The following is the main result of this paper.

######  Theorem 1.2.

Let $\Omega$ be a bounded quasiconvex Lipschitz domain and $A$ satisfy the
standard assumptions. Let $\varphi_{\lambda}$ be a Dirichlet eigenfunction of
$\mathcal{L}=-\nabla\cdot A\nabla$ corresponding to the eigenvalue $\lambda$,
i.e., $\mathcal{L}(\varphi_{\lambda})=\lambda\varphi_{\lambda}$ in $\Omega$
and $\varphi_{\lambda}\in H_{0}^{1}(\Omega)$. Then there exists $\alpha\geq
1/2$ depending only on $d$ such that

$\mathcal{H}^{d-1}(Z(\varphi_{\lambda}))\leq C\lambda^{\alpha},$ (1.2)

where $C$ depends only on $A$ and $\Omega$. If in addition, $A$ is real
analytic, then (1.2) holds with the sharp exponent $\alpha=1/2$.

The proof of Theorem 1.2 relies on the interior results by Donnelly-Fefferman
[6] for real analytic coefficients and Logunov [19] for Lipschitz
coefficients. We follow the ideas developed recently in [21] that use the
almost monotonicity of doubling index and a reduction to the estimate of nodal
sets for $A$-harmonic functions with controlled doubling index. In order to do
so, we prove the almost monotonicity of doubling index in quasiconvex domains.
There is no obvious evidence showing that this property can be extended to
more general domains. The main novelties in our proof are: (1) reveal a
geometric property of convex domains concerning the existence of
(quantitative) almost flat spots, and apply this property to find large
nonzero portions of $A$-harmonic functions near convex boundaries; (2) develop
a general boundary perturbation argument in Lipschitz domains that allows us
to pass from convex boundaries to quasiconvex boundaries.

The rest of the paper is organized as follows. In section 2, we introduce
useful geometric properties of convex domains and quasiconvex domains. In
section 3, we establish the almost monotonicity of doubling index in
quasiconvex domains. In section 4, we prove the quantitative Cauchy uniqueness
in arbitrary Lipschitz domains and a boundary layer lemma on the drop of large
maximal doubling index. In section 5, we develop a boundary perturbation
argument in finding nonzero portions of $A$-harmonic functions. In section 6,
we estimate the nodal sets and prove the main theorem. Finally, some related
results and useful lemmas are proved in Appendix A.

Acknowledgement. The first author is partially supported by NSF DMS-2154506.
The second author is supported by grants from NSFC and AMSS-CAS.

##  2\. Convex and quasiconvex domains

###  2.1. Almost flat spots of convex functions

In this subsection, we show that the convex boundaries have (quantitative)
almost flat spots at every scale. By a rigid transformation, it suffices to
consider a general convex function $\phi$ defined in
$Q_{1}=(-\frac{1}{2},\frac{1}{2})^{d}\subset\mathbb{R}^{d}$ with bounded
Lipschitz character and $\phi(0)=0$. It has been shown in [8] that if $\phi$
is a convex function, then $\nabla^{2}\phi$ is a matrix-valued nonnegative
Radon measure. In particular, for each $j$, $\partial_{j}^{2}\phi$ is a scalar
nonnegative Radon measure. This property is the essential reason for the
existence of almost flat spots for convex functions. Nevertheless, the results
in this subsection are proved in an elementary way without using measure
theory. We will begin with smooth convex functions.

######  Lemma 2.1.

Let $\phi$ be a smooth convex function in $Q_{1}$ with Lipschitz constant
bounded by $L$. For any $r\in(0,1/2)$, there exists a point $y\in Q_{1/2}$
such that for every $j$,

$\int_{-r}^{r}\partial_{j}^{2}\phi(y+te_{j})dt\leq Cr,$ (2.1)

where $C$ depends only on $d$ and $L$.

###### Proof.

In the proof, we will make sure that the constant $C$ is independent of the
smoothness of $\phi$. Since $\phi$ is convex, we know that each
$\partial_{j}^{2}\phi(x)$ is a nonnegative scalar function on $Q_{1}$ and

$\sum_{j=1}^{d}\int_{Q_{1}}\partial_{j}^{2}\phi(x)dx\leq 2dL.$

The last inequality follows from the fundamental theorem of calculus and the
fact that $|\nabla\phi|\leq L$. Now consider the function

$\psi(x)=\int_{-r}^{r}\sum_{j=1}^{d}\partial_{j}^{2}\phi(x+te_{j})dt.$

By the Fubini theorem and the nonnegativity of $\partial_{j}^{2}\phi$,

$\displaystyle\int_{Q_{1/2}}\psi(x)dx$
$\displaystyle=\int_{-r}^{r}\sum_{j=1}^{d}\int_{Q_{1/2}}\partial_{j}^{2}\phi(x+te_{j})dxdt$
$\displaystyle\leq 2r\sum_{j=1}^{d}\int_{Q_{1}}\partial_{j}^{2}\phi(x)dx\leq
4dLr.$

Due to the continuity of $\psi$, this implies that there exists some $y\in
Q_{1/2}$ such that

$\psi(y)\leq Cr.$

In view of the definition of $\psi$, this gives the desired result. ∎

Let $x_{d}=\phi(x^{\prime})$ be a convex graph (function over
$\mathbb{R}^{d-1}$) in $\mathbb{R}^{d}$. For any
$(x^{\prime}_{0},\phi(x_{0}^{\prime}))$ on the graph, we say an affine
function $x_{d}=P(x^{\prime})$ is a support plane of $\phi$ at
$x^{\prime}_{0}$ if $P(x^{\prime}_{0})=\phi(x^{\prime}_{0})$ and
$\phi(x^{\prime})\geq P(x^{\prime})$ for all $x^{\prime}$. Note that if
$\nabla\phi(x^{\prime}_{0})$ exists, then the support plane is unique and
$\nabla\phi(x^{\prime}_{0})=\nabla P$.

######  Lemma 2.2 (Existence of almost flat spots, quantitative version).

Let $x_{d}=\phi(x^{\prime})$ be a convex function in $Q_{1}$ with Lipschitz
constant $L$. There exists $C=C(d,L)>0$ such that for every $r\in(0,1/2)$, we
can find a point $x_{0}^{\prime}\in\overline{Q_{1/2}}$ and its support plane
$P$ such that

$\sup_{|x^{\prime}-x^{\prime}_{0}|<r}|\phi(x^{\prime})-P(x^{\prime})|\leq
Cr^{2}.$ (2.2)

###### Proof.

Temporarily we assume that $\phi$ is smooth with Lipschitz constant $L$.
First, by Lemma 2.1 we can find a point $x_{0}^{\prime}$ such that (2.1) holds
with $y=x_{0}^{\prime}$. Let $P$ be the support plane of $\phi$ at
$x_{0}^{\prime}$. Note that $\nabla^{2}P=0$. This implies that

$\sup_{t\in(-r,r)}|\partial_{j}\phi(x^{\prime}_{0}+te_{j})-\partial_{j}P|\leq
Cr.$ (2.3)

In fact, since $\phi$ is smooth, then
$\partial_{j}\phi(x^{\prime}_{0})=\partial_{j}P$. Note that
$\phi(x^{\prime}_{0}+se_{j})$ is a 1-dimensional convex function in $s$ and
$x_{d}=P(x^{\prime}_{0}+se_{j})$ is the support line at $s=0$. Observe that
$\partial_{j}\phi(x^{\prime}_{0}+se_{j})$ is nondecreasing in $s$. By the
definition of support line (which could be self-explained), it is easy to see
that for every $s>0$,
$\partial_{j}\phi(x^{\prime}_{0}-se_{j})\leq\partial_{j}P\leq\partial_{j}\phi(x^{\prime}_{0}+se_{j})$.
Now for any $t>0$ with $-r<-t<t<r$, by Lemma 2.1,

$\partial_{j}\phi(x^{\prime}_{0}+te_{j})-\partial_{j}\phi(x^{\prime}_{0}-te_{j})=\int_{-t}^{t}\partial_{j}^{2}\phi(x^{\prime}_{0}+se_{j})ds\leq
Cr.$

As a result,

$|\partial_{j}\phi(x^{\prime}_{0}\pm
te_{j})-\partial_{j}P|\leq\partial_{j}\phi(x^{\prime}_{0}+te_{j})-\partial_{j}\phi(x^{\prime}_{0}-te_{j})\leq
Cr.$

This is exactly (2.3).

Integrating (2.3) in $t$ again, we see that

$\sup_{t\in(-r,r)}|\phi(x^{\prime}_{0}+te_{j})-P(x^{\prime}_{0}+te_{j})|\leq
Cr^{2}.$ (2.4)

By the convexity of $\phi-P$ and the fact $\phi\geq P$, for any
$t_{j}\in(-r,-r)$ with $|t|_{1}:=\sum_{j=1}^{d-1}|t_{j}|<r$, we have

$\displaystyle 0$
$\displaystyle\leq\phi(x^{\prime}_{0}+\sum_{j=1}^{d-1}t_{j}e_{j})-P(x^{\prime}_{0}+\sum_{j=1}^{d-1}t_{j}e_{j})$
$\displaystyle\leq\sum_{j=1}^{d-1}\frac{|t_{j}|}{|t|_{1}}\big{(}\phi(x_{0}^{\prime}+{\rm
sgn}(t_{j})|t|_{1}e_{j})-P(x_{0}^{\prime}+{\rm
sgn}(t_{j})|t|_{1}e_{j})\big{)}$
$\displaystyle\leq\sum_{j=1}^{d-1}\frac{|t_{j}|}{|t|_{1}}Cr^{2}=Cr^{2},$

where ${\rm sgn}(a)=a/|a|$ is the sign function (with ${\rm sgn}(0)=0$) and we
have used (2.4) in the third inequality. This implies the desired result with
possibly a smaller radius $cr$. But this can be fixed by adjusting the initial
radius to $c^{-1}r$.

Finally, we need to pass from smooth convex functions to arbitrary convex
functions. For any given convex function $\phi$ with Lipschitz constant $L$,
it is well-known that we can find a sequence of smooth convex functions
$\\{\phi_{k}:k\geq 1\\}$, with the same Lipschitz constant, that converges
uniformly to the given convex function $\phi$ as $k\to\infty$. Meanwhile, we
can find a sequence of $x^{\prime}_{k}\in\overline{Q_{1/2}}$ and their support
planes $P_{k}$ satisfying (2.2), namely, for each $k\geq 1$,

$\sup_{|x^{\prime}-x_{k}^{\prime}|<r}|\phi_{k}(x^{\prime})-P_{k}(x^{\prime})|\leq
Cr^{2},$ (2.5)

where $C$ depends only on $d$ and $L$. Due to the compactness of
$\overline{Q_{1/2}}$ and the set of support planes (since $|\nabla P_{k}|\leq
L$), we can find subsequences of $x_{k}^{\prime}$ and $P_{k}$ such that
$x_{k}^{\prime}\to x_{0}^{\prime}\in Q_{1/2}$ and $P_{k}$ converges uniformly
to a plane $P_{0}$, which turns out to be a support plane of $\phi$ at
$x^{\prime}_{0}$. Hence, by taking limit in (2.5), we obtain (2.2) for a
general convex function. ∎

###  2.2. Quasiconvex domains

The following lemma shows that a quasiconvex Lipschitz domain can be well
approximated locally by convex domains at all small scales.

######  Lemma 2.3.

Let $\Omega$ be a quasiconvex domain given by Definition 1.1. Let
$x\in\partial\Omega$ and $r<r_{0}/2$. Let $V_{r}(x)$ be the convex hull of
$B_{r,+}(x)=B_{r}(x)\cap\Omega$. Then

${\rm dist}(\partial B_{r,+}(x),\partial V_{r}(x))\leq 2r\omega(2r).$

###### Proof.

Let $y\in\partial B_{r,+}(x)$. If $y\in\partial B_{r}(x){\cap\Omega}$, then
$y\in\partial V_{r}(x)$ is trivial. Suppose $y\in\partial\Omega\cap B_{r}(x)$.
Since $\Omega$ is quasiconvex, by definition, we can find a vector $n$
(coinciding with the outward normal if exists) such that $B_{2r}(y)\cap\Omega$
is entirely contained in the convex domain

$\widetilde{V}_{2r}(y)=\\{z:(y-z)\cdot n\ >-2r\omega(2r)\\}\cap B_{2r}(y).$

This implies $\text{dist}(y,\partial\widetilde{V}_{2r}(y))\leq 2r\omega(2r)$.
Also, observe that $y\in\overline{V_{r}(x)}$ and
$V_{r}(x)\subset\widetilde{V}_{2r}(y)$. This is because
$\widetilde{V}_{2r}(y)$ is a convex set containing $B_{r,+}(x)$ while the
convex hull $V_{r}(x)$ is the smallest convex set containing $B_{r,+}(x)$.
Hence,

$\text{dist}(y,\partial
V_{r}(x))\leq\text{dist}(y,\partial\widetilde{V}_{2r}(y))\leq 2r\omega(2r),$

as desired. ∎

##  3\. Almost monotonicity of doubling index

In this section, we will show that for quasiconvex Lipschitz domains and
variable coefficients, we have the almost monotonicity for the doubling index.
We begin with two important facts.

Fact 1: Affine transformations (a combination of linear transformations and
translations) will not change the quasiconvexity (or convexity) of a domain
(up to a constant on $\omega(r)$).

Fact 2: A rescaling $x\to rx$ for small $r<1$ will make $\omega$ and the
Lipschitz constant of $A$ small. This is because $\omega(r\cdot)\to 0$ and
$\lVert\nabla(A(r\cdot))\rVert_{L^{\infty}}=r\lVert\nabla
A\rVert_{L^{\infty}}\to 0$ as $r\to 0$. This fact is crucial for us as we will
assume $\omega_{0}=\omega(1)$ and $\gamma_{0}\simeq\lVert\nabla
A\rVert_{L^{\infty}}$ are sufficiently small later on. Note that for convex
domains, $\omega_{0}=0$; for constant coefficients, $\gamma_{0}=0$.

###  3.1. Frequency function and three-ball inequality

Let $\Omega$ be a Lipschitz domain and $A$ satisfy the standard assumptions.
In this subsection, we further assume (up to an affine transformation) that
$0\in\overline{\Omega}$ and $A(0)=I$. Note that under this normalization, the
constants for $\Omega$ and $A$ will change. In particular, the new
quasiconvexity modulus $\tilde{\omega}(r)$ will satisfy
$\tilde{\omega}(r)\leq\Lambda\omega(\Lambda^{\frac{1}{2}}r)$ which is harmless
up to a constant. Similarly $\tilde{\gamma}\leq\Lambda\gamma$ and
$\tilde{L}\leq\Lambda L$.

Let $R>0$ and $u$ be an $A$-harmonic function in $B_{R,+}=B_{R}(0)\cap\Omega$
and satisfy the Dirichlet boundary condition $u=0$ on
$B_{R}(0)\cap\partial\Omega$. Define

$\mu(x)=\frac{x\cdot A(x)x}{|x|^{2}}\quad\text{and}\quad H(r)=\int_{\partial
B_{r}(0)\cap\Omega}\mu u^{2}d\sigma.$

Define

$D(r)=\int_{B_{r,+}}A\nabla u\cdot\nabla u.$

Define the frequency function of $u$ centered at $0$ as

$\mathcal{N}_{u}(0,r)=\frac{rD(r)}{H(r)}.$ (3.1)

The following monotonicity property is standard. For the reader’s convenience,
we provide a proof in Appendix A.2.

######  Proposition 3.1.

Under the above assumptions, if in addition,

$n(x)\cdot A(x)x\geq 0,\quad\text{for almost every }x\in
B_{R}(0)\cap\partial\Omega,$ (3.2)

then there exists $C=C(\Lambda,d)>0$ such that $e^{C\gamma
r}\mathcal{N}_{u}(0,r)$ is nondecreasing in $(0,R)$.

We say $B_{R,+}$ is $A$-starshaped with respect to $0$ if (3.2) holds. This
additional geometric assumption on $\partial\Omega$ is necessary in the proof
of Proposition 3.1 and will be removed later.

The relationship between $H(r)$ and the frequency function can be seen from

$|\frac{H^{\prime}(r)}{H(r)}-{\frac{d-1}{r}}-\frac{2}{r}\mathcal{N}_{u}(0,r)|\leq
C\gamma.$ (3.3)

The proof of this estimate is contained in the proof of Proposition 3.1; see
(A.7). Integrating (3.3) over $r\in(r_{1},r_{2})$, we have

$\Big{|}\log\frac{H(r_{2})}{H(r_{1})}-(d-1)\log\frac{r_{2}}{r_{1}}-2\int_{r_{1}}^{r_{2}}\frac{\mathcal{N}_{u}(0,r)}{r}dr\Big{|}\leq
C\gamma r_{2}.$ (3.4)

Using Proposition 3.1, $\mathcal{N}_{u}(0,r)\leq e^{C\gamma
r_{2}}\mathcal{N}_{u}(0,r_{2})\leq e^{C\gamma r_{3}}\mathcal{N}_{u}(0,s)$ for
any $r_{1}<r<r_{2}<s<r_{3}$. Thus, for any $r_{1}<r_{2}<r_{3}<r_{0}$, it
follows from (3.4) that

$\log\frac{H(r_{2})}{H(r_{1})}\leq\beta\log\frac{H(r_{3})}{H(r_{2})}+(d-1)\log\frac{r_{2}^{1+\beta}}{r_{1}r_{3}^{\beta}}+C\gamma
r_{3},$

where $\beta=e^{C\gamma r_{3}}\frac{\log(r_{2}/r_{1})}{\log(r_{3}/r_{2})}$.
Then

$H(r_{2})\leq\exp\Big{(}(d-1)\log\frac{r_{2}^{1+\beta}}{r_{1}r_{3}^{\beta}}+C\gamma
r_{3}\Big{)}H(r_{3})^{\frac{\beta}{1+\beta}}H(r_{1})^{\frac{1}{1+\beta}}.$

This is a three-sphere inequality. We would like to convert this into a three-
ball inequality. The following argument, more or less, should be standard.
Replacing in the above inequality $r_{j}$ with $tr_{j}$ and $t\in(0,1)$, and
then integrating in $t$ on (0,1), we arrive at

$\displaystyle\int_{0}^{1}H(tr_{2})dt$ (3.5)
$\displaystyle\leq\exp\Big{(}(d-1)\log\frac{r_{2}^{1+\beta}}{r_{1}r_{3}^{\beta}}+C\gamma
r_{3}\Big{)}\int_{0}^{1}H(tr_{3})^{\frac{\beta}{1+\beta}}H(tr_{1})^{\frac{1}{1+\beta}}dt$
$\displaystyle\leq\exp\Big{(}(d-1)\log\frac{r_{2}^{1+\beta}}{r_{1}r_{3}^{\beta}}+C\gamma
r_{3}\Big{)}\Big{(}\int_{0}^{1}H(tr_{3})dt\Big{)}^{\frac{\beta}{1+\beta}}\Big{(}\int_{0}^{1}H(tr_{1})dt\Big{)}^{\frac{1}{1+\beta}},$

where we also used the Hölder inequality. Note that

$\int_{B_{r,+}}\mu u^{2}=\int_{0}^{r}H(s)ds=r\int_{0}^{1}H(tr)dt.$

It follows from (3.5) that

$\log\frac{\int_{B_{r_{2},+}}\mu u^{2}}{\int_{B_{r_{1},+}}\mu
u^{2}}\leq\beta\log\frac{\int_{B_{r_{3},+}}\mu u^{2}}{\int_{B_{r_{2},+}}\mu
u^{2}}+d\log\frac{r_{2}^{1+\beta}}{r_{3}^{\beta}r_{1}}+C\gamma r_{3}.$ (3.6)

The last two terms on the right side will be small if $\gamma$ or $r_{0}$ is
sufficiently small and $r_{2}/r_{1}$ is sufficiently close to $r_{3}/r_{2}$.

###  3.2. Non-identity matrix

This subsection is devoted to removing the assumption $A(0)=I$ in order to
define the doubling index at any point. Suppose now $A(0)\neq I$. Since $A(0)$
is symmetric and positive definite, we can write
$A(0)=\mathcal{O}D\mathcal{O}^{T}$, where $\mathcal{O}$ is an (constant)
orthogonal matrix and $D$ is a (constant) diagonal matrix with entries
contained in $[\Lambda^{-1},\Lambda]$. Let
$S=\mathcal{O}{D^{\frac{1}{2}}}\mathcal{O}^{T}$. Thus $A(0)=S^{2}$ (or
formally $S=A^{\frac{1}{2}}(0)$) and $S$ is also symmetric. Let $u$ be
$A$-harmonic in $B_{R}(0)\cap\Omega$. Then we can normalize the matrix $A$
such that $A(0)=I$ by a change of variable $x\to Sx$. Precisely, let

$\tilde{u}(x)=u(Sx)\quad\text{ and }\quad\tilde{A}(x)=S^{-1}A(Sx)S^{-1}.$

Then $\tilde{u}$ is $\tilde{A}$-harmonic in
$S^{-1}(B_{R}\cap\Omega)=S^{-1}(B_{R})\cap S^{-1}(\Omega)$. Obviously now
$\tilde{A}(0)=I$. Note that the $A$-starshape condition
$\tilde{n}(x)\cdot\tilde{A}(x)x\geq 0$ on $S^{-1}(B_{R}\cap\partial\Omega)$ is
equivalent to

$n(x)\cdot A(x)A^{-1}(0)x\geq 0,\quad\text{on }B_{R}\cap\partial\Omega.$ (3.7)

This condition is consistent with (3.2) if $A(0)=I$ and invariant under linear
transformations.

Let $\widetilde{\Omega}=S^{-1}(\Omega)$. We would like to find the right form
of the doubling index under a change of variable. By definition of the weight
$\tilde{\mu}$ corresponding to $\tilde{A}$ and the change of variable $y=Sx$,

$\displaystyle\int_{B_{r}(0)\cap\widetilde{\Omega}}\tilde{\mu}\tilde{u}^{2}$
$\displaystyle=\int_{B_{r}(0)\cap\widetilde{\Omega}}\frac{x\cdot
S^{-1}A(Sx)S^{-1}x}{|x|^{2}}(u(Sx))^{2}dx$ $\displaystyle=|\det
A(0)|^{-\frac{1}{2}}\int_{A^{\frac{1}{2}}(0)(B_{r}(0))\cap{\Omega}}\frac{y\cdot
A(0)^{-1}A(y)A(0)^{-1}y}{y\cdot A(0)^{-1}y}u(y)^{2}dy.$

Therefore, taking translations into consideration, we define

$\mu(x_{0},y)=\frac{(y-x_{0})\cdot
A(x_{0})^{-1}A(y)A(x_{0})^{-1}(y-x_{0})}{(y-x_{0})\cdot
A(x_{0})^{-1}(y-x_{0})},$ $E(x_{0},r)=x_{0}+A^{\frac{1}{2}}(x_{0})(B_{r}(0)),$
(3.8)

and

$J_{u}(x_{0},r)=|\det
A(x_{0})|^{-\frac{1}{2}}\int_{E(x_{0},r)\cap{\Omega}}\mu(x_{0},y)u(y)^{2}dy.$
(3.9)

Note that $\Lambda^{d}\geq\det A(x_{0})\geq\Lambda^{-d}$ for any $x_{0}$ and
$J_{u}(x_{0},r)$ is nondecreasing in $r$. We point out that $J_{u}(x_{0},r)$,
invariant under affine transformations, is the right form of the weighted
$L^{2}$ norm for defining doubling index.

In the next lemma, we show the continuous dependence of $J_{u}(x_{0},r)$ on
both $x_{0}$ and $r$. This will be useful when we shift the center of a
doubling index from one point to a nearby point.

######  Lemma 3.2.

There exists $C=C(\Lambda,d)>0$ such that if $\theta=|x_{0}-x_{1}|<r/C$, then

$(1-C\gamma\theta)J_{u}(x_{1},r-C\theta)\leq
J_{u}(x_{0},r)\leq(1+C\gamma\theta)J_{u}(x_{1},r+C\theta).$ (3.10)

###### Proof.

The proof follows from the Lipschitz continuity of $\mu(x_{0},y)$ and
$A^{\frac{1}{2}}(x_{0})$ in $x_{0}$. Note that

$\mu(x_{0},y)=1+\frac{(y-x_{0})\cdot
A(x_{0})^{-1}(A(y)-A(x_{0}))A(x_{0})^{-1}(y-x_{0})}{(y-x_{0})\cdot
A(x_{0})^{-1}(y-x_{0})}.$

Then fixing $y$, one can show that $|\nabla_{x_{0}}\mu(x_{0},y)|\leq C\gamma.$
Thus,

$|\mu(x_{0},y)-\mu(x_{1},y)|\leq C\gamma\theta\mu(x_{1},y),$

which yields

$\mu(x_{0},y)\leq(1+C\gamma\theta)\mu(x_{1},y),$ (3.11)

provided that $\theta$ is suffciently small so that $C\gamma\theta<1/2$.

On the other hand, by the power series, the unique square root of $A$ can be
written as

$A^{\frac{1}{2}}(x_{0})=\Lambda^{\frac{1}{2}}\sum_{n=0}^{\infty}\binom{1/2}{n}(-1)^{n}\big{(}I-\Lambda^{-1}A(x_{0})\big{)}^{n},$
(3.12)

where $\binom{1/2}{n}$ are the binomial coefficients in the Taylor series
$(1+X)^{1/2}=\sum_{n=0}^{\infty}\binom{1/2}{n}X^{n}$. The power series (3.12)
converges, in view of the Gelfand formula, since the all the eigenvalues of
$I-\Lambda^{-1}A(x_{0})$ are contained in $[0,1-\Lambda^{-2}]$. By taking
derivatives in $x_{0}$, we see that $|\nabla A^{\frac{1}{2}}|\leq C|\nabla
A|\leq C\gamma$, where $C$ depends only on $d$ and $\Lambda$. By choosing
$\gamma$ to be small enough if necessary, we have

$E(x_{0},r)\subset E(x_{1},r(1+C_{0}\gamma\theta)+{C_{1}}\theta)\subset
E(x_{1},r+C\theta).$ (3.13)

Moreover,

$|\det A(x_{0})^{\frac{1}{2}}-\det A(x_{1})^{\frac{1}{2}}|\leq
C\gamma\theta\leq C\gamma\theta\det A(x_{1})^{\frac{1}{2}}.$

This implies

$\det A(x_{0})^{\frac{1}{2}}\leq(1+C\gamma\theta)\det A(x_{1})^{\frac{1}{2}}.$
(3.14)

It follows from (3.11), (3.13) and (3.14) that

$\displaystyle J_{u}(x_{0},r)$ $\displaystyle=|\det
A(x_{0})|^{-\frac{1}{2}}\int_{E(x_{0},r)\cap{\Omega}}\mu(x_{0},y)u(y)^{2}dy$
$\displaystyle\leq(1+C\gamma\theta)^{2}|\det
A(x_{1})|^{-\frac{1}{2}}\int_{E(x_{1},r+C\theta)\cap{\Omega}}\mu(x_{1},y)u(y)^{2}dy$
$\displaystyle\leq(1+C\gamma\theta)J_{u}(x_{1},r+C\theta).$

This proves one side of (3.10) and the other side follows from symmetry of the
inequality. ∎

###  3.3. Remove $A$-starshape condition

In the proof of the monotonicity of frequency function, we require the
$A$-starshape condition (3.7) on the boundary, instead of quasiconvexity. To
remove this geometric condition, the classical way is to introduce a nonlinear
change of variables, see [1] or [14] for example. In this paper, however, we
will use a different approach in which the quasiconvexity of the domain will
play an essential role. The key idea of our approach is that if $\Omega\cap
B_{r}$ is close to be convex, we can recover the $A$-starshape condition if we
shift the center from the boundary to an interior point, which is still
relatively close to the boundary. This allows us to prove an almost
monotonicity formula for the frequency function with small errors.

By a rigid transformation, we may assume $0\in\partial\Omega$ and the local
graph of $B_{r}(0)\cap\partial\Omega$ is given by $x_{d}=\phi(x^{\prime})$
such that $B_{r}(0)\cap\Omega=B_{r}(0)\cap\\{x_{d}>\phi(x^{\prime})\\}$, whose
support plane at $0$ is $x_{d}=0$.

######  Lemma 3.3.

There exists $C=C(L,\Lambda)>0$ such that if $r_{0}$ satisfies
$C({\omega}(2r_{0})+{\gamma}r_{0})\leq 1$, then for every $r\in(0,r_{0})$,
$\Omega\cap B_{r}(x_{1})$ is $A$-starshaped with respect to its center
$x_{1}=C(r{\omega}(2r)+{\gamma}r^{2})e_{d}$, namely,

$n(x)\cdot A(x)A^{-1}(x_{1})(x-x_{1})\geq 0,\quad\text{for almost every
}x\in\partial\Omega\cap B_{r}(x_{1}).$ (3.15)

###### Proof.

Consider a point $x_{0}\in\partial\Omega\cap B_{2r}(0)$ such that the normal
$n=n(x_{0})$ exists. By the definition of the quasiconvexity, the support
plane at $x_{0}$, can be written as $x_{d}=P(x^{\prime})$ whose downward
normal direction is $n$. The length of the projection of vector $x_{0}$ onto
the support plane is $t=|(I-n\otimes n)x_{0}|\leq|x_{0}|$. By the definition
of quasiconvexity centered at $x_{0}$ (up to a rotation), we have

$(x_{0}-0)\cdot n\geq-t{\omega}(t).$ (3.16)

Now, we would like to find $k=k(r)>0$ (independent of $x_{0}$) such that
$x_{1}=ke_{d}$ and

$n\cdot A(x_{0})A^{-1}(x_{1})(x_{0}-x_{1})\geq 0.$ (3.17)

In fact, using (3.16), $A(x_{1})A^{-1}(x_{1})=I$ and
$|A(x_{0})-A(x_{1})|\leq{\gamma}|x_{0}-x_{1}|$, we have

$\displaystyle n\cdot A(x_{0})A^{-1}(x_{1})(x_{0}-x_{1})$ $\displaystyle\geq
n\cdot(x_{0}-x_{1})-{\gamma}\Lambda|x_{0}-x_{1}|^{2}$ (3.18)
$\displaystyle\geq-t{\omega}(t)-kn_{d}-{\gamma}\Lambda(|x_{0}|+k)^{2}$

Since $t\leq|x_{0}|<2r$ and $n_{d}=\frac{-1}{\sqrt{1+|\nabla
P|^{2}}}\leq\frac{-1}{\sqrt{1+{L}^{2}}}\leq\frac{-1}{1+{L}}$, where
${L}\geq|\nabla P|$ is the Lipschitz constant of $\partial\Omega\cap
B_{r_{0}}(0)$, we obtain

$n\cdot
A(x_{0})A^{-1}(x_{1})(x_{0}-x_{1})\geq\frac{k}{1+L}-2r\omega(2r)-\gamma\Lambda(2r+k)^{2}.$
(3.19)

We would like $B_{r}(x_{1})$ to contain a part of the boundary including the
origin, which means $k\leq r$. Note that the right-hand side of (3.18) is
independent of $x_{0}\in\partial\Omega\cap B_{2r}(0)$. Now, choosing

$k={9}\Lambda(1+{L})(r{\omega}(2r)+{\gamma}r^{2}),$

we obtain the desired result (3.17) from (3.19) for almost every point on
$B_{2r}(0)\cap\partial\Omega$. Therefore we pick $C=9\Lambda(1+{L})$ and
$x_{1}=C(r{\omega}(2r)+{\gamma}r^{2})e_{d}$ in the statement of the lemma.
Finally it suffices to make sure that $k\leq r$ and
$B_{r}(x_{1})\cap\partial\Omega\subset B_{2r}(0)\cap\partial\Omega$. Clearly
this is satisfied if $r_{0}$ is small enough such that
$C({\omega}(2r_{0})+{\gamma}r_{0})\leq 1$. ∎

Given $r>0$ and $x_{1}$ as defined in Lemma 3.3 and by (3.6), if
$r_{1}<r_{2}<r_{3}<r<r_{0}$,

$\log\frac{J_{u}(x_{1},r_{2})}{J_{u}(x_{1},r_{1})}\leq\beta\log\frac{J_{u}(x_{1},r_{3})}{J_{u}(x_{1},r_{2})}+d\log\frac{r_{2}^{1+\beta}}{r_{3}^{\beta}r_{1}}+C\gamma
r_{3}.$ (3.20)

Now, we would like to obtain a monotonicity formula of the doubling index
centered at $0$. By Lemma 3.2, for any $s<r/2$ and
$s>\theta(r):=C(r{\omega}(2r)+\gamma r^{2})$,

$(1-\theta(r))J_{u}(0,s-\theta(r))\leq
J_{u}(x_{1},s)\leq(1+\theta(r))J_{u}(0,s+\theta(r)).$ (3.21)

Combining (3.20) and (3.21), we obtain

$\displaystyle\log\frac{J_{u}(0,r_{2}-\theta)}{J_{u}(0,r_{1}+\theta)}$
$\displaystyle\leq\beta\log\frac{J_{u}(0,r_{3}+\theta)}{J_{u}(0,r_{2}-\theta)}$
$\displaystyle\qquad+(1+\beta)\log\frac{1+\theta}{1-\theta}+d\log\frac{r_{2}^{1+\beta}}{r_{3}^{\beta}r_{1}}+C\gamma
r_{3}.$

Without loss of generality, assume that $\theta=\theta(r)<\frac{r}{20}$. Let

$r_{1}=\frac{1}{8}r-\theta,\quad r_{2}=\frac{1}{4}r+\theta,\quad
r_{3}=\frac{1}{2}r-\theta.$

It follows that $|\beta-1|\leq C(\gamma r+\theta/r)\leq C(r+{\omega}(2r))$ and

$(1+\beta)\log\frac{1+\theta}{1-\theta}+d\log\frac{r_{2}^{1+\beta}}{r_{3}^{\beta}r_{1}}+C\gamma
r_{3}\leq C(r+{\omega}(2r)).$

Consequently, we obtain

$\log\frac{J_{u}(0,\frac{r}{4})}{J_{u}(0,\frac{r}{8})}\leq(1+C(\gamma
r+{\omega}(2r)))\log\frac{J_{u}(0,\frac{r}{2})}{J_{u}(0,\frac{r}{4})}+C(\gamma
r+{\omega}(2r)).$

Define the doubling index by

$N_{u}(x_{0},r)=\log\frac{J_{u}(x_{0},2r)}{J_{u}(x_{0},r)}.$

Then we have

######  Lemma 3.4.

Let $x_{0}\in\overline{\Omega}$. Then there exist $C=C(\Lambda,L)>0$ and
$r_{0}=r_{0}(\Lambda,L,\gamma,\omega)>0$ (as in Lemma 3.3) such that for
$r<r_{0}$

$N_{u}(x_{0},r)\leq(1+C(\gamma r+\omega(16r)))N_{u}(x_{0},2r)+C(\gamma
r+\omega(16r)).$ (3.22)

######  Remark 3.5.

For any $\varepsilon>0$, if $r$ is small enough (depending only on
$\Lambda,\gamma,L$ and $\omega$), then

$N_{u}(x_{0},r)\leq(1+\varepsilon)N_{u}(x_{0},2r)+\varepsilon.$

Or equivalently,

$N_{u}(x_{0},r)+1\leq(1+\varepsilon)(N_{u}(x_{0},2r)+1).$

This is a useful version of almost monotonicity for the doubling index.

######  Remark 3.6.

The argument in this section actually gives stronger estimates if provided
stronger assumptions on the domain or the coefficients. For example, if
$\omega(r)$ satisfies a Dini-type condition, then one can show

$\sup_{0<s<r}N_{u}(x_{0},s)\leq C(N_{u}(x_{0},r)+1).$

If $\Omega$ is convex (i.e., $\omega(r)=0$) or $C^{1,1}$ (i.e., $\omega(r)\leq
Cr$), then one can show

$\sup_{0<s<r}N_{u}(x_{0},s)\leq(1+C\gamma r)N_{u}(x_{0},r)+C\gamma r.$

If $A$ is constant (i.e., $\gamma=0$) and $\Omega$ is convex, then the above
inequality recover the precise doubling inequality for harmonic functions over
convex domains, i.e., $N(x_{0},r)$ is nondecreasing in $r$.

##  4\. Drop of maximal doubling index

The main result of this section is Lemma 4.2.

###  4.1. Quantitative Cauchy uniqueness

We first introduce a quantitative Cauchy uniqueness over Lipschitz domains. A
different version of Cauchy uniqueness over Lipschitz boundary was proved in
[3], which unfortunately is not suitable for our purpose as we do not have the
estimate for tangential derivatives on the boundary and our estimate is
$L^{\infty}$-based.

We briefly recall the regularity theory of elliptic equation over Lipschitz
domain; see [11, 13] and references therein. Assume that $A$ is Hölder
continuous and let $u$ be an $A$-harmonic function in $\Omega\cap B_{2}$ and
$u|_{\partial\Omega\cap B_{2}}\in H^{1}(\partial\Omega\cap B_{2})$ in the
sense of trace. Then the nontangential maximal function $(\nabla
u)^{*}|_{\partial\Omega\cap B_{1}}\in L^{2}(\partial\Omega\cap B_{1})$. This
particularly implies that for almost every $x\in\partial\Omega\cap B_{1}$,
$\nabla u(y)\to\nabla u(x)$ as $\Omega\ni y\to x$ nontangentially. Therefore,
$\frac{\partial u}{\partial\nu}=n\cdot A\nabla u$ exists almost everywhere.
This fact will justify the following lemma and its proof.

######  Lemma 4.1.

Let $\Omega$ be a Lipschitz domain and $0\in\partial\Omega$. There exists
$0<\tau=\tau(d,L,A)<1$ such that if $u$ is an $A$-harmonic function in
$B_{1,+}$ satisfying $\lVert u\rVert_{L^{2}(B_{1,+})}=1$ and $|u|+|n\cdot
A\nabla u|\leq\varepsilon\leq 1$ on $B_{1}\cap\partial\Omega$, then

$\lVert u\rVert_{L^{\infty}(B_{1/2,+})}\leq C\varepsilon^{\tau},$

where $C$ depends only on $d,L$ and $A$.

###### Proof.

The proof is inspired by [21]. We first extend the Lipschitz coefficient
matrix $A$ from $B_{1,+}$ to $B_{1}$, still denoted by $A$, such that it still
satisfies the same assumptions as in $B_{1,+}$. Since $B_{1}$ is a smooth
domain, we may construct the Green function in $B_{1}$ such that
$\mathcal{L}(G(x,\cdot))=\delta_{x}$ and $G(x,\cdot)=0$ on $\partial B_{1}$.
It is known (see, e.g., [12]) that $G(x,y)$ satisfies (for $d\geq 3$)

$|G(x,y)|\leq\frac{C}{|x-y|^{d-2}},$ (4.1)

and

$|\nabla_{y}G(x,y)|\leq\min\bigg{\\{}\frac{C}{|x-y|^{d-1}},\frac{C\text{dist}(x,\partial
B_{1})}{|x-y|^{d}}\bigg{\\}}.$ (4.2)

For $d=2$, (4.1) should be replaced by $|G(x,y)|\leq C(1+\ln|x-y|)$.

Let $\Gamma_{1}=\partial\Omega\cap B_{1}$ and $\Gamma_{2}=\partial
B_{1}\cap\Omega$. Now for the $A$-harmonic function $u$ in $B_{1,+}$, by the
Green formula, for $x\in B_{1,+}$

$\displaystyle u(x)$ $\displaystyle=\int_{\partial
B_{1,+}}\bigg{\\{}u\frac{\partial
G}{\partial\nu}(x,\cdot)-G(x,\cdot)\frac{\partial
u}{\partial\nu}\bigg{\\}}d\sigma$
$\displaystyle=\int_{\Gamma_{1}}\bigg{\\{}u\frac{\partial
G}{\partial\nu}(x,\cdot)-G(x,\cdot)\frac{\partial
u}{\partial\nu}\bigg{\\}}d\sigma+\int_{\Gamma_{2}}\bigg{\\{}u\frac{\partial
G}{\partial\nu}(x,\cdot)-G(x,\cdot)\frac{\partial
u}{\partial\nu}\bigg{\\}}d\sigma$ $\displaystyle=:u_{1}(x)+u_{2}(x).$

Note that the above calculation also makes sense for $x\in
B_{1,-}=B_{1}\setminus\overline{B_{1,+}}$ and $u(x)=0$ for $x\in B_{1,-}$.
Moreover, $u_{1}$ is $A$-harmonic in $B_{1}\setminus\Gamma_{1}$ with a jump
across $\Gamma_{1}$ (this is a well-known fact in layer potential theory),
while $u_{2}$ is $A$-harmonic in the entire $B_{1}$.

We now estimate $u_{1}$ and $u_{2}$ separately. For $u_{1}(x)$ with $x\in
B_{1}\setminus\Gamma_{1}$, by the assumption

$|u_{1}(x)|\leq\varepsilon\int_{\Gamma_{1}}|G(x,y)|+|\frac{\partial
G}{\partial\nu}(x,y)|d\sigma_{y}.$

It follows from (4.1) and the first part of (4.2) that

$\int_{\Gamma_{1}}|G(x,y)|d\sigma_{y}\leq
C,\quad\text{and}\quad\int_{\Gamma_{1}}|\nabla_{y}G(x,y)|d\sigma_{y}\leq
C(1+|\ln\delta(x)|),$

where $\delta(x)=\text{dist}(x,\Gamma_{1})$. Consequently, we obtain

$|u_{1}(x)|\leq C{\varepsilon}(1+|\ln\delta(x)|),\quad\text{for any }x\in
B_{1}\setminus\Gamma_{1},$

which leads to

$\lVert u_{1}\rVert_{L^{2}(B_{1,\pm})}\leq C\varepsilon,$ (4.3)

where $B_{1,-}=B_{1}\setminus\overline{B_{1,+}}$.

For $u_{2}$, since $u=u_{1}+u_{2}=0$ in $B_{1,-}$, we have

$\lVert u_{2}\rVert_{L^{2}(B_{1,-})}\leq
C\varepsilon,\quad\text{and}\quad\lVert u_{2}\rVert_{L^{2}(B_{1,+})}\leq\lVert
u\rVert_{L^{2}(B_{1,+})}+C\varepsilon.$

Recall that $u_{2}$ is $A$-harmonic in $B_{1}$. By the three-ball inequality
in $B_{1}$, there exists $\tau$ such that

$\displaystyle\lVert u_{2}\rVert_{L^{2}(B_{3/4})}$ $\displaystyle\leq C\lVert
u_{2}\rVert_{L^{2}(B_{1/2,-})}^{\tau}\lVert
u_{2}\rVert_{L^{2}(B_{1})}^{1-\tau}$ $\displaystyle\leq
C\varepsilon^{\tau}(\lVert
u\rVert_{L^{2}(B_{1,+})}^{1-\tau}+\varepsilon^{1-\tau})$ $\displaystyle\leq
C\varepsilon+C\varepsilon^{\tau}\lVert u\rVert_{L^{2}(B_{1,+})}^{1-\tau}.$

Combined with (4.3) and $\lVert u\rVert_{L^{2}(B_{1,+})}=1$, we obtain

$\lVert u\rVert_{L^{2}(B_{3/4,+})}\leq C\varepsilon^{\tau}.$

Finally, the De Giorgi-Nash estimate implies the desired estimate. ∎

###  4.2. Domain decomposition

Let $L\geq 0$ be the Lipschitz constant associated to the Lipschitz domain
$\Omega$. Define a standard rectangular cuboid by
$Q_{0}=[-\frac{1}{2},\frac{1}{2})^{d-1}\times\\{\sqrt{d-1}(1+L)[-1,1)\\}$.
Throughout this paper, all the cuboids considered are translated and rescaled
versions of $Q_{0}$. Let $\pi(Q)$ be the vertical projection of $Q$ on
$\mathbb{R}^{d-1}$. For example,
$\pi(Q_{0})=[-\frac{1}{2},\frac{1}{2})^{d-1}$. Denote the side length of
$Q_{0}$ and $\pi(Q_{0})$ by $s(Q_{0})=s(\pi(Q_{0}))=1$.

Let $0\in\partial\Omega$ and $Q$ be a cuboid centered at $0$. Assume that
$\partial\Omega\cap Q$ is given by the local graph $x_{d}=\phi(x^{\prime})$
with $x^{\prime}\in Q^{\prime}=\pi(Q)$. By our assumption, $|\nabla\phi|\leq
L$ in $Q^{\prime}$. Let $k\geq 3$. We partition $Q^{\prime}$ into $2^{k(d-1)}$
small equal cubes $\\{q^{\prime}\\}$ with side length
$s(q^{\prime})=2^{-k}s(Q^{\prime})$. For each small cube $q^{\prime}$ as
above, we further cover $\pi^{-1}(q^{\prime})\cap Q$ by at most $2^{k}+1$
disjoint cuboids $\\{q\\}$ similar to $Q_{0}$ such that $s(q)=s(q^{\prime})$
and one of these cuboids centers on $\partial\Omega\cap Q$. Let
$\mathcal{B}_{k}(Q)$ be the collection of boundary cuboids $q$ that centered
on $\partial\Omega\cap Q$ and $\mathcal{I}_{k}(Q)$ be the interior cuboids
that intersect with $\Omega\cap Q$. Clearly, there are exactly $2^{k(d-1)}$
boundary cuboids in $\mathcal{B}_{k}(Q)$. See Figure 1 for a demonstration.

Figure 1. Standard domain decomposition

There is a simple fact about the above decomposition:

$\min_{q\in\mathcal{I}_{k}(Q)}\text{dist}(q,Q\cap\partial\Omega)\geq\frac{1}{2}\sqrt{d-1}\cdot
s(q).$

This implies that a thin boundary layer of $Q\cap\partial\Omega$ with width
$\frac{1}{2}\sqrt{d-1}\cdot s(q)$ is entirely contained in
$\cup_{q\in\mathcal{B}_{k}(Q)}q$.

###  4.3. A boundary layer lemma

Let $Q$ be a cuboid centered at a point in $\overline{\Omega}$. Denote by
$\ell(Q)$ the diameter of $Q$ (i.e., length of diagonal). Note that if $x_{Q}$
is the center of $Q$, then $2Q\subset B_{\ell(Q)}(x_{Q})$. As in [21], we
define the maximal doubling index of $u$ in $Q$ by

$N_{u}^{*}(Q)=\sup_{x\in Q\cap\overline{\Omega},\ell(Q)/2\leq
r\leq\ell(Q)}N_{u}(x,r).$

The following is the key lemma of this section which shows that the maximal
doubling index drops when going down to some smaller boundary cuboids.

######  Lemma 4.2.

Let $\Omega$ be a Lipschitz domain whose local boundary near
$0\in\partial\Omega$ is given by $x_{d}=\phi(x^{\prime})$ as in Section 4.2
and $|\nabla\phi|\leq L$. There exist $k_{0}\geq 5$ and $N_{0}\geq 1$ such
that for any integer $k\geq k_{0}$, there exist $\gamma_{0}>0$ and
$\omega_{0}>0$ (both only depending on $k,d$ and $L$) such that the following
statement holds. Suppose $\frac{1}{2}I\leq A\leq 2I,|\nabla A|\leq\gamma_{0}$,
$\omega(1)\leq\omega_{0}$ and $u$ is an $A$-harmonic function in $B_{1,+}$
with $u=0$ on $B_{1}\cap\partial\Omega$. Let $Q\subset B_{1/64}$ be a boundary
cuboid centered on $\partial\Omega$. If $N_{u}^{*}(Q)>N_{0}$, then there
exists $q\in\mathcal{B}_{k}(Q)$ such that
$N_{u}^{*}(q)\leq\frac{1}{2}N_{u}^{*}(Q)$.

###### Proof.

Let $\ell_{Q}=\ell(Q)$ and $x_{Q}$ be the center of $Q$ on $\partial\Omega$.
Given $k\geq 5$, let $\mathcal{B}_{k}(Q)$ be the set of boundary cuboids in
the standard decomposition of $Q$ as in Section 4.2. Then
$\ell_{q}:=\ell(q)=2^{-k}\ell_{Q}$. Put $N=N^{*}_{u}(Q)$ and
$M^{2}=J_{u}(x_{Q},\ell_{Q})$.

We prove by contradiction. Assume for any $q\in\mathcal{B}_{k}(Q)$,
$N_{u}^{*}(q)>N/2$. This means that for any $q\in\mathcal{B}_{k}(Q)$, there
exists $x_{q}\in q$ and $r_{q}\in[\ell_{q}/2,\ell_{q}]$ such that
$N_{u}(x_{q},r_{q})>N/2$. For the given $k$, we may let
$\varepsilon=\varepsilon(k)$ be sufficiently small so that
$(1+\varepsilon)^{k}\leq 2$. By the almost monotonicity in Lemma 3.4 and
Remark 3.5, for such $\varepsilon>0$, by letting $\gamma_{0}$ and $\omega_{0}$
be small enough, we have for all $1\leq j\leq k$,

$N_{u}(x_{q},r_{q})+1\leq(1+\varepsilon)^{j}(N_{u}(x_{q},2^{j}r_{q})+1)\leq
2(N_{u}(x_{q},2^{j}r_{q})+1).$

As a result,

$N_{u}(x_{q},2^{j}r_{q})>\frac{N-2}{4},\quad\text{for }1\leq j\leq k.$

Thus,

$\displaystyle J_{u}(x_{q},4\ell_{q})\leq J_{u}(x_{q},8r_{q})$
$\displaystyle=J_{u}(x_{q},2^{k-1}r_{q})\exp\Big{(}{-\sum_{j=3}^{k-2}N_{u}(x_{q},2^{j}r_{q})}\Big{)}$
$\displaystyle\leq J_{u}(x_{q},\ell_{Q}/2)e^{-(N-2)(k-4)/4}$
$\displaystyle\leq 2M^{2}e^{-(N-2)(k-4)/4},$

where the last inequality follows from Lemma 3.2. In view of the definition of
$J_{u}$ in (3.9) and (3.8), as well as our assumption $\frac{1}{2}I\leq A\leq
2I$, we have

$\int_{B_{2\ell_{q},+}(x_{q})}u^{2}\leq CM^{2}e^{-Nk/5},\quad\text{for each
}q\in\mathcal{B}_{k}(Q),$ (4.4)

where we have chosen $k_{0},N_{0}$ large such that $(N-2)(k-4)/4\geq Nk/5$ for
all $k\geq k_{0},N\geq N_{0}$.

Now, let
$\Gamma=Q\cap\partial\Omega=\\{x=(x^{\prime},x_{d}):x^{\prime}\in\pi(Q),x_{d}=\phi(x^{\prime})\\}$.
Then (4.4) implies

$\sup_{y\in\Gamma}\sup_{B_{\ell_{q}}(y)}|u|\leq CM\ell_{q}^{-d/2}e^{-Nk/10}.$
(4.5)

Define
$\Gamma_{1}=\Gamma+\frac{1}{2}\ell_{q}e_{d}=\\{x=(x^{\prime},x_{d}):x^{\prime}\in\pi(Q),x_{d}=\phi(x^{\prime})+\frac{1}{2}\ell_{q}\\}$.
Note that the distance from $\Gamma_{1}$ to $\partial\Omega$ is comparable to
$\ell_{q}$. By the interior gradient estimate and (4.5), we have

$\sup_{\Gamma_{1}}|u|+\ell_{q}\sup_{\Gamma_{1}}|\nabla u|\leq
CM\ell_{q}^{-d/2}e^{-Nk/10}.$ (4.6)

Let $\hat{x}_{Q}=x_{Q}+\frac{1}{2}\ell_{q}e_{d}\in\Gamma_{1}$ and
$\hat{B}_{r,+}(\hat{x}_{Q})=B_{r,+}(x_{Q})+\frac{1}{2}\ell_{q}e_{d}=\\{x=(x^{\prime},x_{d}):|x-\hat{x}_{Q}|<r,\text{
and }x_{d}>\phi(x^{\prime})+\frac{1}{2}\ell_{q}\\}$. Therefore, applying Lemma
4.1 in $\hat{B}_{s(Q)/2,+}(\hat{x}_{Q})$ with Lipschitz boundary $\Gamma_{1}$,
we obtain from (4.6) that

$\sup_{\hat{B}_{s(Q)/4,+}(\hat{x}_{Q})}|u|\leq CMs(Q)^{-d/2}2^{\tau
kd/2+k}e^{-\tau Nk/10},$ (4.7)

where we also used the fact $s(Q)=C(d,L)2^{k}\ell_{q}$. Combined with (4.5)
again, (4.7) yields

$J_{u}(x_{Q},s(Q)/8)\leq C\int_{{B}_{s(Q)/4,+}({x}_{Q})}u^{2}\leq
CM^{2}2^{\tau kd+2k}e^{-\tau Nk/5}.$ (4.8)

Finally, we relate $J_{u}(x_{Q},s(Q)/8)$ to $J_{u}(x_{Q},\ell_{Q})$ by a
sequence of doubling inequalities. Let $m$ be the smallest integer such that
$2^{m}s(Q)/8>\ell_{Q}$. Note that $m$ depends only on $d$ and $L$. By the
almost monotonicity of the doubling index, for $1\leq j\leq m$,

$N_{u}(x_{Q},2^{-j}\ell_{Q})\leq 2N_{u}(x_{Q},\ell_{Q})\leq 2N.$

Thus

$\displaystyle J_{u}(x_{Q},s(Q)/8)$ $\displaystyle\geq
J_{u}(x_{Q},2^{-m}\ell_{Q})$ (4.9)
$\displaystyle=J_{u}(x_{Q},\ell_{Q})\exp(-\sum_{j=1}^{m}N_{u}(x_{Q},2^{-j}\ell_{Q}))$
$\displaystyle\geq M^{2}e^{-2mN}.$

Comparing (4.8) and (4.9), we obtain

$(\frac{1}{5}-\frac{2m}{k})N\leq\frac{C}{k}+(\tau d+1)\log 2.$

Clearly, this is a contradiction if $k\geq k_{0}$ and $N\geq N_{0}$ for
$k_{0}$ and $N_{0}$ sufficiently large. Note that $k_{0}$ and $N_{0}$ depend
only on $d$ and $L$. ∎

######  Remark 4.3.

In Lemma 4.2 and several lemmas in the rest of the paper, we assume
$\frac{1}{2}I\leq A\leq 2I$. This assumption is only for technical convenience
and can be achieved by a linear transformation in a neighborhood of
$0\in\partial\Omega$.

##  5\. Absence of nodal points

In this section, we will establish more boundary layer lemmas that show the
absence of zeros near relatively large portion of the boundary for small
maximal doubling index. The existence of flat spots for convex domains will be
crucial in this section.

For a point $y$ in $\Omega$, denote by $\delta(y)$ the distance from $y$ to
$\partial\Omega$. The following boundary layer lemma rules out the possibility
of concentration of zeros near any Lipschitz boundary.

######  Lemma 5.1.

Let $\Omega$ be a Lipschitz domain (with Lipschitz constant $L$ and
$\ell=\frac{1}{1+L}$) and $0\in\partial\Omega$. Let $u$ be $A$-harmonic in
$B_{2,+}=\Omega\cap B_{2}$ and $u=0$ on $\partial\Omega\cap B_{2}$. Suppose
for any $B$ centered in $\overline{B_{2,+}}$ with $2B_{+}\subset B_{2,+}$ and
$\text{diam}(B)\geq\frac{\ell}{100}$,

$\frac{\int_{2B_{+}}u^{2}}{\int_{B_{+}}u^{2}}\leq M.$ (5.1)

Assume that $u$ has zeros in $B_{1/16,+}=\Omega\cap B_{1/16}$. Then $u$ has at
least one zero in $B_{1/4,+}\cap\\{\delta(y)>\rho\\}$, where $\rho$ is a
positive constant depending only on $A,M$ and $L$.

###### Proof.

Assume $\int_{B_{1/2,+}}|u|^{2}=1$. Let $\rho\in(0,\ell/100)$. Suppose that
$u$ has no zero in $B_{1/4,+}\cap\\{\delta(y)>\rho\\}$. Without loss of
generalilty, assume $u>0$ in $B_{1/4,+}\cap\\{\delta(y)>\rho\\}$ (otherwise,
replace $u$ by $-u$). We show that if $\rho$ is sufficiently small, then $u$
has no zeros in $B_{1/16,+}$, which is a contradication.

By the local $L^{\infty}$ estimate, we know $|u|\leq C_{0}$ in $B_{1/4,+}$.
Let $z_{0}$ be a point on $\partial B_{1/8,+}\cap\Omega$ and
$\delta(z_{0})\geq\frac{\ell}{16}$ (The existence of such $z_{0}$ follows from
a simple geometric observation). By our assumption that $u>0$ in
$B_{1/4,+}\cap\\{\delta(y)>\rho\\}$ and the fact $B_{\ell/32}(z_{0})\subset
B_{1/4,+}\cap\\{\delta(y)>\rho\\}$, we can apply the Hanack inequality to
obtain

$\inf_{y\in B_{\ell/64}(z_{0})}u(y)\geq
c\bigg{(}\fint_{B_{\ell/32}(z_{0})}u^{2}\bigg{)}^{1/2}\geq c_{M},$

where the last inequality follows from (5.1) and our assumption
$\int_{B_{1/2,+}}|u|^{2}=1$. Note that $B_{\ell/64}(z_{0})\cap B_{1/8,+}$
contains a surface ball of $\partial B_{1/8,+}\cap\Omega$ with positive
measure greater than $c=c(d,L)>0$.

Now we construct a barrier function $v$ in $B_{1/8,+}$ such that $v$ is
$A$-harmonic in $B_{1/8,+}$, $v=c_{M}$ on $\partial B_{1/8,+}\cap
B_{\ell/64}(z_{0})$, $v=-C_{0}$ on $\partial
B_{1/8,+}\cap\\{0<\delta(y)<\rho\\}$ and $v=0$ otherwise on $\partial
B_{1/8,+}$. By the maximum principle, we see that $u\geq v$ in $B_{1/8,+}$. We
show that $v>0$ in $B_{1/16,+}\subset B_{1/8,+}$, provided that $\rho$ is
small enough.

Note that $B_{1/8,+}$ is a Lipschitz domain. Using the elliptic measure
$\omega^{x}_{A}$, we can express $v$ as

$v(x)=c_{M}v_{1}(x)-C_{0}v_{2}(x)\quad\text{for }x\in B_{1/8,+},$

where $v_{1}(x)=\omega_{A}^{x}(\partial B_{1/8,+}\cap B_{\ell/64}(z_{0}))$ and
$v_{2}(x)=\omega_{A}^{x}(\partial B_{1/8,+}\cap\\{0<\delta(y)<\rho\\})$. We
refer to [11] for the definition and properties of elliptic measures. Observe
that both $v_{1},v_{2}$ are positive $A$-harmonic functions. Let $z_{1}$ be an
interior point of $B_{1/8,+}$ such that $\text{dist}(z_{1},\partial
B_{1/8,+})\geq\ell/100$. Then by the estimate of $A$-harmonic measure (and
Harnack inequality, if necessary),

$v_{1}(z_{1})=\omega_{A}^{z_{1}}(\partial B_{1/8,+}\cap
B_{\ell/64}(z_{0}))\geq c_{1},$

for some positive constant $c_{1}$ depending only on $d$ and $L$; see [11,
Lemma 1.3.2]. On the other hand, the $L^{2}$ estimate of the nontangential
maximal function $(v_{2})^{*}$ (see, e.g., [13, Theorem 1.1] with
$\varepsilon=1$) implies that

$\displaystyle\lVert v_{2}\rVert_{L^{2}(B_{1/8,+})}$ $\displaystyle\leq
C\lVert(v_{2})^{*}\rVert_{L^{2}(\partial B_{1/8,+})}{\leq C\lVert
v_{2}\rVert_{L^{2}(\partial B_{1/8,+})}}$ $\displaystyle=C|\partial
B_{1/8,+}\cap\\{0<\delta(y)<\rho\\}|^{\frac{1}{2}}\leq
C_{1}\rho^{\frac{1}{2}},$

where $C_{1}$ depends only on $d$ and $L$. By the local $L^{\infty}$ estimate,
this implies that $0<v_{2}(z_{1})\leq C_{2}\rho^{\frac{1}{2}}$. Then it
follows from the comparison principle for positive solutions (see, e.g., [11,
Lemma 1.3.7]) that

$\frac{v_{2}(x)}{v_{1}(x)}\leq
C_{3}\frac{v_{2}(z_{1})}{v_{1}(z_{1})}\leq\frac{C_{2}C_{3}\rho^{\frac{1}{2}}}{c_{1}}\quad\text{for
all }x\in B_{1/16,+}.$

Now if $\rho$ is sufficiently small (depending on $A,M$ and $L$) such that

$c_{M}-\frac{C_{0}C_{2}C_{3}\rho^{\frac{1}{2}}}{c_{1}}>0,$

then $v=c_{M}v_{1}-C_{0}v_{2}>0$ in $B_{1/16,+}$. This implies $u>0$ in
$B_{1/16,+}$ and contradicts to the assumption that $u$ has zeros in
$B_{1/16,+}$. ∎

######  Lemma 5.2.

Let $\Omega$ be a convex domain (with Lipschitz constant $L$) and
$0\in\partial\Omega$. Assume $\frac{1}{2}I\leq A\leq 2I$. Let $u$ be
$A$-harmonic in $B_{1,+}$ and $u=0$ on $\partial\Omega\cap B_{1}$. Assume

$\qquad N_{u}(0,\frac{1}{4})\leq N.$ (5.2)

Then there exist $\gamma_{0}=\gamma_{0}(d,L)>0,\mu=\mu(d,L,N)>0$,
$\alpha=\alpha(L)\geq 1$ and $z\in\partial\Omega\cap B_{1/80}$ such that if
$|\nabla A|\leq\gamma_{0}$, then

$|u(x)|\geq c\delta(x)^{\alpha}\lVert
u\rVert_{L^{2}(B_{1/4,+})},\quad\text{for all }x\in B_{\mu,+}(z),$

where $c$ depends only on $d,L$ and $N$.

###### Proof.

Recall that for convex domains, $\omega=0$. By Lemma 3.4, we see that the
almost monotonicity formula (3.22) holds for $r_{0}=1$ if $\gamma_{0}$ is
small enough. By an iteration, this implies that for integers $0\leq k\leq 10$

$N_{u}(0,2^{-k-2})\leq N_{1},$

with $N_{1}$ depending on $N$. By a standard argument, this further implies

$N_{u}(x,r)\leq N_{2},$

for all $x\in B_{1/100,+}$ and $r\in(0,1/25)$, which is equivalent to

$\frac{J_{u}(x,2r)}{J_{u}(x,r)}\leq e^{N_{2}}.$

In view of the definition of $J_{u}$, if $\frac{1}{2}I\leq A\leq 2I$, we have
$B_{t/2}(x)\subset E(x,t)\subset B_{2t}(x)$. Consequently, for any $x\in
B_{1/100,+}$ and $t\in(0,1/50)$

$\frac{\int_{B_{2t}(x)}u^{2}}{\int_{B_{t}(x)}u^{2}}\leq
C\frac{J_{u}(x,4t)}{J_{u}(x,2t)}\frac{J_{u}(x,2t)}{J_{u}(x,t)}\frac{J_{u}(x,t)}{J_{u}(x,t/2)}\leq
Ce^{3N_{2}}.$ (5.3)

Let $r\in(0,1/1000)$ (to be determined later). By Lemma 2.2, we can find a
boundary point $z_{r}\in\partial\Omega\cap B_{1/100}$ such that the distance
from $\partial\Omega\cap B_{r}(z_{r})$ to its support plane
$x_{d}=P(x^{\prime})$ at $z_{r}$ is at most $Cr^{2}$. It is crucial to point
out that $C$ is independent of $r$.

Let $D_{r}(z_{r})=B_{r}(z_{r})\cap\\{x_{d}>P(x^{\prime})\\}$ and
$B_{r,+}(z_{r})=\Omega\cap B_{r}(z_{r})$. Notice that both $D_{r}(z_{r})$ and
$B_{r,+}(z_{r})$ are convex, $B_{r,+}(z_{r})\subset D_{r}(z_{r})$ and
$D_{r}(z_{r})\setminus B_{r,+}(z_{r})$ is a thin region. Let $v$ be an
$A$-harmonic function in $D_{r}(z_{r})$ such that $v=u$ on $\\{\partial
B_{r,+}(z_{r})=\partial D_{r}(z_{r})\\}$ and $v=0$ on $\partial
D_{r}(z_{r})\setminus\partial B_{r,+}(z_{r})$.

Without loss of generality, assume $r^{-d/2}\lVert
u\rVert_{L^{2}(B_{2r,+}(z_{r}))}=1$. The gradient estimate in convex domain
$B_{2r,+}(z_{r})$ implies that $\lVert\nabla
u\rVert_{L^{\infty}(B_{r,+}(z_{r}))}\leq C_{1}r^{-1}$. This can be shown by a
standard barrier argument; see Lemma A.1 or [5, Theorem 1.4] for a related
result. Thus, $|u(x)|\leq C_{1}\delta(x)r^{-1}$ for any $x\in B_{r,+}(z_{r})$.
Let $\tilde{\delta}(x)$ be the distance from $x$ to the support plane
$x_{d}=P(x^{\prime})$. Then Lemma A.5 implies that $|v(x)|\leq
C_{2}\tilde{\delta}(x)r^{-1}$ in $D_{r,+}(z_{r})$. Note that for
$x\in\partial\Omega\cap B_{r}(z_{r})$, $\tilde{\delta}(x)\leq Cr^{2}$ and thus
$|v(x)|\leq C_{3}r$. Now, comparing $u$ and $v$, we obtain

$\lVert u-v\rVert_{L^{\infty}(B_{r,+}(z_{r}))}\leq C_{3}r.$ (5.4)

Since $r^{-d/2}\lVert u\rVert_{L^{2}(B_{2r,+}(z_{r}))}=1$, by (5.3), there
exists $M>1$ depending on $N$ such that

$r^{-d/2}J_{u}(z_{r},r/4)\geq M^{-1}.$

Consequently,

$r^{-d/2}J_{v}(z_{r},r/4)\geq\frac{1}{2}M^{-1}-C_{3}r^{2}.$

Choose $r$ small so that $\frac{1}{2}M^{-1}-C_{3}r^{2}\geq\frac{1}{4}M^{-1}$.
As a result,

$N_{v}(z_{r},r/4)=\log\frac{J_{v}(z_{r},r/2)}{J_{v}(z_{r},r/4)}\leq N_{3},$
(5.5)

where $N_{3}$ depends on $N$. Note that, applying the almost monotonicity of
doubling index, we have $N_{v}(z_{r},r/2^{k})\leq CN_{3}$ at least for $2\leq
k\leq 10$.

Since $D_{r,+}(z_{r})$ has flat boundary on the support plane, it is proved in
Lemma A.4 that there exists $t_{0}>0$ (depending only on $N$ and $A$) and
$w\in\partial D_{r,+}(z_{r})\cap B_{r/16}(z_{r})$, such that $v$ does not
change sign in $B_{t_{0}r}(w)\cap D_{r,+}(z_{r})$. Moreover, for every $x\in
B_{t_{0}r}(w)\cap D_{r,+}(z_{r})$,

$|v(x)|\geq c_{N}\tilde{\delta}(x)r^{-1}\lVert
v\rVert_{L^{\infty}(D_{r/2,+}(z_{r}))}\geq c_{N}\tilde{\delta}(x)r^{-1}.$

In view of (5.4), for any $x\in B_{t_{0}r}(w)\cap B_{r,+}(z_{r})$,

$|u(x)|\geq c_{N}\tilde{\delta}(x)r^{-1}-C_{3}r.$

Hence, $u(x)$ has no zeros if $\tilde{\delta}(x)>C_{3}c_{N}^{-1}r^{2}$. Since
$\delta(x)\leq\tilde{\delta}(x)$. Thus $u(x)$ has no zeros if
$\delta(x)>C_{3}c_{N}^{-1}r^{2}$ and $x\in B_{t_{0}r}(w)\cap B_{r,+}(z_{r})$.

Next, since the distance between the flat boundary of $D_{r,+}(z_{r})$ and
$\partial\Omega\cap B_{r}(z_{r})$ is bounded by $Cr^{2}$, we can find a point
$w^{\prime}\in\partial\Omega\cap B_{r}(z_{r})$ such that $|w-w^{\prime}|\leq
Cr^{2}$. Therefore, if $r<\frac{1}{2}C^{-1}t_{0}$, then
$B_{\frac{1}{2}t_{0}r,+}(w^{\prime})=B_{\frac{1}{2}t_{0}r}(w^{\prime})\cap\Omega\subset
B_{t_{0}r}(w)\cap D_{r,+}(z_{r})$. Consequently, $u$ has no zeros in
$B_{\frac{1}{2}t_{0}r}(w^{\prime})\cap\\{\delta(x)>C_{3}c_{N}^{-1}r^{2}\\}$.
Due to this fact, (5.3) and Lemma 5.1 (with rescaling), there exists a
constant $\rho>0$ such that if

$C_{3}c_{N}^{-1}r^{2}<\rho\frac{1}{2}t_{0}r,$

then $u$ has no zeros in $B_{\frac{1}{2}t_{0}r,+}(w^{\prime})$. Definitely,
this is possible if we take

$r=r_{0}:=\min\\{\frac{1}{3}C^{-1}t_{0},\frac{1}{3}\rho
t_{0}c_{N}C_{3}^{-1}\\}.$

Note that when we take different $r$, the point $z_{r}$, as well as $w$ and
$w^{\prime}$ will also vary. But as in the previous argument if we fix $r_{0}$
as above, we can find a particular $z=w^{\prime}\in\partial\Omega\cap
B_{1/80}$ such that $u$ has no zeros in $B_{\frac{1}{2}t_{0}r_{0},+}(z)$.

Finally, let $4\mu=\frac{1}{2}t_{0}r_{0}$. By the Harnack inequality and the
doubling inequality (applied finitely many times depending on $\mu$), we can
find $x_{1}\in B_{3\mu,+}(z)$, away from the boundary, such that
$|u(x_{1})|\geq c\lVert u\rVert_{L^{2}(B_{1/4,+})}$ where $c=c(d,A,L,N)>0$.
Another use of Harnack inequality in inward cones (see Lemma A.3) leads to

$|u(x)|\geq c\delta(x)^{\alpha}u(x_{1})\geq c\delta(x)^{\alpha}\lVert
u\rVert_{L^{2}(B_{1/4,+})},$

for all $x\in B_{\mu,+}(z)$, where $\alpha$ depends only on $L$. This
completes the proof. ∎

Using a similar argument, we can generalize the same result from convex
domains to quaisconvex domains. The idea is that we can approximate
quasiconvex domains by convex domains at every point on the boundary and at
every small scale. Lemma 5.1 will again be useful in this argument.

######  Lemma 5.3.

Let $\Omega$ be a quasiconvex Lipschitz domain and $0\in\partial\Omega$.
Moreover, $\partial\Omega\cap B_{32}(0)$ can be expressed as a Lipschitz graph
with constant $L$. Assume $\frac{1}{2}I\leq A\leq 2I$. Let $u$ be $A$-harmonic
in $B_{16,+}=\Omega\cap B_{16}$ and $u=0$ on $\partial\Omega\cap B_{16}$.
Suppose

$\qquad N_{u}(0,\frac{1}{2})\leq N.$ (5.6)

Then there exist $\omega_{0}>0,\gamma_{0}>0,\theta>0$, depending only on $N,L$
and $d$, such that if $\omega(32)\leq\omega_{0}$ and $|\nabla
A|\leq\gamma_{0}$, then $|u|>0$ in $B_{\theta,+}(y)$ for some
$y\in\partial\Omega\cap B_{1/8}$.

###### Proof.

Assume $J_{u}(0,4)=1$. Let $V$ be the convex hull of
$B_{1,+}=B_{1}\cap\Omega$. By Lemma 2.3,

$\text{dist}(\partial B_{1,+},V)\leq\omega(2)\leq\omega_{0}.$ (5.7)

Let $\tilde{u}$ be the $A$-harmonic function in $V$ such that $\tilde{u}=u$ on
$\partial V\cap\Omega$ and $\tilde{u}=0$ on $\partial V\cap\Omega^{c}$. By
(5.7), the maximum principle and Hölder continuity of $\tilde{u}$ (i.e., De
Giorgi-Nash estimate),

$\displaystyle\lVert\tilde{u}-u\rVert_{L^{\infty}({B}_{1,+})}$
$\displaystyle\leq\lVert\tilde{u}\rVert_{L^{\infty}(\partial\Omega\cap
B_{1})}$ $\displaystyle\leq
C\omega_{0}^{\sigma}\lVert\tilde{u}\rVert_{L^{\infty}(V)}$ $\displaystyle\leq
C\omega_{0}^{\sigma}\lVert{u}\rVert_{L^{\infty}(\Omega\cap\partial B_{1})}$
$\displaystyle\leq C\omega_{0}^{\sigma}J_{u}(0,4)^{1/2}\leq
C\omega_{0}^{\sigma}.$

Hence, if $\omega_{0}$ is sufficiently small (depending on $N$), we have

$\qquad N_{\tilde{u}}(0,\frac{1}{4})\leq N_{1},$

where $N_{1}$ depends only on $N$ (indeed, similar to the proof of (5.5), one
can show $N_{1}\leq 2N+3$). By Lemma 3.2, if $\omega_{0}$ is sufficiently
small, there exists $x_{0}$ on $\partial V\cap B_{1/16}$ such that
$|x_{0}|\leq\omega_{0}$ and

$\qquad N_{\tilde{u}}(x_{0},\frac{1}{8})\leq N_{2},$

where $N_{2}$ depends only on $N$. It follows from Lemma 5.2 that there exists
$\mu=\mu(d,L,N)$, $\alpha=\alpha(L)$ and $z\in\partial V\cap B_{1/40}$ such
that

$|\tilde{u}(x)|\geq c_{N}\tilde{\delta}(x)^{\alpha},\quad\text{for all
}x\in\tilde{B}_{\mu,+}(z),$

where $\tilde{\delta}(x)=\text{dist}(x,\partial V)$ and
$\tilde{B}_{\mu,+}(z)=V\cap B_{\mu}(z)$.

Consequently, we have

$|u(x)|\geq c_{N}\tilde{\delta}^{\alpha}(x)-C_{N}\omega_{0}^{\sigma}.$

Thus $|u(x)|>0$ if $\tilde{\delta}(x)>C_{N}\omega_{0}^{\sigma/\alpha}$. Note
that $\delta(x)<\tilde{\delta}(x)$. Let $\rho$ be the constant in Lemma 5.1
corresponding to $M=C_{d}e^{72N+C_{d}}$, where $C_{d}$ is an absolute constant
depending only on $d$ and $M$ will be shown how to be determined. Now choose
$\omega_{0}$ sufficiently small such that
$C_{N}\omega_{0}^{\sigma/\alpha}<\frac{1}{2}\rho\mu$, then

$|u(x)|>0,\quad\text{if }x\in\tilde{B}_{\mu,+}(z)\text{ and
}\delta(x)>\frac{1}{2}\rho\mu.$

Due to (5.7), for $\omega_{0}<\frac{1}{4}\rho\mu$, there exists some
$y\in\partial\Omega\cap B_{1/40}$ such that
$|y-z|\leq\omega_{0}<\frac{1}{4}\rho\mu$. Thus,
$B_{\frac{1}{2}\mu,+}(y)\subset B_{\mu,+}(z)$. It follows that

$|u(x)|>0\quad\text{if }x\in B_{\frac{1}{2}\mu,+}(y)\text{ and
}\delta(x)>\frac{1}{2}\rho\mu.$ (5.8)

Finally, fixing $\mu$ and $\rho$ above, we show how $M$ is determined and
Lemma 5.1 is applied to conclude the result. By (5.6) and a standard argument
(the details are given in Lemma A.6), we have

$\sup_{1/10<r\leq 1/5}N_{u}(x,r)\leq 12N+C_{d},\quad\text{for all }x\in
B_{1/40,+}(0),$ (5.9)

where $C_{d}$ depending only on $d$. Let $m=m(d,N,L)$ be the smallest integer
such that $2^{-m}\leq\mu\ell/100$, where $\ell=\frac{1}{1+L}$. Let
$\varepsilon=\varepsilon(d,N,L)>0$ be a small number such that
$(1+\varepsilon)^{m}\leq 2$. Now let $\omega_{0},\gamma_{0}$ be sufficiently
small such that the almost monotonicity of doubling index and (5.9) imply

$\displaystyle\sup_{\mu\ell/1000<r\leq 1/5}N_{u}(x,r)$
$\displaystyle\leq\sup_{2^{-m}/10<r\leq 1/5}N_{u}(x,r)$
$\displaystyle\leq(1+\varepsilon)^{m-1}\sup_{1/10<r\leq 1/5}(N_{u}(x,r)+1)$
$\displaystyle\leq 24N+C_{d}.$

Applying the same argument as (5.3), we obtain

$\sup_{\mu\ell/500<r\leq
1/10}\frac{\int_{B_{2r}(x)}u^{2}}{\int_{B_{r}(x)}u^{2}}\leq
C_{d}e^{72N+C_{d}}=M,\quad\text{for all }x\in B_{1/40,+}(0).$ (5.10)

In view of our choice of $\rho$ (corresponding to $M$), (5.8) and (5.10), we
can apply Lemma 5.1 to conclude that $|u|>0$ in $B_{\frac{1}{8}\mu,+}(y)$. ∎

To end this section, we restate the above lemma in terms of cuboids.

######  Lemma 5.4.

Let $\Omega$ be a quasiconvex Lipschitz domain whose local boundary near
$0\in\partial\Omega$ is given by $x_{d}=\phi(x^{\prime})$ as in Section 4.2
and $|\nabla\phi|\leq L$. Let $N>0$. There exist $k_{0}\geq 5$, $\gamma_{0}>0$
and $\omega_{0}>0$ (depending on $N,d$ and $L$) such that the following
statement holds. Assume $\frac{1}{2}I\leq A\leq 2I,|\nabla
A|\leq\gamma_{0},\omega(1)\leq\omega_{0}$ and $u$ is an $A$-harmonic function
in $B_{1,+}$ with $u=0$ on $B_{1}\cap\partial\Omega$. Let $Q\subset B_{1/64}$
be a boundary cuboid centered on $\partial\Omega$ and $N_{u}^{*}(Q)\leq N$.
Then for any $k\geq k_{0}$, there exists $q\in\mathcal{B}_{k}(Q)$ such that
$Z(u)\cap q=\varnothing$.

##  6\. Estimate of nodal sets

In this section, we prove the main theorem by combining Lemma 4.2 and Lemma
5.4. The proof is similar to [21].

###  6.1. Nodal sets of $A$-harmonic functions

We first recall the interior estimate of nodal sets for analytic or Lipschitz
coefficients.

######  Lemma 6.1.

There exists $r_{0}=r_{0}(A)>0$ such that if $q$ is a cuboid with $s(q)<r_{0}$
and $u$ is $A$-harmonic in $(2d\Lambda)q$, then

$\mathcal{H}^{d-1}(Z(u)\cap q)\leq C_{0}N_{u}^{*}(q)^{\beta}s(q)^{d-1},$ (6.1)

where $\beta=1$ if $A$ is real analytic and $\beta>1$ if $A$ is Lipschitz.

######  Remark 6.2.

In the case of Lipschitz coefficients in the above lemma, the conclusion in
(6.1) follows from [19, Theorem 6.1]. In the case of analytic coefficients,
the estimate is essentially contained in [6]. Here we briefly sketch the proof
in our setting. Let the origin to be the center of $q$. By the analytic
hypoellipticity, we have

$\displaystyle\frac{|\partial^{\alpha}u(0)|}{\alpha!}\leq{\Big{(}\frac{C_{1}}{r}\Big{)}^{|\alpha|}}\|u\|_{L^{\infty}(B_{r})},$
(6.2)

for all $r\in(0,r_{0}]$, where $\alpha$ is a multi-index, and $C_{1}$ and
$r_{0}$ depend on the analyticity of $A$. In terms of the Taylor series of
$u(x)$ in $B_{r}$, we can extend $u$ to be a holomorphic function in
$\mathbb{B}_{c_{1}r}:=\\{z\in\mathbb{C}^{d}:|z|<c_{1}r\\}$ such that

$\displaystyle\sup_{z\in\mathbb{B}_{c_{1}r}}|u(z)|\leq C_{2}\lVert
u\rVert_{L^{\infty}(B_{r})},$ (6.3)

where $c_{1}<1$ and $C_{2}$ depending on $C_{1}$. By a finite number of
iteration of interior doubling inequalities, we have

$\displaystyle\|u\|_{L^{\infty}(B_{r})}\leq
e^{CN_{u}^{*}(q)}\|u\|_{L^{\infty}(B_{cr/2})}.$

Combining the above two estimates, we obtain

$\displaystyle\sup_{z\in\mathbb{B}_{c_{1}r}}|u(z)|\leq
e^{CN_{u}^{*}(q)}\|u\|_{L^{\infty}(B_{c_{1}r/2})}.$ (6.4)

Thus, an application of [6, Proposition 6.7] yields

$\mathcal{H}^{d-1}(Z(u)\cap B_{c_{2}r})\leq C_{3}N_{u}^{*}(q){r^{d-1}},$ (6.5)

where $c_{2}<1$ is a constant depending on $C_{1}$. Note that (6.5) holds for
all $r\leq\min\\{r_{0},s(q)\\}$. Hence, applying (6.5) to a finite number of
balls that cover $q$, we obtain (6.1) for real analytic coefficients.

The constant $C_{0}$ in (6.1) (depending on $C_{1}$ in (6.2), not on specific
$u$) can be quantified in terms of the quantitative analyticity property of
$A$, such as the radius of convergence for the Taylor series. In our later
application of Lemma 6.1, $q$ could be small cuboids very close to the
boundary. To guarantee that we have a uniform constant $C_{0}$ in (6.1), we
need a control of the analyticity of the coefficients as $q$ approaching the
boundary. A simple way to achieve this is to assume that $A$ is analytic in
$\overline{\Omega}$, or in other words, $A$ is analytic up to the boundary
(the Taylor series of $A$ converges in a neighborhood of any boundary point).
Thus, the compactness of $\overline{\Omega}$ will give a uniform bound of
$C_{0}$ in (6.1). Nevertheless, for the sake of brevity, in this paper we will
simply say that $A$ is analytic (or real analytic).

Define $N_{u}^{**}(Q)=N_{u}^{*}(Q)+1.$

######  Theorem 6.3.

Let $u$ be $A$-harmonic in $B\cap\Omega$ and vanishing on
$B\cap\partial\Omega$. Let $Q$ be a standard boundary cuboid such that
$Q\subset\frac{1}{10}B$. Then there exists $r_{0}=r_{0}(A,\Omega)>0$ and
$C_{0}=C_{0}(A,\Omega)>0$ such that if $s(Q)\leq r_{0}$,

$\mathcal{H}^{d-1}(Z(u)\cap Q)\leq C_{0}N_{u}^{**}(Q)^{\beta}s(Q)^{d-1}.$
(6.6)

where $\beta=1$ if $A$ is real analytic and $\beta>1$ if $A$ is Lipschitz.

###### Proof.

Without loss of generality, assume the center of $Q$ is $0\in\partial\Omega$.
By an affine transformation, we may assume $A(0)=I$ and then $Q$ is turned
into a rescaled cuboid that is contained in some standard cuboid with
comparable size, still denoted by $Q$. Therefore, it is sufficient to prove
(6.6) in this situation for $s(Q)\leq r_{0}$.

First let us explain how $r_{0}$ is determined. Let $N_{0}$ and $k_{0}$ be
given by Lemma 4.2, which depends only on $A$ and $\Omega$. Also let $N=N_{0}$
in Lemma 5.4 and pick $k$ to be the maximum of $k_{0}$ in Lemma 4.2 and Lemma
5.4. Now, we let $\gamma_{0}$ and $\omega_{0}$ be given by Lemma 4.2 and Lemma
5.4, whichever is smaller. Note that all these parameters depend only on $A$
and $\Omega$. Now, we can perform a rescaling $x\to x/r_{1}$ such that
$\frac{1}{2}I\leq A\leq 2I,|\nabla A|\leq\gamma_{0}$ in $Q$ and
$\omega(32)\leq\omega_{0}$ in $B_{1,+}(0)$, where $r_{1}=Cr_{0}$ and $Q\subset
B_{1/64}(0)$. Note that $r_{0}$ now depends only on $A$ and $\Omega$.
Consequently, both Lemma 4.2 and Lemma 5.4 apply to $Q$ and all its boundary
subcuboids. Again, we only need to show (6.6) in this rescaled case since it
is scale invariant.

Let $K$ be a compact set $K\subset\Omega$. We would like to show

$\mathcal{H}^{d-1}(Z(u)\cap Q\cap K)\leq
C_{0}N_{u}^{**}(Q)^{\beta}s(Q)^{d-1},$ (6.7)

for some constant $C_{0}$ independent of $K$. Definitely if $Q$ is small
enough such that $Q\cap K=\varnothing$, then $\mathcal{H}^{d-1}(Z(u)\cap Q\cap
K)=0$. We prove (6.7) by induction from small boundary cuboids to large
boundary cuboids.

Let $k\geq 1$ to be determined. Let $(Q,\mathcal{B}_{k},\mathcal{I}_{k})$ be a
standard decomposition as in Section 4.2. Assume for each small boundary
cuboid $q\in\mathcal{B}_{k}$,

$\mathcal{H}^{d-1}(Z(u)\cap q\cap K)\leq
C_{0}N_{u}^{**}(q)^{\beta}s(q)^{d-1}.$

Note that the base case of induction $q\cap K=\varnothing$ is trivial.

Now consider

$\mathcal{H}^{d-1}(Z(u)\cap Q\cap
K)\leq\sum_{q\in\mathcal{I}_{k}}\mathcal{H}^{d-1}(Z(u)\cap
q)+\sum_{q\in\mathcal{B}_{k}}\mathcal{H}^{d-1}(Z(u)\cap q\cap K).$

By the interior result in Lemma 6.1,

$\sum_{q\in\mathcal{I}_{k}}\mathcal{H}^{d-1}(Z(u)\cap q)\leq
C_{1}\sum_{q\in\mathcal{I}_{k}}N_{u}^{**}(q)^{\beta}s(q)^{d-1}\leq
C_{k}N_{u}^{**}(Q)^{\beta}s(Q)^{d-1}.$ (6.8)

where $\beta=1$ if $A$ is real analytic and $\beta>1$ if $A$ is Lipschitz. For
all other boundary cuboids $q\in\mathcal{B}_{k}$, we have by Remark 3.5

$N_{u}^{**}(q)\leq(1+\varepsilon)^{k}N_{u}^{**}(Q),$

if $Q$ is small enough. Combining Lemma 4.2 and Lemma 5.4 (with $N=N_{0}$), we
know that there exists $k\geq 1$, depending only on $d,L$, such that there is
a cube $q_{0}\in\mathcal{B}_{k}$ such that either
$N_{u}^{*}(q_{0})\leq\frac{1}{2}N_{u}^{*}(Q)$ or $Z(u)\cap q_{0}=\varnothing$.
Without loss of generality, assume $N_{0}\geq 10$. Since the first case takes
place if $N\geq N_{0}$, thus $N_{u}^{*}(q_{0})\leq\frac{1}{2}N_{u}^{*}(Q)$
implies $N_{u}^{**}(q_{0})\leq\frac{2}{3}N_{u}^{**}(Q).$ Recall that
$\mathcal{B}_{k}$ has $2^{k(d-1)}$ subcuboids. Applying the induction argument
to each boundary subcuboids, we have

$\displaystyle\sum_{q\in\mathcal{B}_{k}}\mathcal{H}^{d-1}(Z(u)\cap q\cap K)$
(6.9) $\displaystyle=\sum_{q\in\mathcal{B}_{k},q\neq
q_{0}}\mathcal{H}^{d-1}(Z(u)\cap q\cap K)+\mathcal{H}^{d-1}(Z(u)\cap q_{0}\cap
K)$ $\displaystyle\leq\sum_{q\in\mathcal{B}_{k},q\neq
q_{0}}C_{0}N_{u}^{**}(q)^{\beta}s(q)^{d-1}+C_{0}(2/3)^{\beta}N_{u}^{**}(Q)^{\beta}s(q_{0})^{d-1}$
$\displaystyle\leq\bigg{\\{}\frac{2^{k(d-1)}-1}{2^{k(d-1)}}(1+\varepsilon)^{\beta
k}+\frac{(2/3)^{\beta}}{2^{k(d-1)}}\bigg{\\}}C_{0}N_{u}^{**}(Q)^{\beta}S(Q)^{d-1}.$

Since $\beta\geq 1$ and $k\geq 3$ are fixed, we choose $\varepsilon$ small
(here we need to make $r_{0}=r_{0}(A,\Omega)>0$ small enough) and $C_{0}$
large enough such that

$C_{k}+\bigg{\\{}\frac{2^{k(d-1)}-1}{2^{k(d-1)}}(1+\varepsilon)^{\beta
k}+\frac{(2/3)^{\beta}}{2^{k(d-1)}}\bigg{\\}}C_{0}\leq C_{0}.$

Note that $C_{0}$ does not depend on the compact set $K$. Thus, the
combination of (6.8) and (6.9) gives (6.7), which implies (6.6) by letting $K$
exhaust $Q\cap\Omega$. ∎

###  6.2. Dirichlet eigenfunctions

Let $\Omega$ be a bounded quasiconvex Lipschitz domain. Let
$\varphi_{\lambda}$ be the Dirichlet eigenfunction of $\mathcal{L}$
corresponding to the eigenvalue $\lambda>0$, namely,
$\mathcal{L}(\varphi_{\lambda})=\lambda\varphi_{\lambda}$ in $\Omega$ and
$\varphi_{\lambda}=0$ on $\partial\Omega$. Let

$u_{\lambda}(x,t)=e^{t\sqrt{\lambda}}\varphi_{\lambda}(x).$

Then $u_{\lambda}$ is $\widetilde{A}$-harmonic in
$\widetilde{\Omega}:=\Omega\times\mathbb{R}$, where

$\widetilde{A}(x,t)=\begin{bmatrix}A(x)&0\ \\\ 0&1\ \end{bmatrix}.$

######  Lemma 6.4.

If a Lipschitz $\Omega$ is quasiconvex, then $\widetilde{\Omega}$ is also
quasiconvex.

###### Proof.

First note $\partial\widetilde{\Omega}=\partial\Omega\times\mathbb{R}$. Let
$(x_{0},t_{0})\in\partial\widetilde{\Omega}$. Then $x_{0}\in\partial\Omega$.
Since $\Omega$ is quasiconvex, we can rotate the domain $\Omega$ such that the
local graph of $\partial\Omega$ is given by a Lipschitz graph
$x_{d}=\phi(x^{\prime})$ with $\phi(x^{\prime}_{0})=0$ and for
$|x^{\prime}-x^{\prime}_{0}|<r_{0}$,

$\phi(x^{\prime})\geq-|x^{\prime}-x^{\prime}_{0}|\omega(|x^{\prime}-x^{\prime}_{0}|).$

Clearly the local graph of $\partial\widetilde{\Omega}$ is given by
$x_{d}=\widetilde{\phi}(x^{\prime},t):=\phi(x^{\prime})$. Thus, for any
$|(x^{\prime},t)-(x^{\prime}_{0},t_{0})|<r_{0}$

$\displaystyle\widetilde{\phi}(x^{\prime},t)$
$\displaystyle\geq-|x^{\prime}-x^{\prime}_{0}|\omega(|x^{\prime}-x^{\prime}_{0}|)$
$\displaystyle\geq-|(x^{\prime},t)-(x^{\prime}_{0},t_{0})|\omega(|(x^{\prime},t)-(x^{\prime}_{0},t_{0})|),$

which, by definition, shows that $\widetilde{\Omega}$ is quasiconvex with the
same quasiconvexity modulus. ∎

The following lemma gives the bound of doubling index for the
$\widetilde{A}$-harmonic extension $u_{\lambda}$.

######  Lemma 6.5.

Let $\Omega$ be a bounded quasiconvex Lipschitz domain. Let $u_{\lambda}$ be
the $\widetilde{A}$-harmonic extension in $\widetilde{\Omega}$ of the
Dirichlet eigenfunction $\varphi_{\lambda}$. There exists
$r_{0}=r_{0}(A,\Omega)>0$ such that if $(x,t)\in\widetilde{\Omega}$ and
$0<r<r_{0}$, then $N_{\varphi_{\lambda}}((x,t),r)\leq C_{r}\sqrt{\lambda}$,
where $C_{r}$ depends only on $r,A$ and $\Omega$.

###### Proof.

The proof is standard. We refer to [21] for Lipschitz domains by using the
three-ball inequality on a chain of balls and to [6, 7] for the case of smooth
domains. We skip the details. ∎

Finally, we prove Theorem 1.2.

###### Proof of Theorem 1.2.

Let $\varphi_{\lambda}$ be the Dirichlet eigenfunction corresponding to the
eigenvalue $\lambda>0$. Let $u_{\lambda}=e^{t\sqrt{\lambda}}\varphi_{\lambda}$
be the $\widetilde{A}$-harmonic extension of $\varphi_{\lambda}$ in
$\widetilde{\Omega}=\Omega\times\mathbb{R}$. Note that
$Z(u_{\lambda})=Z(\varphi_{\lambda})\times\mathbb{R}$. Thus it suffices to
estimate the nodal set of $u_{\lambda}$ in
$\Omega\times[-1,1]\subset\widetilde{\Omega}$.

Let $r_{0}$ be as in Theorem 6.3 or Lemma 6.5, whichever is smaller. Let
$\Omega_{I}=\\{x\in\Omega:\delta(x)>cr_{0}\\}$. Then $\Omega_{I}\times[-1,1]$
can be covered (with finite overlaps) by a sequence of balls $B_{i}$ with
radius $cr_{0}/100$ such that $10B_{i}\subset\Omega\times[-2,2]$. Note that
the number of these balls depends only on $cr_{0},d$ and $\Omega$. By Lemma
6.1 and Lemma 6.5

$\mathcal{H}^{d}(Z(u_{\lambda})\cap\Omega_{I}\times[-1,1])\leq\sum_{i}\mathcal{H}^{d}(Z(u_{\lambda})\cap
B_{i})\leq C\lambda^{\beta/2}.$

On the other hand, let $\Omega_{B}=\Omega\setminus\Omega_{I}$. Then
$\Omega_{B}\times[-1,1]$ can be covered by a family of boundary balls
$B_{k}^{\prime}$ centered on $\partial\Omega\times[-1,1]$ with radius
$10cr_{0}$. The number of these balls depends only on $cr_{0},d$ and $\Omega$.
For each of these ball $B_{k}^{\prime}$,
$B^{\prime}_{k}\cap\partial\widetilde{\Omega}$ can be rotated into a local
graph such that $B^{\prime}_{k}$ is entirely contained in a cuboid $Q_{k}$
with $s(Q_{k})\leq r_{0}$. Thus, Theorem 6.3 and Lemma 6.5 show

$\displaystyle\mathcal{H}^{d}(Z(u_{\lambda})\cap\Omega_{B}\times[-1,1])$
$\displaystyle\leq\sum_{k}\mathcal{H}^{d}(Z(u_{\lambda})\cap Q_{k})$
$\displaystyle\leq\sum_{k}CN^{**}(Q_{k})^{\beta}s(Q_{k})^{d}$
$\displaystyle\leq C(1+\lambda)^{\beta/2}$ $\displaystyle\leq
C\lambda^{\beta/2}.$

Hence, $\mathcal{H}^{d}(Z(u_{\lambda})\cap\Omega\times[-1,1])\leq
C\lambda^{\beta/2}.$ This implies the desired estimate of
$\mathcal{H}^{d-1}(Z(\varphi_{\lambda}))$. ∎

## Appendix A

###  A.1. A quasiconvex Lipschitz curve that is nowhere convex or $C^{1}$

We construct a quasiconvex Lipschitz curve in $(0,1)$ which is neither $C^{1}$
nor convex in any subinterval of $(0,1)$. First let
$\\{q_{k}\\}_{k=1}^{\infty}$ be the list of all rational number in $[0,1]$ and
define a nonnegative Radon measure by

$\mu=\sum_{k=1}^{\infty}2^{-k}\delta_{q_{k}},$

where $\delta_{q_{k}}$ is the Dirac measure at $q_{k}$. Now let
$f(t)=\mu((0,t))$. Note that $f$ is a bounded nondecreasing function with
$f(0)=0$ and $f(1)=1$. Moreover, $f$ is not continuous in any subinterval of
$(0,1)$. Let

$\phi(x)=\int_{0}^{x}f(t)dt-x^{2}.$

Since $\int_{0}^{x}f(t)dt$ is convex and $x^{2}$ is $C^{1}$, it is easy to see
that $\phi(x)$ is quasiconvex and Lipschitz.

Since $f$ is not continuous in any subinterval of $(0,1)$, then clearly,
$\phi$ is not $C^{1}$ in any subinterval of $(0,1)$. Now we show that $\phi$
is not convex in any subinterval of $(0,1)$. It is sufficient to show that
$\phi^{\prime\prime}$ is not a nonnegative Radon measure on any subinterval of
$(0,1)$. In fact,

$\phi^{\prime\prime}=-2+\sum_{k=1}^{\infty}2^{-k}\delta_{q_{k}}.$

Divided $(0,1)$ into $2^{j}$ equal subintervals with length $2^{-j}$. For any
such interval $I$, if there is no $q_{k}$ with $1\leq k\leq j$ dropping in
$I$, then

$\phi^{\prime\prime}(I)\leq-2|I|+\sum_{k=j+1}^{\infty}2^{-k}\leq-2^{-j}<0.$

In other words, there are at most $j$ subintervals (out of $2^{j}$
subintervals in total) such that $\phi^{\prime\prime}(I)\geq 0$. As $j$
approaching infinity, we see that $\phi^{\prime\prime}$ cannot be nonnegative
in any nonempty subintervals.

###  A.2. Proof of Proposition 3.1

We adapt the arguments in [1], [18] with emphasis on the role of Lipschitz
coefficient. Recall that we assumed $A(0)=I$ and defined

$\displaystyle\mu(x)=\frac{x\cdot A(x)x}{|x|^{2}}.$ (A.1)

Using the standard assumption on $A$, we can check that

$\displaystyle|\nabla\mu|\leq C\gamma\quad\ \mbox{and}\
\quad\Lambda^{-1}\leq\mu(x)\leq\Lambda.$ (A.2)

We choose $w(x)=\frac{A(x)x}{\mu(x)}$ and verify directly that

$\displaystyle\partial_{j}w_{i}(x)=\delta_{ij}+O(\gamma|x|).$

Differentiating $H(r)$ with respect to $r$, we have

$\displaystyle H^{\prime}(r)=\frac{d-1}{r}\int_{\partial B_{r}\cap\Omega}\mu
u^{2}\,d\sigma+2\int_{\partial B_{r}\cap\Omega}\mu u\frac{\partial u}{\partial
r}\,d\sigma+\int_{\partial B_{r}\cap\Omega}\frac{\partial\mu}{\partial
r}u^{2}\,\,d\sigma,$ (A.3)

where $\frac{\partial}{\partial r}=\frac{x}{|x|}\cdot\nabla$. The conormal
derivative associated with $\mathcal{L}=-\nabla\cdot(A\nabla)$ is defined as
$\frac{\partial u}{\partial\nu}=n\cdot A\nabla u$, where $n$ is the outer
normal of the boundary. Since $\mathcal{L}(u)=0$ in $B_{r}\cap\Omega$ and
$u=0$ on $B_{r}\cap\partial\Omega$, an integration by parts gives

$\displaystyle\int_{B_{r}\cap\Omega}A\nabla u\cdot\nabla u=\int_{\partial
B_{r}\cap\Omega}u\frac{\partial u}{\partial\nu}\,d\sigma.$ (A.4)

Note that on $\partial B_{r}\cap\Omega$, $n(x)=\frac{x}{|x|}$. Let

$\displaystyle\tau=A\frac{x}{|x|}-\mu\frac{x}{|x|}.$

Observe that $\tau$ is a tangent vector field on $\partial B_{r}$ since
$\tau\cdot n=0$ on $\partial B_{r}$. Moreover, using $n=\frac{x}{|x|}$ on
$\partial B_{r}\cap\Omega$, we can rewrite $\tau=(\tau_{i})$ as

$\displaystyle\tau_{i}=a_{ij}n_{j}-a_{ml}n_{m}n_{l}n_{i}=a_{ij}n_{j}n_{m}n_{m}-a_{ml}n_{m}n_{l}n_{i}.$

Thus, we have

$\displaystyle\begin{aligned}
\tau\cdot\nabla&=a_{ij}n_{j}n_{m}n_{m}\partial_{i}-a_{ml}n_{m}n_{l}n_{i}\partial_{i}\\\
&=a_{ij}n_{j}n_{m}n_{m}\partial_{i}-a_{ij}n_{i}n_{j}n_{m}\partial_{m}\\\
&=a_{ij}n_{j}n_{m}(n_{m}\partial_{i}-n_{i}\partial_{m}),\end{aligned}$ (A.5)

where we have interchanged the index $m$ and $i$, and $l$ and $j$ in the
second equality. The key is that $(n_{m}\partial_{i}-n_{i}\partial_{m})$ is a
tangential derivative on $\partial B_{r}\cap\Omega$ that allows for
integration by parts on the boundary. Since $u=0$ on
$B_{r}\cap\partial\Omega$, integrating by parts shows that

$\displaystyle\begin{aligned} \int_{\partial
B_{r}\cap\Omega}\nabla(u^{2})\cdot\tau\,d\sigma&=\int_{\partial
B_{r}\cap\Omega}a_{ij}n_{j}n_{m}(n_{m}\partial_{i}-n_{i}\partial_{m})u^{2}d\sigma\\\
&=-\int_{\partial
B_{r}\cap\Omega}u^{2}(n_{m}\partial_{i}-n_{i}\partial_{m})(a_{ij}n_{j}n_{m})d\sigma\\\
&=\int_{\partial
B_{r}\cap\Omega}u^{2}(n_{m}\partial_{i}-n_{i}\partial_{m})(n_{i}n_{m})d\sigma\\\
&\qquad-\int_{\partial
B_{r}\cap\Omega}u^{2}(n_{m}\partial_{i}-n_{i}\partial_{m})((a_{ij}-\delta_{ij})n_{j}n_{m})d\sigma\\\
&=O(\gamma)H(r),\end{aligned}$ (A.6)

where we have used $A(0)=I$, and the Lipschtiz continuity of $A$ and $n$ on
the smooth boundary $\partial B_{r}\cap\Omega$, as well as the observation
$(n_{m}\partial_{i}-n_{i}\partial_{m})(n_{i}n_{m})=0$. Hence, from (A.6), we
have

$\displaystyle D(r)=\int_{\partial B_{r}\cap\Omega}u\frac{\partial
u}{\partial\nu}\,d\sigma=$ $\displaystyle\int_{\partial B_{r}\cap\Omega}\mu
u\frac{\partial u}{\partial r}\,d\sigma+\int_{\partial B_{r}\cap\Omega}u\nabla
u\cdot\tau\,d\sigma$ $\displaystyle=$ $\displaystyle\int_{\partial
B_{r}\cap\Omega}\mu u\frac{\partial u}{\partial
r}\,d\sigma+\frac{1}{2}\int_{\partial
B_{r}\cap\Omega}\nabla(u^{2})\cdot\tau\,d\sigma$ $\displaystyle=$
$\displaystyle\int_{\partial B_{r}\cap\Omega}\mu u\frac{\partial u}{\partial
r}\,d\sigma+O(\gamma)H(r).$

It follows from (A.3) that

$\displaystyle H^{\prime}(r)=\frac{d-1}{r}H(r)+2D(r)+O(\gamma)H(r).$ (A.7)

Next we consider $D^{\prime}(r)$. We use the following Rellich-Necas identity

$\displaystyle\begin{aligned} &\nabla\cdot(w(A\nabla u\cdot\nabla u))\\\
&=2\nabla\cdot((w\cdot\nabla u)A\nabla u)+(\nabla\cdot w)A\nabla u\cdot\nabla
u\\\ &\qquad-2\partial_{i}w_{k}a_{ij}\partial_{j}u\partial_{k}u-2(w\cdot\nabla
u)\nabla\cdot(A\nabla
u)+w_{k}\partial_{k}a_{ij}\partial_{i}u\partial_{j}u.\end{aligned}$ (A.8)

Direct calculations show that

$\displaystyle w\cdot\frac{x}{|x|}=r\quad\mbox{and}\quad(w\cdot\nabla
u)A\nabla u\cdot\frac{x}{|x|}=\frac{r}{\mu(x)}(\frac{\partial
u}{\partial\nu})^{2}\quad\mbox{for}\ x\in\partial B_{r}\cap\Omega.$ (A.9)

On $B_{r}\cap\partial\Omega$, since $n=\frac{\nabla u}{|\nabla u|}$ (if
$\nabla u\neq 0$), we have

$\displaystyle(w\cdot n)A\nabla u\cdot\nabla u=(w\cdot\nabla u)A\nabla u\cdot
n=\frac{(Ax\cdot n)(An\cdot n)}{\mu(x)}|\nabla u|^{2}.$ (A.10)

This identity definitely holds if $\nabla u=0$. We integrate the Rellich-Necas
identity (A.8) to have

$\displaystyle\begin{aligned} &\int_{\partial B_{r}\cap\Omega}A\nabla
u\cdot\nabla u\\\ &=\frac{1}{r}\int_{B_{r}\cap\partial\Omega}\frac{(Ax\cdot
n)(An\cdot n)}{\eta_{1}(x)}(\frac{\partial u}{\partial
n})^{2}\,d\sigma+2\int_{\partial B_{r}\cap\Omega}\frac{1}{\mu}(\frac{\partial
u}{\partial\nu})^{2}\\\ &\qquad+\frac{d-2}{r}\int_{B_{r}\cap\Omega}A\nabla
u\cdot\nabla u+O(\gamma)\int_{B_{r}\cap\Omega}A\nabla u\cdot\nabla
u,\end{aligned}$ (A.11)

where we have used the fact that $\nabla\cdot(A\nabla u)=0$ in
$B_{r}\cap\Omega$, $|\nabla u|=|n\cdot\nabla u|=|\frac{\partial u}{\partial
n}|$ almost everywhere on $B_{r}\cap\partial\Omega$, and the Lipschitz
continuity of $A$. Note that

$\displaystyle D^{\prime}(r)=\int_{\partial B_{r}\cap\Omega}A\nabla
u\cdot\nabla u.$ (A.12)

Using the assumption that $Ax\cdot n\geq 0$ on $B_{r}\cap\partial\Omega$ and
(A.12), on one hand, we have

$\displaystyle\frac{D^{\prime}(r)}{D(r)}\geq\frac{d-2}{r}+\frac{2\int_{\partial
B_{r}\cap\Omega}\mu^{-1}(\frac{\partial
u}{\partial\nu})^{2}\,d\sigma}{\int_{B_{r}\cap\Omega}A\nabla u\cdot\nabla
u}+O(\gamma).$ (A.13)

On the other hand, with the aid of (A.7), we obtain that

$\displaystyle\frac{H^{\prime}(r)}{H(r)}=\frac{d-1}{r}+\frac{2\int_{B_{r}\cap\Omega}A\nabla
u\cdot\nabla u\,d\sigma}{\int_{\partial
B_{r}\cap\Omega}\mu(x)u^{2}\,d\sigma}+O(\gamma).$ (A.14)

Now we can establish the monotonicity property of the frequency function.
Thanks to (A.13), (A.14), (A.4) and the definition of $\mathcal{N}_{u}(0,r)$
in (3.1), we get

$\displaystyle\begin{aligned}
\frac{\mathcal{N}_{u}^{\prime}(0,r)}{\mathcal{N}_{u}(0,r)}&=\frac{1}{r}+\frac{D^{\prime}(r)}{D(r)}-\frac{H^{\prime}(r)}{H(r)}\\\
&\geq\frac{2\int_{\partial B_{r}\cap\Omega}\mu^{-1}(\frac{\partial
u}{\partial\nu})^{2}\,d\sigma}{\int_{\partial B_{r}\cap\Omega}u\frac{\partial
u}{\partial\nu}d\sigma}-\frac{2\int_{\partial B_{r}\cap\Omega}u\frac{\partial
u}{\partial\nu}d\sigma}{\int_{\partial B_{r}\cap\Omega}\mu
u^{2}\,d\sigma}+O(\gamma)\\\ &\geq-C\gamma,\end{aligned}$ (A.15)

where $\mathcal{N}_{u}^{\prime}(0,r)=\frac{d}{dr}\mathcal{N}_{u}(0,r)$ and we
have used the Cauchy-Schwarz inequality in the last inequality. Thus,
$e^{C\gamma r}\mathcal{N}(0,r)$ is nondecreasing with respect to $r$.

###  A.3. Some useful results for elliptic equations

######  Lemma A.1.

Let $\Omega$ be a convex domain and $0\in\partial\Omega$. Suppose that $u$ is
$A$-harmonic in $B_{1,+}=B_{1}(0)\cap\Omega$ with $u=0$ on $\partial\Omega\cap
B_{1}(0)$. Then

$\lVert\nabla u\rVert_{L^{\infty}(B_{1/2,+})}\leq C\lVert
u\rVert_{L^{2}(B_{1,+})}.$

###### Proof.

Without loss of generality, assume that $x_{d}=0$ is the support plane of
$\Omega$ at $0$. Let $D_{3/4,+}:=B_{1}(0)\cap\\{x_{d}>0\\}$. By the De Giorgi-
Nash estimate,

$K:=\lVert u\rVert_{L^{\infty}(B_{3/4,+})}\leq C\lVert
u\rVert_{L^{2}(B_{1,+})}.$ (A.16)

Consider a positive barrier function $v$ which is $A$-harmonic in $D_{3/4,+}$
and satisfies $v=K$ on $\partial B_{3/4}\cap\\{x_{d}>0\\}$ and $v=0$ on
$B_{3/4}\cap\\{x_{d}=0\\}$. We show that $v(x)\leq CKx_{d}$ for all $x\in
D_{5/8,+}=B_{5/8}\cap\\{x_{d}>0\\}$. This follows easily from the gradient
estimate of $v$ over the flat boundary, i.e.,

$\lVert\nabla v\rVert_{L^{\infty}(D_{5/8,+})}\leq C\lVert
v\rVert_{L^{\infty}(D_{3/4,+})}\leq CK.$

Here we need $A$ to be at least Hölder continuous.

Now, observe that $-v\leq u\leq v$ in $B_{3/4,+}$, due to the fact $|u|\leq v$
on $\partial B_{3/4,+}$ and the comparison principle. Then we have

$|u(x)|\leq CKx_{d},\quad\text{for all }x\in B_{5/8,+}.$

In particular, for any $x=(x^{\prime},x_{d})$ with $x^{\prime}=0$, we have
$|u(x)|\leq CK{\rm dist}(x,\partial\Omega)$. Note that this argument actually
works for any points in $B_{1/2,+}$ as any interior point can be connected to
the nearest point on the boundary with a perpendicular support plane.
Therefore, we obtain

$|u(x)|\leq CK{\rm dist}(x,\partial\Omega),\quad\text{for all }x\in
B_{5/8,+}.$

Finally, for any $x\in B_{1/2,+}$, let $\delta(x)={\rm
dist}(x,\partial\Omega)\leq 1/8$ (the case $\delta(x)>1/8$ is trivial). By the
interior gradient estimate

$|\nabla u(x)|\leq\frac{C}{\delta(x)}\lVert
u\rVert_{L^{\infty}(B_{\delta(x)}(x))}\leq\frac{C}{\delta(x)}CK\delta(x)\leq
CK.$

This, combined with (A.16), gives the desired estimate. ∎

######  Lemma A.2.

Let $\beta>0$ and $\mathcal{C}=\\{x=(x^{\prime},x_{d}):0\leq|x^{\prime}|<\beta
x_{d}\text{ and }x_{d}<1\\}$. Let $u$ be a positive $A$-harmonic function in
$\mathcal{C}$. Then there exists $\alpha=\alpha(\beta,\Lambda)>0$ such that
for any $0<t<1/2$

$u(0,t)\geq t^{\alpha}u(0,1/2).$

###### Proof.

Let $t\in(0,1/2)$ and assume $u(0,1/2)=1$ (without loss of generality). Note
that $(0,t)$ is an interior point of $\mathcal{C}$ with ${\rm
dist}((0,t),\partial\mathcal{C})\geq\frac{\beta t}{1+\beta}$. Then for
$t\in[1/4,1/2)$, ${\rm
dist}((0,t),\partial\mathcal{C})\geq\frac{\beta}{4(1+\beta)}$. By a Harnack
chain of balls with radius $\frac{\beta}{8(1+\beta)}$, one can connect
$(0,1/2)$ to any point $(0,t)$ for $1/4\leq t<1/2$. The number of such balls
is at most $N=N(\beta)\geq 1$. Then the Harnack inequality implies

$u(0,t)\geq c^{N}u(0,1/2)=c(\beta,\Lambda)>0,\quad\text{for all }1/4\leq
t<1/2.$

Note that $c(\beta,\Lambda)<1$. Repeating this argument, we can obtain that,
for $k\geq 1$,

$u(0,t)\geq c^{k}(\beta,\Lambda),\quad\text{for all }2^{-k}\leq t<2^{-k+1}.$

It follows that for all $t\in(0,1/2)$,

$u(0,t)\geq
2^{(\log_{2}c(\beta,\Lambda))(-2\log_{2}t)}=t^{-2\log_{2}(c(\beta,\Lambda))}.$

This gives the desired estimate with $\alpha=-2\log_{2}(c(\beta,\Lambda))>0$.
∎

######  Lemma A.3.

Let $\Omega$ be a Lipschitz domain and $0\in\partial\Omega$. Assume
$\partial\Omega\cap B_{1}(0)$ is given by the Lipschitz graph
$x_{d}=\phi(x^{\prime})$ and $B_{1,+}=\Omega\cap
B_{1}(0)=B_{1}(0)\cap\\{x=(x,x_{d}):x_{d}>\phi(x^{\prime})\\}$. Then, there
exists $\alpha,c>0$ (depending only on $\Lambda$ and the Lipschitz constnat of
$\phi$) such that for any positive $A$-harmonic function $u$ in $B_{1}^{+}$,
we have

$u(x)\geq c\delta(x)^{\alpha}u(0,1/2),\quad\text{for all }x\in B_{1/2,+}.$

###### Proof.

Assume $u(0,1/2)=1$. First of all, in view of the Harnack inequality, we have
$u(x)\geq c>0$ for all $x\in
B_{7/8,+}\cap\\{x=(x^{\prime},x_{d}):x_{d}>\phi(x^{\prime})+1/8\\}$. Thus, it
suffices to consider an arbitrary point $x\in
B_{1/2,+}\cap\\{x=(x^{\prime},x_{d}):\phi(x^{\prime})<x_{d}<\phi(x^{\prime})+1/8\\}$.
Fix such an $x=(x^{\prime},x_{d})$ and let
$x_{0}=(x^{\prime},\phi(x^{\prime}))\in\partial\Omega$. Let

$\mathcal{C}_{1/4}(x_{0})=\\{y=(y^{\prime},y_{d}):0\leq|y^{\prime}-x^{\prime}|<\beta(y_{d}-\phi(x^{\prime}))\text{
and }y_{d}-\phi(x^{\prime})<1/4\\},$

where $\beta=\min\\{1,L^{-1}\\}$ and $L$ is the Lipschitz constant of $\phi$.
Note that $\mathcal{C}_{1/4}(x_{0})$ is a rescaled and translated version of
the cone in Lemma A.2 and is entirely contained in $B_{1,+}$. Also note that
$x$ in on the vertical axis of the cone $\mathcal{C}_{1/4}(x_{0})$.

Let $x_{1}=(x^{\prime},\phi(x^{\prime})+1/8)$. Using $u(x_{1})\geq c$ and
applying Lemma A.2 in $\mathcal{C}_{1/4}(x_{0})$, we obtain $u(x)\geq
c(x_{d}-\phi(x^{\prime}))^{\alpha}=c|x-x_{0}|^{\alpha}\geq
c\delta(x)^{\alpha}$, where the last inequality follows from the fact that $x$
is on the vertical axis of $\mathcal{C}_{1/4}(x_{0})$ whose distance from the
boundary is comparable to $|x-x_{0}|$. This completes the proof. ∎

The following lemma is a generalization of [21, Lemma 9] from harmonic
functions (proved by using reflection) to $A$-harmonic functions. We point out
that the assumptions $\frac{1}{2}I\leq A\leq 2I$ and $|\nabla
A|\leq\gamma_{0}$ below are purely for technical convenience.

######  Lemma A.4.

There exists $\gamma_{0}=\gamma_{0}(d)>0$ such that the following statement
holds. If $\frac{1}{2}I\leq A\leq 2I,|\nabla A|\leq\gamma_{0}$ and $u$ is a
continuous $A$-harmonic function in $B_{2,+}:=B_{2}\cap\\{x_{d}>0\\}$
vanishing on $B_{2}\cap\\{x_{d}=0\\}$ and satisfying $N_{u}(0,1/2)\leq N$.
Then there exists $w\in B_{1/2}\cap\\{x_{d}=0\\}$ and $t>0,C>0$ depending only
on $N$ and $A$ such that $u$ does not change sign in $B_{t,+}(w)$. Moreover,
for all $x\in B_{t,+}(w)$,

$|u(x)|\geq Cx_{d}J_{u}(0,1)^{1/2}.$

###### Proof.

By normalization, assume $J_{u}(0,1)=1$. If $|\nabla A|$ is smaller than some
absolute constant $\gamma_{0}=\gamma_{0}(d)>0$, Lemma 3.4 implies the almost
monotonicity of $N_{u}(0,r)$. In particular, we have $N_{u}(0,2^{-k})\leq
2N_{u}(0,1/2)\leq 2N$ for $k=2,3$. Consequently,

$\displaystyle\lVert u\rVert_{L^{2}(B_{1/4,+})}^{2}\gtrsim
J_{u}(0,\frac{1}{8})$
$\displaystyle=J_{u}(0,1)\exp\big{(}-\sum_{k=1}^{3}N_{u}(0,2^{-k})\big{)}$
$\displaystyle\geq Ce^{-5N}.$

On the other hand, by Lemma 4.1, we have

$\displaystyle\lVert u\rVert_{L^{2}(B_{1/4,+})}$ $\displaystyle\leq
C\Big{(}\sup_{B_{1/2}\cap\\{x_{d}=0\\}}|n\cdot A\nabla u|\Big{)}^{\tau}\lVert
u\rVert_{L^{2}(B_{1/2,+})}^{1-\tau}$ $\displaystyle\leq
C\sup_{B_{1/2}\cap\\{x_{d}=0\\}}|\nabla u|^{\tau}.$

Combining the above two estimates, we know that there exists $w\in
B_{1/2}\cap\\{x_{d}=0\\}$ such that $|\nabla u(w)|=|\partial_{d}u(w)|\geq
Ce^{-5N/2\tau}$. Without loss of generality, assume $\partial_{d}u(w)>0$. Now,
using the fact that $u\in C^{1,\alpha}$ in $B_{3/4,+}$, we see that there
exists $t>0$ (depending on $N$ and $A$) such that
$\partial_{d}u(x)\geq\frac{1}{2}Ce^{-5N/2\tau}$ for all $x\in B_{t,+}(w)$.
This implies that $u$ is positive in $B_{t,+}(w)$ and satisfies the desired
estimate by the fundamental theorem of calculus. ∎

######  Lemma A.5.

Assume $\Lambda^{-1}I\leq A\leq\Lambda I$. There exists
$\gamma_{0}=\gamma_{0}(d,\Lambda)>0$ such that the following statement holds.
If $|\nabla A|\leq\gamma_{0}$ and $u$ is a continuous $A$-harmonic function in
$B_{1,+}:=B_{1}(0)\cap\\{x_{d}>0\\}$ satisfying $|u(x)|\leq x_{d}$ for all
$x\in\partial B_{1,+}$, then $|u(x)|\leq 2x_{d}$ for all $x\in B_{1,+}$.

###### Proof.

Consider $w=2x_{d}-x_{d}^{2}$. Clearly $w\geq u$ on $\partial B_{1,+}$. We
check that $\nabla\cdot(A\nabla w)\leq 0$ if $\gamma_{0}$ is sufficiently
small. In fact,

$\nabla\cdot(A\nabla
w)=-2a_{dd}+\sum_{i=1}^{d}\partial_{i}a_{id}(2-2x_{d})\leq-2\Lambda^{-1}+2d\gamma_{0}.$

Therefore, if $\gamma_{0}=1/(\Lambda d)$, we have $\nabla\cdot(A\nabla w)\leq
0$ and thus $\nabla\cdot(A\nabla(w-u))\leq 0$. By the weak maximum principle,

$\inf_{B_{1,+}}(w-u)\geq\inf_{\partial B_{1,+}}{(w-u)}\geq 0.$

Consequently, $u\leq w\leq 2x_{d}$ in $B_{1,+}$. Similarly, one can show
$u\geq-2x_{d}$ in $B_{1,+}$. This ends the proof. ∎

######  Lemma A.6.

Let $\Omega,A$ and $u$ be the same as in Lemma 5.3. If $\omega_{0}$ and
$\gamma_{0}$ are sufficiently small, then for any $x\in B_{1/40,+}$,

$\sup_{\frac{1}{10}<r\leq\frac{1}{5}}N_{u}(x,r)\leq 12N+C_{d},$

where $C_{d}>0$ is an absolute constant depending only on $d$.

###### Proof.

Let $\varepsilon>0$ be small such that $(1+\varepsilon)^{5}\leq 2$. By the
assumption (5.6) and Lemma 3.4, for $\omega_{0}$ and $\gamma_{0}$ sufficiently
small, $1\leq k\leq 6$

$N_{u}(0,2^{-k})+1\leq(1+\varepsilon)^{k-1}(N_{u}(0,\frac{1}{2})+1)\leq
2(N+1).$

Let $x\in B_{1/40,+}$ and $1/10<r\leq 1/5$. If $\gamma_{0}$ is sufficiently
small, then the difference between $A(0)$ and $A(x)$ is small. Thus, in view
of the definition of the ellipsoids $E(x,r)$, we have $E(0,1/64)\subset
E(x,1/10)\subset E(x,2/5)\subset E(0,1)$. It follows that

$\displaystyle N_{u}(x,r)$
$\displaystyle\leq\log\frac{J_{u}(x,2/5)}{J_{u}(x,1/10)}\leq\log\frac{CJ_{u}(0,1)}{J_{u}(0,1/64)}$
$\displaystyle=\log C+\sum_{k=1}^{6}N_{u}(0,2^{-k})$ $\displaystyle\leq
12N+C_{d},$

as desired. ∎

## References

  * [1] V. Adolfsson and L. Escauriaza.  $C^{1,\alpha}$ domains and unique continuation at the boundary.  Comm. Pure Appl. Math., 50(10):935–969, 1997.
  * [2] V. Adolfsson, L. Escauriaza, and C. Kenig.  Convex domains and unique continuation at the boundary.  Rev. Mat. Iberoamericana, 11(3):513–525, 1995.
  * [3] G. Alessandrini, L. Rondi, E. Rosset, and S. Vessella.  The stability for the Cauchy problem for elliptic equations.  Inverse Problems, 25(12):123004, 47, 2009.
  * [4] J. Bourgain and T. Wolff.  A remark on gradients of harmonic functions in dimension $\geq 3$.  Colloq. Math., 60/61(1):253–260, 1990.
  * [5] A. Cianchi and V. G. Maz’ya.  Global Lipschitz regularity for a class of quasilinear elliptic equations.  Comm. Partial Differential Equations, 36(1):100–133, 2011.
  * [6] H. Donnelly and C. Fefferman.  Nodal sets of eigenfunctions on Riemannian manifolds.  Invent. Math., 93(1):161–183, 1988.
  * [7] H. Donnelly and C. Fefferman.  Nodal sets of eigenfunctions: Riemannian manifolds with boundary.  In Analysis, et cetera, pages 251–262. Academic Press, Boston, MA, 1990.
  * [8] R. M. Dudley.  On second derivatives of convex functions.  Math. Scand., 41(1):159–174, 1977.
  * [9] J. M. Gallegos.  Size of the zero set of solutions of elliptic PDEs near the boundary of Lipschitz domains with small Lipschitz constant.  arXiv:2201.12307, 2022.
  * [10] H. Jia, D. Li, and L. Wang.  Global regularity for divergence form elliptic equations on quasiconvex domains.  J. Differential Equations, 249(12):3132–3147, 2010.
  * [11] C. Kenig.  Harmonic analysis techniques for second order elliptic boundary value problems, volume 83 of CBMS Regional Conference Series in Mathematics.  Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994.
  * [12] C. Kenig, F. Lin, and Z. Shen.  Periodic homogenization of Green and Neumann functions.  Comm. Pure Appl. Math., 67(8):1219–1262, 2014.
  * [13] C. Kenig and Z. Shen.  Layer potential methods for elliptic homogenization problems.  Comm. Pure Appl. Math., 64(1):1–44, 2011.
  * [14] C. Kenig and Z. Zhao.  Boundary unique continuation on $C^{1}$-Dini domains and the size of the singular set.  Arch. Ration. Mech. Anal., 245(1):1–88, 2022.
  * [15] C. Kenig and Z. Zhao.  Examples of non-Dini domains with large singular sets.  arXiv:2212.01541, 2022.
  * [16] I. Kukavica and K. Nyström.  Unique continuation on the boundary for Dini domains.  Proc. Amer. Math. Soc., 126(2):441–446, 1998.
  * [17] F. Lin and J. Zhu.  Upper bounds of nodal sets for eigenfunctions of eigenvalue problems.  Math. Ann., 382(3-4):1957–1984, 2022.
  * [18] F.-H. Lin.  Nodal sets of solutions of elliptic and parabolic equations.  Comm. Pure Appl. Math., 44(3):287–308, 1991.
  * [19] A. Logunov.  Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure.  Ann. of Math. (2), 187(1):221–239, 2018.
  * [20] A. Logunov.  Nodal sets of Laplace eigenfunctions: proof of Nadirashvili’s conjecture and of the lower bound in Yau’s conjecture.  Ann. of Math. (2), 187(1):241–262, 2018.
  * [21] A. Logunov, E. Malinnikova, N. Nadirashvili, and F. Nazarov.  The sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions.  Geom. Funct. Anal., 31(5):1219–1244, 2021.
  * [22] S. Mccurdy.  Unique continuation on convex domains.  Rev. Mat. Iberoam., to appear.
  * [23] D. Mitrea, M. Mitrea, and L. Yan.  Boundary value problems for the Laplacian in convex and semiconvex domains.  J. Funct. Anal., 258(8):2507–2585, 2010.
  * [24] X. Tolsa.  Unique continuation at the boundary for harmonic functions in ${C}^{1}$ domains and Lipschitz domains with small constant.  Communications on Pure and Applied Mathematics, to appear.
  * [25] J. Zhu.  Boundary doubling inequality and nodal sets of robin and neumann eigenfunctions.  Potential Anal, 2022.
  * [26] J. Zhuge.  Weak maximum principle for biharmonic equations in quasiconvex Lipschitz domains.  J. Funct. Anal., 279(12):108786, 36, 2020.