# The sharp upper bound for the area of the nodal sets of Dirichlet Laplace
eigenfunctions

A. Logunov  A.L.: Department of Mathematics, Princeton University, Princeton,
NJ, USA<EMAIL_ADDRESS>,  E. Malinnikova  E.M.: Department of Mathematics,
Stanford University, Stanford, CA, USA Department of Mathematical Sciences,
NTNU, Trondheim, Norway<EMAIL_ADDRESS>,  N. Nadirashvili  N.N: CNRS,
Institut de Mathématiques de Marseille, Marseille, France
<EMAIL_ADDRESS>and  F. Nazarov  F.N.: Department of
Mathematics, Kent State University, Kent, OH, USA<EMAIL_ADDRESS>
###### Abstract.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ with $C^{1}$ boundary and
let $u_{\lambda}$ be a Dirichlet Laplace eigenfunction in $\Omega$ with
eigenvalue $\lambda$. We show that the $(n-1)$-dimensional Hausdorff measure
of the zero set of $u_{\lambda}$ does not exceed $C(\Omega)\sqrt{\lambda}$.
This result is new even for the case of domains with $C^{\infty}$-smooth
boundary.

##  1\. Introduction

Let $\Delta_{M}$ be the Laplace operator on a compact $n$-dimensional
Riemannian manifold and let $u_{\lambda}$ be an eigenfunction of $-\Delta_{M}$
with the eigenvalue $\lambda$, i.e., $\Delta_{M}u_{\lambda}+\lambda
u_{\lambda}=0$. Denote by $Z(u_{\lambda})=\\{u_{\lambda}=0\\}$ the zero set of
$u_{\lambda}$. S. T. Yau [20] conjectured that the surface area of the zero
set of $u_{\lambda}$ satisfies the following inequalities

$c\sqrt{\lambda}\leq\mathcal{H}^{n-1}(Z(u_{\lambda}))\leq C\sqrt{\lambda},$

where the constants $c,C$ depend on $M$. This conjecture was proved by
Donnelly and Fefferman in [6] under the assumption that the metric is real
analytic. The lower bound and a polynomial in $\lambda$ upper bound were
obtained recently by the first author in [15] and [14] respectively.

In this article we consider the case of eigenfunctions of the Euclidean
Laplace operator on a bounded domain with sufficiently regular boundary and
the Dirichlet boundary condition. One of our results is the following.

######  Theorem 1.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ with $C^{1}$ boundary and
let $u_{\lambda}$ be an eigenfunction of the Laplace operator with the
Dirichlet boundary condition, $\Delta u_{\lambda}+\lambda u_{\lambda}=0$ and
$u_{\lambda}|_{\partial\Omega}=0$. Then

(1) $\mathcal{H}^{n-1}(Z(u_{\lambda}))\leq C\sqrt{\lambda},$

where $C$ depends only on $\Omega$.

The lower bound

$\mathcal{H}^{n-1}(Z(u_{\lambda}))\geq c\sqrt{\lambda},$

for sufficiently large $\lambda$, follows from the results of Donnelly and
Fefferman in [6] combined with Lemma 10 below. We remark that this bound also
holds for any solution of the equation $\Delta u_{\lambda}+\lambda
u_{\lambda}=0$ and the boundary condition plays no role. This follows from the
fact that the zero set is $C\lambda^{-1/2}$ dense and a non-trivial result of
[15]. The inequality (1) was also proved by Donnelly and Fefferman in [7] for
the case of real analytic boundary $\partial\Omega$. Their result was
generalized to eigenfunctions of elliptic operators with real analytic
coefficients by Kukavica [9]. Similar estimates were recently obtained by Lin
and Zhu [13] for eigenfunctions of the bi-Laplace operator with various
boundary conditions under the assumption that the boundary is real analytic.
Also, the polynomial (in the eigenvalue) upper bounds for the area of the zero
set of the Dirichlet, Neumann, and Robin eigenfunctions in smooth bounded
domains in $\mathbb{R}^{n}$ were proved by Zhu in [21].

Our proof of Theorem 1 is based on the results of Donnelly and Fefferman and
the ideas developed in [16, 14, 15]. In particular, we reduce the statement of
the theorem to an estimate of the size of the nodal set of a harmonic function
with controlled doubling index (the doubling index in defined in Section 3
below). The novelty of the current work is the treatment of domains with non-
analytic boundaries. More precisely, we work with Lipschitz domains in the
Euclidean space and assume that (locally) the Lipschitz constant is small
enough; the precise definition and the formulation of the main result are
given in the next section. This class of domains was recently considered by
Tolsa [19] in a different problem.

The rest of the article is organized in the following way. In Section 3 we
first discuss the doubling index of harmonic functions and its (weak)
monotonicity properties near the boundary of Lipschitz domains with small
Lipschitz constant, and then we formulate the main estimate for the size of
the zero set of harmonic functions in terms of the doubling index, see Theorem
2 below. Two auxiliary results are contained in Section 4, where the low
regularity of the boundary requires some careful considerations. We prove
Theorem 2 for harmonic functions in Section 5, and explain how Theorem 1
follows from Theorem 2 in Section 6.

Acknowledgements.The authors are grateful to Misha Sodin for constant
encouragement and motivation, as well as for the organization of research
visits and workshops that allowed the authors to meet and discuss the current
work, including the workshop on nodal sets of eigenfunctions at IAS,
Princeton, in February 2017. Our special thanks also go to the Tel Aviv
University, where this work started. A.L. was supported in part by the Packard
Fellowship and Sloan Fellowship. E.M. was partially supported by NSF grant
DMS-1956294 and by Research Council of Norway, Project 275113. F.N. was
partially supported by NSF grant DMS-1900008.

##  2\. Preliminaries

###  2.1. Smoothness of the boundary

Some of the tools used in the current paper should be compared to those in
[19], where the following boundary uniqueness conjecture is studied.

Let $h$ be a bounded harmonic function in a Lipschitz domain $\Omega$. Assume
that $h$ vanishes on a relatively open set $U\subset\partial\Omega$ and
$\nabla h$ vanishes on a subset of $U$ of positive surface measure. Then
$h=0$.

Recently Xavier Tolsa verified the conjecture for Lipschitz domains with small
Lipschitz constant, see [19]. We use the following definition.

######  Definition 1.

Let $\Omega$ be a domain in $\mathbb{R}^{d}$, $\tau\in(0,1)$, and let
$B=B(x,r)$ be a ball centered on $\partial\Omega$. We say that
$\partial\Omega$ is $\tau$-Lipschitz in $B$ if there is an isometry
$T:\mathbb{R}^{d}\to\mathbb{R}^{d}$ and a function
$f:B^{d-1}(0,r)\to\mathbb{R}$ such that $T(0)=x$, $f$ is a Lipschitz function
with the Lipschitz constant bounded by $\tau$, $f(0)=0$, and

$\Omega\cap B=T\left(\\{(y^{\prime},y^{\prime\prime})\in
B^{d}(0,r)\subset\mathbb{R}^{d-1}\times\mathbb{R}:y^{\prime\prime}>f(y^{\prime})\\}\right).$

In this case we write $\partial\Omega\cap B\in Lip(\tau)$.

######  Remark 1.

Our considerations are mostly local. When considering the part of the boundary
$\partial\Omega\cap B\in Lip(\tau)$, we choose local coordinates in
$B=B(x,r)$, $x\in\partial\Omega$, so that the isometry $T$ in the definition
is the identity. We denote by $e_{d}$ the unit vector in the direction of the
last coordinate, so that $x+\varepsilon e_{d}\in\Omega$ for $0<\varepsilon<r$.

######  Remark 2.

Note that if $\partial\Omega\cap B\in Lip(\tau)$ and $B_{1}\subset B$ is a
ball centered on $\partial\Omega$, then $\partial\Omega\cap B_{1}\in
Lip(\tau)$. Also, rescaling does not change the Lipschitz constant. So if
$\partial\Omega\cap B\in Lip(\tau)$ and $x\in\partial\Omega$ is the center of
$B$, then, denoting $\Omega_{c}=\\{x+c(y-x):y\in\Omega\\}$ and
$B_{c}=cB=\\{x+c(y-x):y\in B\\}$ for some $c>0$, we have
$\partial\Omega_{c}\cap B_{c}\in Lip(\tau)$.

######  Definition 2.

We say that $\Omega$ is a Lipschitz domain with local Lipschitz constant
$\tau$ if there exists $r>0$ such that $\partial\Omega\cap B(x,r)\in
Lip(\tau)$ for any $x\in\partial\Omega$.

Clearly, any bounded $C^{1}$ domain is a domain with local Lipschitz constant
$\tau$ for any positive $\tau$. So Theorem 1 follows from the next result.

######  Theorem $\mathbf{1^{\prime}}$.

For each $n$, there exists $\tau_{n}>0$ such that the following statement
holds. Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{n}$ with
local Lipschitz constant $\tau_{n}$ and let $u_{\lambda}$ be an eigenfunction
of the Laplace operator in $\Omega$ with the Dirichlet boundary condition,
$\Delta u_{\lambda}+\lambda u_{\lambda}=0$ and
$u_{\lambda}|_{\partial\Omega}=0$. Then

$\mathcal{H}^{n-1}(Z(u_{\lambda}))\leq C\sqrt{\lambda},$

where $C$ depends only on $\Omega$.

The constant $C$ depends only on the parameter $r$ for $\Omega$ in the
definition of a Lipschitz domain with local Lipschitz constant $\tau_{n}$, on
the diameter of $\Omega$, and on the dimension $n$. In what follows we assume
that the dimension of the ambient Euclidean space is fixed so usually we will
not emphasize the dependence of our constants on it.

The rest of the article is devoted to a proof of Theorem ${1^{\prime}}$. We
start with the following property of Lipschitz domains.

######  Lemma 1.

Suppose that $\partial\Omega\cap B\in Lip(\tau)$, $\tau<1/4$, where $B=B(x,r)$
and $x\in\partial\Omega$. We choose coordinates as in Remark 1. Let
$x_{0}\in\overline{\Omega}\cap\frac{1}{4}B$, and let $x_{1}=x_{0}+\tau
re_{d}$. Then $\Omega\cap B(x_{1},r/2)$ is star-shaped with respect to
$x_{1}$.

A version of this lemma can be found in [10]. We provide a proof for the
convenience of the reader.

###### Proof.

Let $x_{1}=(x_{1}^{\prime},x_{1}^{\prime\prime})$. Suppose that
$x_{2}=(x_{2}^{\prime},x_{2}^{\prime\prime})\in\Omega\cap B(x_{1},r/2)$. Let
now $x_{3}=(x_{3}^{\prime},x_{3}^{\prime\prime})$ be a point on the interval
$(x_{1},x_{2})$. Clearly $x_{3}\in B(x_{1},r/2)$. We want to check that
$x_{3}^{\prime\prime}>f(x_{3}^{\prime})$.

Let $x_{3}=ax_{1}+(1-a)x_{2}$, $a\in(0,1)$. We have $x_{1}^{\prime\prime}\geq
f(x_{1}^{\prime})+\tau r$ and $x_{2}^{\prime\prime}>f(x_{2}^{\prime})$.
Therefore, we obtain

$x_{3}^{\prime\prime}=ax_{1}^{\prime\prime}+(1-a)x_{2}^{\prime\prime}>af(x_{1}^{\prime})+a\tau
r+(1-a)f(x_{2}^{\prime}).$

Then, since $f(x_{1}^{\prime})\geq
f(x_{2}^{\prime})-\tau|x_{1}^{\prime}-x_{2}^{\prime}|>f(x_{2}^{\prime})-\tau
r/2$, we have

$x_{3}^{\prime\prime}>f(x_{2}^{\prime})+a\tau r-\frac{a\tau
r}{2}=f(x_{2}^{\prime})+\frac{a\tau r}{2}>f(x_{3}^{\prime}),$

where the last inequality holds since
$|x_{3}^{\prime}-x_{2}^{\prime}|=a|x_{1}^{\prime}-x_{2}^{\prime}|<ar/2$ and
$f$ is $\tau$-Lipschitz. ∎

###  2.2. Some observations

In this section we recall some results about harmonic functions.

Suppose that $h$ is a harmonic function in $\Omega$, $h\in
C(\overline{\Omega})$, and $h=0$ on $\partial\Omega\cap B$, where $B=B(x,r)$
and $x\in\overline{\Omega}$. We define the function $v$ in $B$ by $v=h^{2}$ in
$\Omega\cap B$, and $v=0$ in $B\setminus\Omega$. Then $v$ is subharmonic in
$B$ and the mean-value theorem implies that for any $y\in B(x,r/2)\cap\Omega$,

(2)
$h^{2}(y)\leq\frac{1}{|B(y,r/2)|}\int_{B(y,r/2)\cap\Omega}h^{2}\leq\frac{1}{|B(y,r/2)|}\int_{B(x,r)\cap\Omega}h^{2},$

where $|E|$ is the $d$-dimensional Lebesgue measure of the set $E$.

Another known fact that we use is the following quantitative version of the
Cauchy uniqueness theorem.

######  Lemma 2.

Let $B_{+}$ be the half-ball,

$B_{+}=\\{(x^{\prime},x^{\prime\prime})\in\mathbb{R}^{d-1}\times\mathbb{R}:|x^{\prime}|^{2}+(x^{\prime\prime})^{2}<1,x^{\prime\prime}>0\\}.$

There exist $\gamma\in(0,1)$ and $C>0$ such that if $h$ is harmonic in
$B_{+}$, $h\in C^{1}(\overline{B}_{+})$ and satisfies the inequalities
$|h|\leq 1,|\nabla h|\leq 1$ in $B_{+}$ and $|h|\leq\varepsilon$,
$|\partial_{d}h|\leq\varepsilon$ on
$\Gamma=\\{(x^{\prime},x^{\prime\prime})\in\overline{B}_{+},x^{\prime\prime}=0\\}$,
$\varepsilon\leq 1$, then

$|h(x)|\leq C\varepsilon^{\gamma}\quad{\text{when}}\quad
x\in\frac{1}{3}B_{+}=\left\\{(x^{\prime},x^{\prime\prime}):|x^{\prime}|^{2}+(x^{\prime\prime})^{2}<\frac{1}{9},x^{\prime\prime}>0\right\\}.$

The reader can find a proof of a similar statement in [12] and a general
result on second order elliptic PDEs in Lipschitz domains in [5]. A simple
proof is also given in Section A.5 for the convenience of the reader.

##  3\. The doubling index

###  3.1. The doubling index inside the domain

Let $h\in C(\overline{\Omega})$ be a non-zero harmonic function in a domain
$\Omega\subset\mathbb{R}^{d}$. For each $x\in\overline{\Omega}$ and $r>0$, we
define

(3) $H_{h}(x,r)=\int_{B(x,r)\cap\Omega}h^{2}\quad{\text{and}}\quad
N_{h}(x,r)=\log\frac{H_{h}(x,2r)}{H_{h}(x,r)},$

and, with some abuse of language, we call $N_{h}(x,r)$ the doubling index of
$h$ in $B=B(x,r)$.

Assume first that $B(x,2R)\subset\Omega$, then

(4) $N_{h}(x,r)\leq N_{h}(x,R),\quad{\text{when}}\quad r<R.$

An elementary proof can be obtained by decomposing $h$ into spherical
harmonics, see, e.g., [18]. This is a simple and useful result, its various
versions go back to the works of Landis [11], Agmon [1], and Almgren [2].

Suppose that $B(x,4r)\subset\Omega$. Then we rewrite the inequality
$N_{h}(x,r)\leq N_{h}(x,2r)$ as

(5)
$\left(\int_{B(x,2r)}h^{2}\right)^{2}\leq\int_{B(x,r)}h^{2}\int_{B(x,4r)}h^{2}.$

Similarly to (2), for any $y\in B(x,3r/2)$, we have

$h^{2}(y)\leq\frac{1}{|B(y,r/2)|}\int_{B(y,r/2)}h^{2}\leq\frac{1}{|B(y,r/2)|}\int_{B(x,2r)}h^{2}.$

Finally, applying (5) and using the trivial bound of the $L^{2}$ norms by the
$L^{\infty}$ norms, we obtain

(6) $\sup_{B(x,3r/2)}|h|\leq
2^{d}(\sup_{B(x,r)}|h|)^{1/2}(\sup_{B(x,4r)}|h|)^{1/2}.$

###  3.2. The doubling index on the boundary

We need a version of the monotonicity formula (4) and the three ball
inequality (6) near a part of the boundary on which the harmonic function
vanishes. First, we recall a lemma that is proven in [10].

######  Lemma 3 (Kukavica, Nyström).

Let $\Omega$ be a domain in $\mathbb{R}^{d}$ and let $B_{1}$ be a ball
centered on $\partial\Omega$ such that $\partial\Omega\cap B_{1}$ is $C^{3}$
smooth. Let also $x\in\Omega$ be such that $\Omega\cap B(x,R)$ is star-shaped
with respect to $x$, $B(x,R)\subset B_{1}$. Suppose that $h\in
C(\overline{\Omega})$ is a non-zero harmonic function in $\Omega$ and $h=0$ on
$\partial\Omega\cap B_{1}$. Then

(7)
$\log\frac{H_{h}(x,r_{2})}{H_{h}(x,r_{1})}\leq\frac{\log(r_{2}/r_{1})}{\log(r_{3}/r_{2})}\log\frac{H_{h}(x,r_{3})}{H_{h}(x,r_{2})},$

when $0<r_{1}<r_{2}<r_{3}<R$.

The assumption that the boundary of $\Omega$ is $C^{3}$ smooth implies that
$h\in C^{2}(\overline{\Omega}\cap B_{1})$, so every integration by parts in
[10] can be easily justified.

Now we prove the following almost monotonicity property of the doubling index
in Lipschitz domains.

######  Lemma 4.

Let $\Omega$ be a domain in $\mathbb{R}^{d}$. For any $\varepsilon>0$, there
exists $\tau_{\varepsilon}>0$ such that if $\tau<\tau_{\varepsilon}$,
$\partial\Omega\cap B\in Lip(\tau)$, where $B=B(x,R),\ x\in\partial\Omega$,
and $h\in C(\overline{\Omega})$ is a non-zero harmonic function in $\Omega$,
$h=0$ on $\partial\Omega\cap B$, then

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

for any $x_{0}\in\overline{\Omega}\cap\frac{1}{4}B$ and $r<R/16$.

We remark that a stronger result holds when the boundary of the domain is
smooth. For example for the case of a $C^{1,Dini}$ domain the inequality (8)
can be replaced by $N_{h}(x_{0},r_{1})\leq(1+\varepsilon)N(x_{0},r_{2})$ when
$r_{1}<r_{2}<R/8$, see [10]111We refer the reader also to the preceding works
[4] and [3] for related results.. Thus for $C^{1,Dini}$ domains, we know that
the doubling index over balls centered on $\partial\Omega\cap B$ stays
uniformly bounded. We do not know if this is still true for Lipschitz domains.
For the case of the domains with small Lipschitz constant, we can conclude
only that the doubling index $N_{h}(x_{0},r)$ does not grow faster than
$r^{-a}$ with some small positive $a$ as $r\to 0$, which is sufficient for our
purposes.

###### Proof.

First we assume that $\partial\Omega\cap B$ is a graph of a $C^{3}$-smooth
function. Let $e_{d}$ be as in Remark 1 and let $x_{1}=x_{0}+16\tau re_{d}$.
We assume that $\tau<1/16$. Then by Lemma 1 we see that
$B(x_{1},8r)\cap\Omega$ is star-shaped with respect to $x_{1}$. We apply (7)
and obtain

$N_{h}(x_{0},r)=\log\frac{H_{h}(x_{0},2r)}{H_{h}(x_{0},r)}\leq\log\frac{H_{h}(x_{1},(2+16\tau)r)}{H_{h}(x_{1},(1-16\tau)r)}\\\
\leq\frac{\log((2+16\tau)/(1-16\tau))}{\log((4-16\tau)/(2+16\tau))}\log\frac{H_{h}(x_{1},(4-16\tau)r)}{H_{h}(x_{1},(2+16\tau)r)}\\\
\leq(1+O(\tau))\log\frac{H_{h}(x_{0},4r)}{H_{h}(x_{0},2r)}=(1+\varepsilon)N_{h}(x_{0},2r),$

when $\tau$ is small enough.

We want to drop the assumption that $\partial\Omega\cap B$ is $C^{3}$ smooth.
We fix the ball $B$ and assume that $\partial\Omega\cap B$ is given by the
graph of a Lipschitz function $f:B^{d-1}(0,R)\to\mathbb{R}$ with the Lipschitz
constant bounded by $\tau$. In this coordinate system the ball $B$ is
identified with $B^{d}(0,R)$

Let $\varphi$ be a mollifier supported in the unit ball of $\mathbb{R}^{d-1}$
and let, as usual,
$\varphi_{\delta}(x)=\delta^{-(d-1)}\varphi\left(\frac{x}{\delta}\right)$. We
define $f_{n}=f*\varphi_{R/n}+\tau R/n$. Then $\\{f_{n}\\}$ is a sequence of
$C^{3}$ smooth functions such that

$f_{n}:B^{d-1}(0,(1-1/n)R)\to\mathbb{R},\quad
f(y^{\prime})<f_{n}(y^{\prime})<f(y^{\prime})+2\tau R/n,$

and the Lipschitz constant of $f_{n}$ is also bounded by $\tau$. We also
define

$\Omega_{n}=\\{y=(y^{\prime},y^{\prime\prime})\in
B^{d}(0,(1-1/n)R)\subset\mathbb{R}^{d-1}\times\mathbb{R}:y^{\prime\prime}>f_{n}(y^{\prime})\\}.$

Clearly, $\Omega_{n}\subset B^{d}(0,(1-1/n)R)\cap\Omega$. Let

$\Gamma_{n}=\\{y=(y^{\prime},y^{\prime\prime})\in
B^{d}(0,(1-1/n)R):y^{\prime\prime}=f_{n}(y^{\prime})\\}.$

First, we see that $\delta_{n}=\sup_{\Gamma_{n}}|h|$ converge to zero as
$n\to\infty$ since $h$ is uniformly continuous on
$\overline{\Omega}\cap\overline{B}$, $h=0$ on $\partial\Omega$, and
${\operatorname{dist}}(y,\partial\Omega)<2\tau R/n$ when $y\in\Gamma_{n}$.

Next, we consider the harmonic function $h_{n}$ in $\Omega_{n}$ such that on
$\partial\Omega_{n}$

$h_{n}(x)=\begin{cases}h(x)-\delta_{n},\ {\text{if}}\ h(x)>\delta_{n},\\\
0,\quad\quad\quad\ \ \ {\text{if}}\ |h(x)|\leq\delta_{n},\\\ h(x)+\delta_{n},\
{\text{if}}\ h(x)<-\delta_{n}.\end{cases}$

Clearly we have $h_{n}\in C(\overline{\Omega}_{n})$, $h_{n}=0$ on
$\Gamma_{n}$, and, by the maximum principle, $|h-h_{n}|\leq\delta_{n}$ in
$\Omega_{n}$. Thus $h_{n}\rightarrow h$ uniformly on compact subsets of
$B\cap\Omega$.

We fix $x_{0}\in\Omega\cap\frac{1}{4}B$ and $r\in(0,R/16)$. Then
$x_{0}\in\Omega_{n}\cap B(0,(1-1/n)R/4)$ and $r<(1-1/n)R/16$ for $n$ large
enough. Also, $|h_{n}|\leq\max_{\overline{\Omega}\cap\overline{B}}|h|$ and
$|(\Omega\cap B(x,R))\setminus\Omega_{n}|\to 0$ as $n\to\infty$. Then we have

$N_{h}(x_{0},r)=\frac{\int_{B(x_{0},2r)\cap\Omega}h^{2}}{\int_{B(x,r)\cap\Omega}h^{2}}=\lim_{n\to\infty}\frac{\int_{B(x_{0},2r)\cap\Omega_{n}}h_{n}^{2}}{\int_{B(x_{0},r)\cap\Omega_{n}}h_{n}^{2}}=\lim_{n\to\infty}N_{h_{n}}(x_{0},r).$

The inequality (8) is now obtained as the limit of the corresponding
inequalities for $h_{n}$. Finally the required inequality (8) for
$x_{0}\in\partial\Omega\cap\frac{1}{4}B$ follows by taking the limit as
$\varepsilon\to 0+$ of the corresponding inequalities for $x_{0}+\varepsilon
e_{d}$. ∎

######  Corollary 1.

Let $\partial\Omega\cap B\in Lip(\tau)$, $\tau<\tau_{\varepsilon}$,
$B=B(x,R)$, $x\in\partial\Omega$, and let
$x_{1},x_{2}\in\overline{\Omega}\cap\frac{1}{4}B$ with $|x_{1}-x_{2}|<r/4$ and
$r<R/8$. If $h\in C(\overline{\Omega})$ is a non-zero harmonic function in
$\Omega$ such that $h=0$ on $\partial\Omega\cap B$, then

$N_{h}(x_{1},r/2)\leq 3(1+\varepsilon)^{2}N_{h}(x_{2},r).$

###### Proof.

Note that $B(x_{1},r)\subset B(x_{2},2r)$ and $B(x_{1},r/2)\supset
B(x_{2},r/4)$. Thus we obtain

$N_{h}(x_{1},r/2)\leq\log\frac{\int_{B(x_{2},2r)\cap\Omega}h^{2}}{\int_{B(x_{2},r/4)\cap\Omega}h^{2}}=N_{h}(x_{2},r/4)+N_{h}(x_{2},r/2)+N_{h}(x_{2},r).$

Now Lemma 4 implies the required estimate. ∎

###  3.3. Three ball inequality

We apply the monotonicity lemma a number of times. First, we claim that it
implies a version of the three ball theorem for harmonic functions vanishing
on some part of the boundary.

######  Lemma 5.

Let $\Omega$ be a domain in $\mathbb{R}^{d}$ and let $B$ be a ball centered on
$\partial\Omega$. We assume that $\partial\Omega\cap B\in Lip(\tau)$, where
$\tau$ is small enough. Then for any function $h\in C(\overline{\Omega})$
harmonic in $\Omega$ and vanishing on $\partial\Omega\cap B$, we have

$\sup_{\frac{3}{2}B_{0}\cap\Omega}|h|\leq
3^{d}(\sup_{B_{0}\cap\Omega}|h|)^{1/3}(\sup_{4B_{0}\cap\Omega}|h|)^{2/3},$

for any ball $B_{0}$ with the center in $\overline{\Omega}\cap\frac{1}{4}B$
and such that $16B_{0}\subset B$.

###### Proof.

Assume that $\tau<\tau_{1}$ given by Lemma 4 for the case $\varepsilon=1$. Let
$B_{0}=B(x_{0},r)$. We apply Lemma 4. Taking the exponentials, we obtain

$\int_{2B_{0}\cap\Omega}h^{2}\leq\left(\int_{B_{0}\cap\Omega}h^{2}\right)^{1/3}\left(\int_{4B_{0}\cap\Omega}h^{2}\right)^{2/3}.$

Then (2) and the trivial bound of the $L^{2}$-norm by the $L^{\infty}$-norm
imply that for any $y\in\frac{3}{2}B_{0}\cap\Omega$,

$h^{2}(y)\leq\frac{1}{|B(y,r/2)|}\int_{2B_{0}\cap\Omega}h^{2}\leq
8^{d}\left(\sup_{B_{0}\cap\Omega}|h|\right)^{2/3}\left(\sup_{4B_{0}\cap\Omega}|h|\right)^{4/3}.$

∎

###  3.4. The maximal doubling index

Let $\Omega$ be a domain in $\mathbb{R}^{d}$ and let $\partial\Omega\cap B\in
Lip(\tau)$ where $B$ is centered on $\partial\Omega$. We consider a closed
cube $Q\subset\frac{1}{32}B$ such that $Q\cap\Omega\neq\varnothing$. Assume
that a non-zero function $h\in C(\overline{\Omega})$ is harmonic in $\Omega$
and vanishes on $\partial\Omega\cap B$ and let
$\ell={\operatorname{diam}}(Q)$. We define the maximal doubling index of $h$
in $Q$ by

(9) $N^{*}_{h}(Q)=\sup_{x\in Q\cap\overline{\Omega},\frac{\ell}{2}\leq
r\leq\ell}N_{h}(x,r).$

Clearly the function $(x,r)\mapsto N_{h}(x,r)$ is continuous on
$(Q\cap\overline{\Omega})\times[\ell/2,\ell]$. Therefore the supremum above is
finite.

Lemma 4 on the monotonicity of the doubling index implies that if
$\varepsilon>0$ and $\tau<\tau_{\varepsilon}$, then for any cube $Q_{1}\subset
Q\subset\frac{1}{32}B$ and $Q_{1}\cap\Omega\neq\varnothing$, we have

$N_{h}^{*}(Q_{1})\leq\left(\frac{2s(Q)}{s(Q_{1})}\right)^{2\varepsilon}N_{h}^{*}(Q),$

where $s(Q)$ is the side length of the cube $Q$; we have used the inequality
$\log_{2}(1+\varepsilon)\leq 2\varepsilon$.

###  3.5. A version of the main result for harmonic functions

Let $\Omega$ be a domain in $\mathbb{R}^{d}$ and let $h\in
C(\overline{\Omega})$ be a non-zero harmonic function in $\Omega$. We assume
that $h=0$ on the part $\partial\Omega\cap B$ of the boundary, where $B$ is a
ball centered on $\partial\Omega$ and $\partial\Omega\cap B\in Lip(\tau)$. Our
aim is to estimate the $(d-1)$-dimensional measure of the zero set of $h$
using the doubling index of $h$. We define the zero set of $h$ by

$Z(h)=\\{x\in\Omega:h(x)=0\\},$

so that the boundary points are not included into the zero set.

######  Theorem 2.

Let $\Omega\subset\mathbb{R}^{d}$, let $x\in\partial\Omega$ and let $r>0$ be
such that $\partial\Omega\cap B(x,128r)\in Lip(\tau),$ where $\tau$ is small
enough. Then there exists $C$ such that

$\mathcal{H}^{d-1}(Z(h)\cap B(x,r))\leq C(N_{h}(x,4r)+1)r^{d-1},$

for any non-zero function $h\in C(\overline{\Omega})$ that is harmonic in
$\Omega$ and satisfies $h=0$ on $\partial\Omega\cap B(x,128x)$.

Theorem 2 is proved in Section 5.2. We then deduce Theorem
$\ref{th:main}^{\prime}$ in Section 6.2, where we consider the harmonic
extension of the eigenfunction and use Lemma 10 below to estimate the doubling
index of the extension by a multiple of the square root of the eigenvalue.

Theorem 2 allows us to estimate the area of the zero set of a harmonic
function near the part of the boundary where the function vanishes. We remark
also that the estimate for the zero set inside the domain was proved by
Donnelly and Fefferman in [6].

######  Lemma 6 (Donnelly, Fefferman).

Let $h$ be a non-zero harmonic function in $\Omega\subset\mathbb{R}^{d}$.
There exists $C$ such that

$\mathcal{H}^{d-1}(Z(h)\cap B)\leq C(N_{h}(x,4r)+1)r^{d-1},$

for any ball $B=B(x,r)$ that satisfies $\overline{B}(x,8r)\subset\Omega$.

The proof follows from the argument in [6], some versions of this result can
be also found in [12] and [8]. We outline some steps of the proof for the
interested reader in the Appendix, see A.3.

##  4\. Two auxiliary lemmas

###  4.1. A standard construction

In this section we give two versions of the Hyperplane Lemma. We suggest that
the reader compares the statements to the one of [14, Lemma 4.1]. Both
statements refer to the following construction.

Figure 1. The standard construction

Assume that $\Omega\subset\mathbb{R}^{d}$ and $\partial\Omega\cap B\in
Lip(\tau)$, where $B$ is a ball centered on $\partial\Omega$ and
$\tau\in(0,(16\sqrt{d})^{-1})$. We fix a coordinate system as in Remark 1. Let
$Q$ be a cube centered at
$x_{Q}=(x_{Q}^{\prime},x_{Q}^{\prime\prime})\in\partial\Omega\cap B$ whose
sides are parallel to the axes of this coordinate system and such that
$Q\subset B$. As above, the side length of $Q$ is denoted by $s(Q)$. Our
choice of $\tau$ implies that $\partial\Omega$ does not intersect the two
faces of the cube $Q$ which are orthogonal to $e_{d}$, moreover,
$\partial\Omega\cap Q$ is contained in the middle part
$\\{(x^{\prime},x^{\prime\prime})\in
Q:|x^{\prime\prime}-x_{Q}^{\prime\prime}|<s(Q)/4\\}$ of $Q$.

Let $k\geq 3$. We partition the projection $\pi(Q)$ of $Q$ to the hyperplane
$\mathbb{R}^{d-1}\times\\{0\\}$ into $2^{k(d-1)}$ small equal cubes $w$ with
the side length $s(w)=2^{-k}s(Q)$ in the usual way so that any two distinct
small cubes have no common inner points. For each small cube $w$, there is a
uniquely defined $d$-dimensional cube $q$ such that $\pi(q)=w$ and the center
of $q$ lies on $\partial\Omega\cap Q$. Furthermore, we cover
$(\pi^{-1}(w)\cap(\Omega\cap Q))\setminus{q}$ by at most $2^{k}$ cubes $p$
such that $p\subset Q$, $p$, $\pi(p)=w$, $p$ has no common inner points with
$q$, and $s(p)=s(q)=2^{-k}s(Q)$, cubes $p$ may overlap. See Figure 1.

We denote the set of all boundary cubes $q$ by $\mathcal{B}_{k}(Q)$ and the
set of all inner cubes $p$ by $\mathcal{I}_{k}(Q)$. Note that for each
$p\in\mathcal{I}_{k}(Q)$, we have
${\operatorname{dist}}(p,\partial\Omega)>cs(p)$ for some absolute constant
$c$. We call the triple $(Q,\mathcal{B}_{k}(Q),\mathcal{I}_{k}(Q))$ the
standard construction. After we fix a coordinate system, our standard
construction depends on the choice of the cube $Q$ and the parameter $k$, the
family $\mathcal{B}_{k}(Q)$ of the boundary cubes is defined uniquely and we
may fix some choice for the inner cubes $\mathcal{I}_{k}(Q)$.

###  4.2. The first hyperplane lemma

In the first lemma we assume that the maximal doubling index $N_{h}^{*}(Q)$ is
large enough.

######  Lemma 7.

There exist constants $k_{0}\geq 3$ and $N_{0}\geq 1$ such that for any
integer $k\geq k_{0},$ there exists $\tau(k)>0$ for which the following
statement holds. Suppose that $\Omega$ is a domain in $\mathbb{R}^{d}$,
$\partial\Omega\cap B\in Lip(\tau)$, $\tau<\tau(k),$ and
$Q\subset\frac{1}{64}B$ is a cube as above centered on $\partial\Omega$. Then
for any function $h\in C(\overline{\Omega})$ harmonic in $\Omega$, with $h=0$
on $B\cap\partial\Omega$, and $N^{*}_{h}(Q)>N_{0}$, there exists a cube
$q\in\mathcal{B}_{k}(Q)$ such that $N^{*}_{h}(q)\leq N_{h}^{*}(Q)/2$.

###### Proof.

Let $x_{Q}$ be the center of the cube $Q$ and let $B_{1}=B(x_{Q},\ell)$, where
$\ell={\operatorname{diam}}(Q)$. We have $B_{1}\subset B$ and define
$M^{2}=\int_{B_{1}\cap\Omega}h^{2}$.

Denote $N=N_{h}^{*}(Q)$ and suppose that the inequality $N^{*}_{h}(q)>N/2$
holds for each cube $q\in\mathcal{B}_{k}(Q)$. Then for each such $q$, there
exist $y_{q}\in q\cap\overline{\Omega}$ and
$r_{q}\in(2^{-k-1}\ell,2^{-k}\ell)$ such that $N_{h}(y_{q},r_{q})>N/2$.
Suppose that

(10) $\tau<\tau_{\varepsilon},\quad{\text{and}}\quad(1+\varepsilon)^{k}<2,$

where we use the notation of Lemma 4. Then the almost monotonicity of the
doubling index, Lemma 4, implies $N_{h}(y_{q},2^{m}r_{q})>N/4$ when $0\leq
m\leq k$.

Assuming that $k\geq 20$, we apply the estimate of the doubling index $k-4$
times and use that $B(y_{q},\ell/2)\subset B_{1}$ to obtain

$\int_{B(y_{q},2^{-k+2}\ell)\cap\Omega}h^{2}\leq\int_{B(y_{q},8r_{q})\cap\Omega}h^{2}\leq
e^{-N(k-4)/4}\int_{B(y_{q},2^{k-1}r_{q})\cap\Omega}h^{2}\\\ \leq
e^{-N(k-4)/4}\int_{B(y_{q},\ell/2)\cap\Omega}h^{2}\leq e^{-Nk/5}M^{2}.$

Next, we note that the integral estimate above implies a pointwise estimate in
a smaller ball by (2). We have

(11) $\sup_{B(y_{q},2^{-k+1}\ell)\cap\Omega}h^{2}\leq
C2^{dk}\ell^{-d}\int_{B(y_{q},2^{-k+2}\ell)\cap\Omega}h^{2}\leq
C2^{dk}\ell^{-d}e^{-Nk/5}M^{2},$

where $C=C(d)$.

As above, we assume also that $\tau<(16\sqrt{d})^{-1}$. For each cube
$q\in\mathcal{B}_{k}(Q)$, denote by $q^{+}$ its upper quarter, where ”up” is
in the direction of $e_{d}$. Then $q^{+}\subset\Omega$ and
${\operatorname{dist}}(q^{+},\partial\Omega)\geq 2^{-k}s(Q)/10$. For $y\in
q^{+}$, the standard Cauchy estimate implies

$|\nabla h(y)|\leq C2^{k}\ell^{-1}\sup_{B(y,2^{-k}s(Q)/10)}|h|.$

We note that $B(y,2^{-k}s(Q)/10)\subset B(y_{q},2^{-k+1}\ell)\cap\Omega$. Then
combining the above inequality with (11), we obtain

(12) $\sup_{q^{+}}|\nabla h|\leq
C2^{k}\ell^{-1}\sup_{B(y_{q},2^{-k+1}\ell)\cap\Omega}|h|\leq
C2^{k(d+2)/2}\ell^{-(d+2)/2}e^{-Nk/10}M.$

Let $B_{0}=B(x_{Q}+3\cdot 2^{-k-3}s(Q)e_{d},s(Q)/2)$ and let

$B_{0,+}=\\{x=(x^{\prime},x^{\prime\prime})\in B_{0}:x^{\prime\prime}\geq
x^{\prime\prime}_{Q}+3\cdot 2^{-k-3}s(Q)\\}$

be the upper half of $B_{0}$. We denote by $\Gamma_{0}$ the flat part of the
boundary of $B_{0,+}$. We note that $2B_{0}\subset B_{1}$. Assuming that
$\tau<2^{-k-3}$, we have ${\operatorname{dist}}(B_{0,+},\partial\Omega)\geq
2^{-k-2}s(Q)$. Then using (2) and the Cauchy estimate, we get

$\sup_{B_{0}\cap\Omega}|h|\leq C\ell^{-d/2}M,\quad\sup_{B_{0,+}}|\nabla h|\leq
C2^{k}\ell^{-d/2-1}M.$

Also, by (11) and (12), we have

$\sup_{\Gamma_{0}}|h|\leq
C2^{kd/2}\ell^{-d/2}e^{-Nk/10}M,\quad\sup_{\Gamma_{0}}|\nabla h|\leq
C2^{kd/2+k}\ell^{-d/2-1}e^{-Nk/10}M,$

since $\Gamma_{0}\subset\bigcup_{q\in\mathcal{B}_{k}(Q)}q^{+}$.

Applying Lemma 2 to $B_{0,+}$, we get

$\sup_{\frac{1}{3}B_{0,+}}|h|\leq C2^{\gamma kd/2+k}\ell^{-d/2}e^{-\gamma
Nk/10}M.$

Let $y_{Q}=x_{Q}+s(Q)e_{d}/12$ and let $m$ be the least integer such that
$2^{m}>16\sqrt{d}$. Then $B_{2}=B(y_{Q},2^{-m}\ell)\subset\frac{1}{3}B_{0,+}$
when $k$ is large enough (we remark that $B_{0,+}$ depends on $k$).
Integrating the last inequality over $B_{2}$ and using that $vol(B_{2})\leq
C\ell^{d}$, we obtain

$\int_{B_{2}}h^{2}\leq C2^{\gamma kd+2k}e^{-\gamma Nk/5}M^{2}.$

Finally, we compare the last integral to $\int_{B_{1}\cap\Omega}h^{2}=M^{2}$.
Note that $B_{1}\subset B(y_{Q},2\ell)=2^{m+1}B_{2}$. By the almost
monotonicity of the doubling index, recalling that
$\tau<\tau_{\varepsilon}<\tau_{1}$, we have

$2^{m+1}N_{h}(y_{Q},\ell)\geq\sum_{j=0}^{m}N_{h}(y_{Q},2^{-j}\ell)=\log\frac{\int_{B(y_{Q},2\ell)\cap\Omega}h^{2}}{\int_{B_{2}}h^{2}}\\\
\geq\log\frac{\int_{B_{1}\cap\Omega}h^{2}}{\int_{B_{2}}h^{2}}\geq\gamma
Nk/5-\gamma kd-2k-C.$

Since $N_{h}(y_{Q},\ell)\leq N^{*}_{h}(Q)=N$, we get $2^{m+1}N\geq\gamma
Nk/5-\gamma kd-2k-C$. Taking $k$ large enough we may achieve $\gamma
k/5>2^{m+2}$. Then the inequality above implies

$N\leq\frac{10(\gamma kd+2k+C)}{\gamma k}\leq
10\left(d+(2+C)\gamma^{-1}\right).$

Taking $N_{0}=10\left(d+(2+C)\gamma^{-1}\right)$, we obtain a contradiction
for $N>N_{0}$. We also choose $\varepsilon=\varepsilon(k)$ such that
$(1+\varepsilon)^{k}<2$ and finally choose
$\tau(k)=\min\\{\tau_{\varepsilon},2^{-k-3},(16\sqrt{d})^{-1}\\}$.

∎

###  4.3. The second hyperplane lemma: cubes without zeros

For cubes with the maximal doubling index bounded by $N_{0}$, we use the
following version of the above statement. The reader may compare it to
Corollary 3.4.4 in [17].

######  Lemma 8.

For any $N>0$ there exist $\tau(N)$ and $k(N)$ such that the following
statement holds. Suppose that $\Omega$ is a domain in $\mathbb{R}^{d}$,
$\partial\Omega\cap B\in Lip(\tau)$, $\tau<\tau(N)$, and
$Q\subset\frac{1}{64}B$ is a cube centered on $\partial\Omega$. Let also $h\in
C(\overline{\Omega})$ be a non-zero function harmonic in $\Omega$, with $h=0$
on $B\cap\partial\Omega$ and $N^{*}_{h}(Q)\leq N$. Then for any $k\geq k(N)$,
there exists $q\in\mathcal{B}_{k}(Q)$ such that $Z(h)\cap q=\varnothing$.

We remark that in this version both $\tau$ and $k$ depend on $N$. First, we
prove the following version of the lemma for a half ball.

######  Lemma 9.

Let $B$ be the unit ball in $\mathbb{R}^{d}$ and let $B_{+}$ be the half ball,

$B_{+}=\\{y=(y^{\prime},y^{\prime\prime})\in\mathbb{R}^{d-1}\times\mathbb{R}:|y^{\prime}|^{2}+y^{\prime\prime
2}<1,y^{\prime\prime}>0\\}.$

Let $g$ be a function harmonic in $B_{+}$, $g\in C(\overline{B}_{+})$, $g=0$
on $\overline{B}_{+}\cap\\{y^{\prime\prime}=0\\}$, and

$\sup_{\frac{1}{4}B_{+}}|g|=1.$

For any $N>0$, there exist $\rho=\rho(N)\in(0,1/16)$ and $c_{0}=c_{0}(N)>0$
such that if $N_{g}(0,1/4)\leq N$, then there is
$x^{\prime}\in\mathbb{R}^{d-1}$ with $|x^{\prime}|<1/16$ such that

$|g(y)|\geq c_{0}y^{\prime\prime},\quad{\text{for any}}\quad
y=(y^{\prime},y^{\prime\prime})\in B((x^{\prime},0),\rho)\cap B_{+}.$

###### Proof.

Let $B_{-}$ be the reflexion of the half-ball $B_{+}$ with respect to the
hyperplane $y^{\prime\prime}=0$. Then $g$ can be extended to a harmonic
function in $B$ by
$g(y^{\prime},y^{\prime\prime})=-g(y^{\prime},-y^{\prime\prime})$ when
$(y^{\prime},y^{\prime\prime})\in B_{-}$. We denote this extension by $g$ as
well. The normalization $\sup_{\frac{1}{4}B_{+}}|g|=1$ and the standard Cauchy
estimate imply that every partial derivative of $g$ is uniformly bounded in
$B(0,1/8)$.

Let $\delta=\max_{x^{\prime}\in\mathbb{R}^{d-1},|x^{\prime}|\leq 1/16}|\nabla
g(x^{\prime},0)|$. Lemma 2, applied to the half ball $\frac{1}{16}B_{+}$
implies that

$\sup_{B(0,\frac{1}{64})}|g|\leq C\delta^{\gamma}.$

Then $\int_{B(0,\frac{1}{64})}g^{2}\leq C\delta^{2\gamma}$ and
$\int_{B(0,\frac{1}{2})}g^{2}\geq c\sup_{B(0,\frac{1}{4})}g^{2}=c$. On the
other hand,

$\log\frac{\int_{B(0,\frac{1}{2})}g^{2}}{\int_{B(0,\frac{1}{64})}g^{2}}\leq
5N_{g}(0,\frac{1}{4})\leq 5N.$

We have used that the doubling index of $g$ in $B_{+}$ and of the extension
are the same for balls centered at the origin and that the doubling index
inside the domain is monotone by (4). We conclude that $\delta\geq
ce^{-3N\gamma^{-1}}$.

Let $x^{\prime}_{*}\in\mathbb{R}^{d-1}$, $|x^{\prime}_{*}|\leq 1/16$, be such
that $|\nabla g(x^{\prime}_{*},0)|=\delta$. Clearly we have $|\nabla
g(x^{\prime}_{*},0)|=|\partial_{d}g(x^{\prime}_{*},0)|$ and we may assume that
$\partial_{d}g(x_{*}^{\prime},0)=\delta$, otherwise we consider the function
$-g$. Then $\partial_{d}g(x)>\delta/2$ when
${\operatorname{dist}}(x,(x^{\prime}_{*},0))<\rho=\min\\{c_{0}\delta,1/16\\}$,
where $c_{0}$ depends on the constant upper bound for the second derivatives
of $g$ in $B(0,1/8)$. Therefore

$g(y)\geq\delta y^{\prime\prime}/2\geq ce^{-3N\gamma^{-1}}y^{\prime\prime},$

when $y=(y^{\prime},y^{\prime\prime})\in B((x_{*}^{\prime},0),\rho)$. ∎

###### Proof of Lemma 8.

Now we deduce Lemma 8 from Lemma 9. By rescaling, see Remark 2, we can achieve
that $s(Q)=4$. We may also assume that

(13) $\sup_{B(x_{Q},3)\cap\Omega}|h|=1.$

Let $x_{1}=x_{Q}-3\tau e_{d}$, $B_{1}=B(x_{1},1)$, and let $B_{1,+}$ be the
upper half of $B_{1}$. Let also $B_{2}=2B_{1}$. First, we consider the
harmonic function $g_{0}$ such that $g_{0}=1$ on the upper half of the sphere
$\partial B_{2}$ and $g_{0}=-1$ on the lower half of $\partial B_{2}$. We
denote as usual $x_{1}^{\prime\prime}=x_{1}\cdot e_{d}$. Clearly $g_{0}=0$ on

$\Gamma_{0}=\\{x=(x^{\prime},x^{\prime\prime})\in
B_{2}:x^{\prime\prime}=x_{1}^{\prime\prime}\\}$

and $g_{0}\geq 0$ on $B_{2,+}$. We note that $\Gamma_{0}$ does not intersect
$\overline{\Omega}$. Then $|h|\leq g_{0}$ on $\Omega\cap B_{2}\subset B_{2,+}$
by the maximum principle. We also have $g_{0}(x)\leq
C_{1}(x^{\prime\prime}-x^{\prime\prime}_{1})$ when
$x=(x^{\prime},x^{\prime\prime})\in B_{1,+}$, since $g_{0}=0$ on $\Gamma_{0}$
and $g_{0}$ has bounded derivatives in $B_{1}$. Therefore $|h(x)|\leq
C_{1}(x^{\prime\prime}-x^{\prime\prime}_{1})$ when
$x=(x^{\prime},x^{\prime\prime})\in\Omega\cap B_{1}$.

Let now $g$ be the harmonic function in $B_{1,+}$ such that $g=h$ on $\partial
B_{1,+}\cap\Omega$ and $g=0$ on $\partial B_{1,+}\setminus\Omega$. We have
$|g(x)|\leq C_{1}(x^{\prime\prime}-x_{1}^{\prime\prime})$ in $B_{1,+}$ by the
above estimate on $h$ and the maximum principle. We consider the difference
$g-h$. We have $g=h$ on $\Omega\cap\partial B_{1}$ and $|g-h|=|g|\leq
4C_{1}\tau$ on $\partial\Omega\cap B_{1}$. Then, by the maximum principle,
$|g-h|\leq 4C_{1}\tau\quad{\text{in}}\quad\Omega\cap B_{1}.$ We extend $h$ by
zero to $B_{1,+}\setminus\Omega$. Then $|g-h|\leq 4C_{1}\tau$ in $B_{1,+}$.

Let $m$ be an integer such that $2\sqrt{d}\leq 2^{m}<4\sqrt{d}$, clearly
$m\geq 1$. Then the estimate $N^{*}_{h}(Q)\leq N$ implies
$N_{h}(x_{Q},2^{m})\leq N$. We choose $\varepsilon$ such that
$(1+\varepsilon)^{m+3}\leq 2$ and assume that $\tau<\tau_{\varepsilon}$ using
the notation of Lemma 4. Then $N_{h}(x_{Q},2^{j})\leq 2N$ when $-3\leq j\leq
m$. We use (13) and (2) to conclude that

$\int_{B(x_{Q},\frac{1}{8})\cap\Omega}h^{2}\geq
e^{-10N}\int_{B(x_{Q},4)\cap\Omega}h^{2}\geq ce^{-10N}.$

Suppose that $\tau<\frac{1}{24}$. Then
$B(x_{Q},\frac{1}{8})\cap\Omega\subset\frac{1}{4}B_{1,+}$ and we have

$\left(\int_{\frac{1}{4}B_{1,+}}g^{2}\right)^{1/2}\geq\left(\int_{\frac{1}{4}B_{1,+}}h^{2}\right)^{1/2}-C_{2}\tau\geq\left(\int_{B(x_{Q},\frac{1}{8})\cap\Omega}h^{2}\right)^{1/2}-C_{2}\tau.$

Assuming that $\tau(N)$ is small enough, we conclude that

(14) $\int_{\frac{1}{4}B_{1,+}}g^{2}\geq c_{1}e^{-10N}.$

We also have $\sup_{\frac{1}{2}B_{1,+}}|g|\leq\sup_{B_{1}\cap\Omega}|h|\leq 1$
by (13). Then

$N_{g}(x_{1},\frac{1}{4})=\log\frac{\int_{\frac{1}{2}B_{1,+}}g^{2}}{\int_{\frac{1}{4}B_{1,+}}g^{2}}\leq
C(N+1).$

We note that (14) implies $\sup_{\frac{1}{4}B_{1,+}}|g|\geq ce^{-5N}$. Then,
by Lemma 9, there exist $x_{*}\in\Gamma_{0}\cap\frac{1}{16}B_{1}$,
$c_{2}=c_{2}(N)>0$, and $\rho=\rho(C(N+1))$ such that

$|g(x)|\geq c_{2}(x^{\prime\prime}-x_{1}^{\prime\prime})\quad{\text{for}}\quad
x=(x^{\prime},x^{\prime\prime})\in B(x_{*},\rho)\cap B_{1,+}.$

We may assume that $g>0$ in $B(x_{*},\rho)\cap B_{1,+}$, otherwise we consider
$-h$ in place of $h$. Then we obtain

$h(x)\geq g(x)-4C\tau\geq
c_{2}(x^{\prime\prime}-x_{1}^{\prime\prime})-4C\tau\quad{\text{in}}\quad
B(x_{*},\rho)\cap\Omega.$

We note that $\rho$ does not depend on $\tau$ and for $\tau$ small enough we
have $B(x_{*},\frac{\rho}{4})\cap\partial\Omega\neq\varnothing$. We also have
$B(x_{*},\frac{\rho}{2})\subset Q$.

Our goal is to show that $h>0$ on $B(x_{*},\frac{\rho}{2})\cap\Omega$. Let
$y_{*}=(y_{*}^{\prime},y_{*}^{\prime\prime})\in
B(x_{*},\frac{\rho}{2})\cap\partial\Omega$. We note that

(15) $h(x)\geq
c_{2}(x^{\prime\prime}-y_{*}^{\prime\prime})-c_{3}\tau\quad{\text{in}}\quad
B(x_{*},\rho)\cap\Omega,$

where $c_{3}=4C+4c_{2}$. We consider the harmonic function

$h_{*}(x)=\frac{1}{d\rho}\left((d-1)(x^{\prime\prime}-y^{\prime\prime}_{*})^{2}-|x^{\prime}-y^{\prime}_{*}|^{2}\right),$

where $x=(x^{\prime},x^{\prime\prime})$. We claim that $h(x)\geq
c_{2}h_{*}(x),$ when $x\in B\left(y_{*},\frac{\rho}{2}\right)\cap\Omega$ and
$\tau$ is small enough.

First, we note that $h_{*}(x)\leq 0$ if
$|x^{\prime\prime}-y_{*}^{\prime\prime}|\leq(d-1)^{-1/2}|x^{\prime}-y_{*}^{\prime}|$
and therefore $h_{*}\leq 0$ on $\partial\Omega\cap B(y_{*},\frac{\rho}{2})$
when $\tau$ is small enough, while $h=0$ on $\partial\Omega\cap
B(y_{*},\frac{\rho}{2})$. On $\partial B(y_{*},\frac{\rho}{2})\cap\Omega$ we
have

$h_{*}(x)=\frac{(x^{\prime\prime}-y_{*}^{\prime\prime})^{2}}{\rho}-\frac{\rho}{4d}.$

Comparing (15) to the last identity and denoting
$t=x^{\prime\prime}-y_{*}^{\prime\prime}$, we reduce the inequality $h\geq
c_{2}h_{*}$ on $\partial B(y_{*},\frac{\rho}{2})\cap\Omega$ to the following
one:

$c_{2}t-c_{3}\tau\geq c_{2}\left(\frac{t^{2}}{\rho}-\frac{\rho}{4d}\right),$

when $t\in(-\tau\rho/2,\rho/2)$ and $\tau$ is small enough. It sufficies to
check the inequality for $t=-\tau\rho/2$ and $t=\rho/2$. For $t=-\tau\rho/2$
we obtain the inequality

$\frac{c_{2}\rho}{4d}\geq\tau\left(\frac{c_{2}\rho}{2}+\frac{c_{2}\tau\rho}{4}+c_{3}\right),$

which holds when $\tau$ is small enough. On the other hand, for $t=\rho/2$,
the inequality is reduced to

$c_{2}\rho\left(\frac{1}{4}+\frac{1}{4d}\right)\geq c_{3}\tau.$

This one is also satisfied for small $\tau$.

Thus, by the maximum principle, $h\geq c_{2}h_{*}$ in
$B(y_{*},\rho/2)\cap\Omega$. In particular,
$h(y_{*}^{\prime},y^{\prime\prime})\geq
h_{*}(y_{*}^{\prime},y^{\prime\prime})>0$ when
$y_{*}^{\prime\prime}<y^{\prime\prime}<\rho/2$. Therefore $h>0$ on
$B(x_{*},\frac{\rho}{2})\cap\Omega$.

Finally, since $B(x_{*},\rho/2)$ contains a ball of radius $\rho/4$ centered
on $\partial\Omega$, if $k$ is large enough, there is $q\in\mathcal{B}_{k}(Q)$
such that $q\subset B(x_{*},\frac{\rho}{2})$ and then $Z(h)\cap
q=\varnothing$. ∎

##  5\. Proof of Theorem 2

Let $N_{0}$ be as in Lemma 7 and let $\Omega$, $B=B(x,r)$, and $h$ be as in
the statement of Theorem 2. We remind that the maximal doubling index
$N_{h}^{*}(Q)$ of $h$ in a cube $Q$ was defined by (9). For the rest of the
proof we modify the maximal doubling index and write
$N^{**}_{h}(Q)=\max\\{N^{*}_{h}(Q),N_{0}/2\\}$. Then Lemmas 7 and 8 imply that
there is $k$ such that for $\tau$ small enough, if $Q\subset 2B$ and
$(Q,\mathcal{B}_{k}(Q),\mathcal{I}_{k}(Q))$ is a standard construction, then
there is a cube $q_{0}\in\mathcal{B}_{k}(Q)$ such that

(16) ${\text{either}}\quad(i)\
N_{h}^{**}(q_{0})<N_{h}^{**}(Q)/2\quad{\text{or}}\quad(ii)\ Z(h)\cap
q_{0}=\varnothing.$

###  5.1. Reduction to one cube

Let $Q$ be a cube as above. We claim that

(17) $\mathcal{H}^{d-1}(Z(h)\cap Q)\leq CN_{h}^{**}(Q)s(Q)^{d-1}.$

Assume first that (17) holds. We show that Theorem 2 follows. We need to
switch from cubes to balls and from the maximal doubling index to the doubling
index at a single point.

To this end, we cover the ball $B(x,r)$ with cubes $Q_{j}\subset B(x,2r)$ such
that ${\operatorname{diam}}(Q_{j})=r/10$ and either
${\operatorname{dist}}(Q_{j},\partial\Omega)>s(Q_{j})/10$ (inner cubes) or
$Q_{j}$ satisfies the assumptions in the main construction (boundary cubes).
We may assume that there are not more than $C=C(d)$ of such cubes.

First, for each cube $Q=Q_{j}$ in this cover, we have $Q\cap
B(x,r)\neq\varnothing$, and we compare $N_{h}^{*}(Q)$ to $N_{h}(x,4r)$. There
exists $y\in Q\cap\overline{\Omega}$ and $r_{y}\in[r/20,r/10]$ such that
$N_{h}^{*}(Q)=N_{h}(y,r_{y})$. Assuming that $\tau<\tau_{1}$ in the notation
of Lemma 4, we get $N_{h}(y,32r_{y})\geq 2^{-5}N_{h}^{*}(Q)$. We have
${\operatorname{dist}}(x,y)\leq\frac{11}{10}r$ and

$N_{h}(y,32r_{y})=\log\frac{\int_{B(y,64r_{y})\cap\Omega}h^{2}}{\int_{B(y,32r_{y})\cap\Omega}h^{2}}\leq\log\frac{\int_{B(x,8r)\cap\Omega}h^{2}}{\int_{B(x,r/2)\cap\Omega}h^{2}}\leq
16N_{h}(x,4r)$

by Lemma 4. Hence, $N^{*}_{h}(Q)\leq 2^{9}N_{h}(x,4r)$ and $N^{**}_{h}(Q)\leq
C(N_{h}(x,4r)+1)$.

Each inner cube $Q\subset\Omega$ can be covered by at most $C$ balls $b$ with
centers in $Q$ and with radii $s(Q)/100$. Then $8\overline{b}\subset\Omega$.
Moreover, if $b=B(y,s(Q)/100)$, we have $N_{h}(y,s(Q)/25)\leq CN_{h}^{*}(Q)$
by Lemma 4 again. Then we use Lemma 6 to estimate the area of the zero set of
$h$ in each of the balls $b$ and obtain

$\mathcal{H}^{d-1}(Z(h)\cap b)\leq C(N_{h}(y,s(Q)/25)+1)r^{d-1}\\\ \leq
C^{\prime}(N_{h}^{*}(Q)+1)r^{d-1}\leq C^{\prime\prime}(N_{h}(x,4r)+1)r^{d-1}.$

For the boundary cubes, we use the inequality (17). Thus for every $Q_{j}$, we
obtain

$\mathcal{H}^{d-1}(Z(h)\cap Q_{j})\leq C(N_{h}(x,4r)+1)s(Q_{j})^{d-1}.$

Summing these inequalities over all cubes, we obtain the required estimate. It
remains to prove (17).

###  5.2. Proof of (17)

We fix a compact set $K\subset\Omega$ and prove that

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

where $Q\subset 2B$ is a cube as in the standard construction and $C_{0}$ is
independent of $K$. Then (17) follows.

First, note that (18) holds for all cubes $Q$ small enough, since $Q\cap
K=\varnothing$ for such cubes. We prove (18) by induction on the size of $Q$,
going from small cubes to larger ones. Assume that it holds for cubes with
$s(Q)<s$, we want to prove it for cubes with $s(Q)<2^{k}s$, where $k$ is as in
(16).

We consider the standard construction
$(Q,\mathcal{B}_{k}(Q),\mathcal{I}_{k}(Q))$. Each inner cube
$q\in\mathcal{I}_{k}(Q)$ can be covered by balls $b$ centered in $q$ with
radii $s(q)/100$ and such that $8\overline{b}\subset\Omega$, so that the
number of balls is bounded by a dimensional constant. For each such ball
$b=B(y,s(q)/100)$, applying Lemma 4, we get $N_{h}(y,s(q)/25)\leq
C(k)N^{*}_{h}(Q)$ when $\tau$ is small enough. Then by Lemma 6, we have

(19) $\sum_{q\in\mathcal{I}_{k}(Q)}\mathcal{H}^{d-1}(Z(h)\cap q)\leq
C(N_{h}^{*}(Q)+1)s(Q)^{d-1}\leq C_{1}N_{h}^{**}(Q)s(Q)^{d-1},$

where $C$ and $C_{1}$ depend on $k$.

For all other boundary cubes $q$, we have
$N^{**}_{h}(q)\leq(1+\varepsilon)^{k}N^{**}_{h}(Q)$. Also (16) implies that
there is a cube $q_{0}\in\mathcal{B}_{k}(Q)$ such that either
$N^{**}_{h}(q_{0})\leq N_{h}^{**}(Q)/2$ or $Z(h)\cap q_{0}=\varnothing$. We
apply the induction assumption to each boundary cube and obtain

$\mathcal{H}^{d-1}(Z(h)\cap K\cap(\cup_{q\in\mathcal{B}_{k}(Q)}q))\\\
\leq\sum_{q\in\mathcal{B}_{k}(Q),q\neq q_{0}}\mathcal{H}^{d-1}(Z(h)\cap K\cap
q)+\mathcal{H}^{d-1}(Z(h)\cap K\cap q_{0})\\\
\leq\sum_{q\in\mathcal{B}_{k}(Q),q\neq
q_{0}}C_{0}N_{h}^{**}(q)s(q)^{d-1}+\frac{C_{0}}{2}N_{h}^{**}(Q)s(q_{0})^{d-1}\\\
\leq\left(\frac{2^{k(d-1)}-1}{2^{k(d-1)}}(1+\varepsilon)^{k}+\frac{1}{2}\cdot\frac{1}{2^{k(d-1)}}\right)C_{0}N_{h}^{**}(Q)s(Q)^{d-1}.$

Finally, we choose $\varepsilon$ small and $C_{0}$ large enough so that

$C_{1}+\left(\frac{2^{k(d-1)}-1}{2^{k(d-1)}}(1+\varepsilon)^{k}+\frac{1}{2}\cdot\frac{1}{2^{k(d-1)}}\right)C_{0}<C_{0}.$

Note that $C_{0}$ does not depend on $K$. Then, assuming that $\tau$ is small
enough and taking into account (19), we obtain

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

This concludes the induction step and the proof of (17).

##  6\. Dirichlet Laplace eigenfunctions

###  6.1. Harmonic extension and an estimate of the doubling index

Let $\Omega_{0}\subset\mathbb{R}^{n}$ be a bounded Lipschitz domain. Let
$u_{\lambda}$ be an eigenfunction of the Dirichlet Laplace operator,
$u_{\lambda}\in W_{0}^{1,2}(\Omega_{0}),$ $\Delta u_{\lambda}+\lambda
u_{\lambda}=0$. Then $u_{\lambda}\in C(\overline{\Omega}_{0})$. This fact is
well-known, we provide a proof in the Appendix below, see Section A.4.

We consider the harmonic extension of $u_{\lambda}$ to the domain
$\Omega=\Omega_{0}\times\mathbb{R}\subset\mathbb{R}^{n+1}$, given by

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

Then $h\in C(\overline{\Omega})$ and, clearly,
$Z(h)=Z(u_{\lambda})\times\mathbb{R}$, where the zero sets are sets inside the
domains $\Omega$ and $\Omega_{0}$ respectively. We need the following estimate
of the doubling index of this harmonic extension.

######  Lemma 10.

Let $\Omega_{0}$ be a bounded domain in $\mathbb{R}^{n}$ with a sufficiently
small local Lipschitz constant $\tau$. Let $r_{0}>0$ be such that
$\partial\Omega_{0}\cap B(x,r_{0})\in Lip(\tau)$ for any
$x\in\partial\Omega_{0}$. Then for any $r\in(0,r_{0}/16)$, there exists
$C=C(r,\Omega_{0})>0$ such that for any Dirichlet Laplace eigenfunction
$u_{\lambda}$, the corresponding harmonic extension
$h(x,t)=u_{\lambda}(x)e^{\sqrt{\lambda}t}$ satisfies $N_{h}(y,r)\leq
C\sqrt{\lambda}$ when $y=(x,t)\in\overline{\Omega}$.

This result is similar to the results of Donnelly and Fefferman, [6, 7], who
considered eigenfunctions on compact manifolds and on domains with
$C^{\infty}$-smooth boundaries and obtained the above estimate of the doubling
index for eigenfunctions. However, in contrast to the previous results, the
doubling index is allowed to blow up as $r\to 0$ in the above lemma. The
statement of the lemma follows by application of Lemma 5 and inequality (6) to
a chain of balls, the argument is similar to the one in [17, Section 2.4]. For
the convenience of the reader, we provide the details below.

###### Proof.

We consider any $y=(x,t)\in\overline{\Omega}$ and let $y_{0}=(x,0)$. Since
$h(x,t+s)=e^{\sqrt{\lambda}t}h(x,s)$, we have $N_{h}(y,r)=N_{h}(y_{0},r)$. So
it is enough to estimate the doubling index of $h$ in the balls centered on
$\overline{\Omega}_{0}\times\\{0\\}$.

We fix $r\in(0,r_{0}/16)$ and let $\mathcal{S}\in\overline{\Omega}_{0}$ be a
finite $r/8$-net for $\overline{\Omega}_{0}$, i.e.,
$\overline{\Omega}_{0}\subset\bigcup_{p\in\mathcal{S}}B(p,r/8)$. Let
$B_{*}=B(y,r)$ be a ball of radius $r$ centered at
$y=(y_{*},0)\in\overline{\Omega}_{0}\times\\{0\\}$. Assume that
$\max_{\Omega_{0}}|u_{\lambda}|=|u_{\lambda}(x_{0})|=1$. We consider a path
$\gamma:[0,1]\to\overline{\Omega}_{0}$ from $y_{*}$ to $x_{0}$ such that
$\gamma((0,1))\subset\Omega_{0}$. Now we construct a chain of balls
$\\{B_{j}\\}_{j=0}^{J}$. Let $B_{0}=B(y_{*},r/2)$. Assuming that
$B_{j}=B(y_{j},r/2)$ is constructed, we define

$s_{j}=\sup\\{s\in[0,1]:|\gamma(s)-y_{j}|\leq r/8\\}.$

If $s_{j}<1$, we have $|\gamma(s_{j})-y_{j}|=r/8$ and we choose
$y_{j+1}\in\mathcal{S}$ such that $|y_{j+1}-\gamma(s_{j})|<r/8$. If $s_{j}=1$,
we define $y_{j+1}=y_{J}=x_{0}$ and stop the chain. We have
$|y_{j}-y_{j+1}|<r/4$ and define $B_{j+1}=B(y_{j+1},r/2)$. We note that
$s_{j+1}>s_{j}$ when $0\leq j<J-1$ and that
$y_{j+1}\in\mathcal{S}\setminus\\{y_{0},...,y_{j}\\}$ when $0\leq j<J-1$. We
also have $B_{j+1}\subset\frac{3}{2}B_{j}$. The resulting chain is finite,
moreover, the number of balls in the chain is bounded by the number of
elements in $\mathcal{S}$ plus two.

Let now $\widetilde{B}_{j}=B((y_{j},0),r/2)$ be the corresponding ball in
$\mathbb{R}^{n+1}$. Then $\sup_{4\widetilde{B}_{j}\cap\Omega}|h|\leq
e^{2\sqrt{\lambda}r}$. If $4\tilde{B}_{j}\subset\Omega$, then (6) gives

$\sup_{\frac{3}{2}\widetilde{B}_{j}}|h|\leq
2^{n+1}(\sup_{\widetilde{B}_{j}}|h|)^{1/2}(\sup_{4\widetilde{B}_{j}}|h|)^{1/2}\\\
\leq
3^{n+1}(\sup_{\widetilde{B}_{j}}|h|)^{1/3}(\sup_{4\widetilde{B}_{j}}|h|)^{2/3}\leq
3^{n+1}e^{4\sqrt{\lambda}r/3}(\sup_{\tilde{B}_{j}}|h|)^{1/3}.$

Otherwise we have
${\operatorname{dist}}(y_{j},\partial\Omega_{0})<2r<r_{0}/8$. In this case,
there is a ball $\tilde{B}$ of radius $r_{0}$ centered on
$\partial\Omega_{0}\times\\{0\\}$ such that
$(y_{j},0)\in\overline{\Omega}\cap\frac{1}{4}\tilde{B}$ and
$16\tilde{B}_{j}\subset\tilde{B}$. Then Lemma 5, applied to the ball
$\tilde{B}_{j}$, implies that

$\sup_{\frac{3}{2}\tilde{B}_{j}\cap\Omega}|h|\leq
3^{n+1}(\sup_{\tilde{B}_{j}\cap\Omega}|h|)^{1/3}(\sup_{4\tilde{B}_{j}\cap\Omega}|h|)^{2/3}\leq
3^{n+1}e^{4\sqrt{\lambda}r/3}(\sup_{\tilde{B}_{j}\cap\Omega}|h|)^{1/3}.$

Therefore, we obtain for each $j$,

$\sup_{\widetilde{B}_{j}\cap\Omega}|h|\geq
3^{-3(n+1)}(\sup_{\frac{3}{2}\widetilde{B}_{j}\cap\Omega}|h|)^{3}e^{-4\sqrt{\lambda}r}\geq
3^{-3(n+1)}(\sup_{\widetilde{B}_{j+1}\cap\Omega}|h|)^{3}e^{-4\sqrt{\lambda}r}.$

We also have $\sup_{\widetilde{B}_{J}\cap\Omega}|h|=e^{\sqrt{\lambda}r/2}$.
Combining the above inequalities, we get

$\sup_{\widetilde{B}_{0}\cap\Omega}|h|\geq c_{1}e^{-C_{2}\sqrt{\lambda}},$

where $c_{1}$ and $C_{2}$ depend on $r$ and $J$ but not on $\lambda$. We can
choose the $r/8$-net $\mathcal{S}$ so that the number of points in
$\mathcal{S}$ depends only on ${\operatorname{diam}}(\Omega_{0})$, $r$, and
the dimension. Thus we conclude that the constants in the last inequality
depend only on $r$, the diameter of $\Omega_{0}$, and $n$.

Finally, applying (2), we obtain

$N_{h}(y,r)=\log\frac{\int_{4\widetilde{B}_{0}\cap\Omega}h^{2}}{\int_{2\widetilde{B}_{0}\cap\Omega}h^{2}}\leq\log\frac{\sup_{4\widetilde{B}_{0}\cap\Omega}|h|^{2}}{\sup_{\widetilde{B}_{0}\cap\Omega}|h|^{2}}+C\leq(4r+2C_{2})\sqrt{\lambda}+C\leq
C\sqrt{\lambda},$

where $C=C(\Omega_{0},r)$. We remark that
$\lambda\geq\lambda_{1}(\Omega_{0})>0$, where $\lambda_{1}(\Omega_{0})$ is the
first Dirichlet Laplace eigenvalue in $\Omega_{0}$. Moreover, if $B^{*}$ is a
ball of radius ${\operatorname{diam}}(\Omega_{0})$ then
$\lambda_{1}(\Omega_{0})\geq\lambda_{1}(B^{*})$. Thus the constant $C$ in the
conclusion of this Lemma depends only on $r$,
${\operatorname{diam}}(\Omega_{0})$, and $n$. ∎

###  6.2. Proof of Theorem $\ref{th:main}^{\prime}$

Let $\Omega_{0}\subset\mathbb{R}^{n}$ be a bounded domain with a sufficiently
small local Lipschitz constant $\tau$. Let also $r_{0}>0$ be such that
$\partial\Omega_{0}\cap B(x,r_{0})\in Lip(\tau)$ for every
$x\in\partial\Omega_{0}$. We consider the domain
$\Omega=\Omega_{0}\times\mathbb{R}\subset\mathbb{R}^{n+1}$ and let
$\Omega_{1}=\Omega_{0}\times[-1,1]$. For each $x\in\partial\Omega\times[-1,1]$
we consider a ball centered at $x$ of radius $2^{-9}r_{0}$. These balls cover
the closed $2^{-10}r_{0}$-neighborhood of the set
$\partial\Omega\times[-1,1]$. We can choose a disjoint collection of these
balls $b_{j}$ such that the balls $B_{j}=4b_{j}$ cover the same closed
neighborhood of $\partial\Omega\times[-1,1]$. Then for each point of
$\Omega_{1}\setminus\cup_{j}B_{j}$, we choose a ball $b$ centered at the point
of radius $2^{-15}r_{0}$, so that $32b\subset\Omega$. Once again, we find a
finite sub-collection of disjoint balls $b^{\prime}_{k}$ such that
$B_{k}^{\prime}=4b_{k}^{\prime}$ cover $\Omega_{1}\setminus\cup_{j}B_{j}$. We
note that $8B_{k}^{\prime}\subset\Omega$. We fix this covering of $\Omega_{1}$
and remark that radii of all balls depend only on $r_{0}$ and the number of
balls depends on $r_{0}$, the diameter of $\Omega_{0}$, and $n$.

Let now $u_{\lambda}$ be a Dirichlet Laplace eigenfunction in $\Omega_{0}$:
$\Delta u_{\lambda}+\lambda u_{\lambda}=0$ in $\Omega_{0}$ and $u_{\lambda}=0$
on $\partial\Omega_{0}$. We consider its harmonic extension
$h(x,t)=e^{\sqrt{\lambda}t}u_{\lambda}(x)$. Then $h\in C(\overline{\Omega})$
is non-zero, and $h=0$ on $\partial\Omega$. Let
$C_{0}=\max\\{C(2^{-5}r_{0},\Omega_{0}),C(2^{-11}r_{0},\Omega_{0})\\}$, where
$C(r,\Omega_{0})$ is as in Lemma 10. Then for
$B(x,r)\in\\{B_{j}\\}\cup\\{B_{k}^{\prime}\\}$, we have $N_{h}(x,4r)\leq
C_{0}\sqrt{\lambda}$. Finally, we apply Theorem 2 to each of the balls $B_{j}$
and Lemma 6 to each of the inner balls $B_{k}^{\prime}$. We conclude that

$\mathcal{H}^{n}(Z(h)\cap\Omega_{1})\leq\sum_{j}\mathcal{H}^{n}(Z(h)\cap
B_{j})+\sum_{k}\mathcal{H}^{n}(Z(h)\cap B^{\prime}_{k})\\\ \leq
C(C_{0}\sqrt{\lambda}+1)\left(\sum_{j}r(B_{j})^{n}+\sum_{k}r(B^{\prime}_{k})^{n}\right)\leq
C_{1}\sqrt{\lambda}.$

Then $\mathcal{H}^{n-1}(Z(u_{\lambda})\cap\Omega_{0})\leq
C_{1}\sqrt{\lambda}$, which finishes the proof of Theorem
$\ref{th:main}^{\prime}$.

## Appendix: Proofs of some auxiliary results

###  A.3. Estimates for the zero set of harmonic functions inside the domain

We outline some steps of the proof of Lemma 6. First the harmonic function $h$
is extended to a holomorphic function $H$ on a domain in $\mathbb{C}^{d}$, see
Lemma 7.2 in [6]. Our situation is particularly simple, since we only consider
the standard Laplace operator on Euclidean domains. For this case the
holomorphic extension is given by the complexification of the Poisson kernel.
The Poisson kernel in a ball $B(x,r)\subset\mathbb{R}^{d}$ is given by

$P_{r}(z,y)=c_{d}\frac{r^{2}-|z-x|^{2}}{r|z-y|^{d}},\quad|z-x|<r,\ |y-x|=r.$

For any $y\in\partial B(x,r)$, the function
$z=(z_{1},...,z_{d})\mapsto\sum_{j}(z_{j}-y_{j})^{2}$ maps the complex ball
$B_{\mathbb{C}}(x,r/\sqrt{2})\subset\mathbb{C}^{d}$ of radius $r/\sqrt{2}$
centered at $x\in\mathbb{R}^{d}\subset\mathbb{C}^{d}$ to the half-plane
$\Re\xi>0$. Then the Poisson kernel has the holomorphic extension to
$B_{\mathbb{C}}(x,r/\sqrt{2})$. Moreover, for any $a<1/\sqrt{2}$,

$|P_{r}(z,y)|\leq C(a)r^{-(d-1)},\quad z\in B_{\mathbb{C}}(x,r_{0}),\
r_{0}\leq ar.$

We consider a ball $B=B(x,8r)$ such that $\overline{B}\subset\Omega$. Then
there exists a holomorphic extension $H(z)$ of $h$ defined on a ball
$B_{\mathbb{C}}(x,3r)$,

$H(z)=\int_{\partial B(x,6r)}P_{6r}(z,y)h(y)d\sigma(y),$

such that $|H(z)|\leq C\max_{\overline{B}(x,6r)}|h|$. Then

$\sup_{B_{\mathbb{C}}(x,3r)}|H(z)|\leq
C^{\prime}r^{-d/2}\left(\int_{B(x,8r)}h^{2}\right)^{1/2}.$

Now we can cover the set $Z(h)\cap B(x,r)$ by a finite number of balls with
centers in $B(x,r)$ of radii $r/20$ so that the number of the balls is bounded
by a constant depending on the dimension only. Let $B(y,r/20)$ be one of such
balls. By a version of Corollary 1 for the doubling index inside the domain,
we have $N_{h}(y,2r)\leq 3N$, where $N=N_{h}(x,4r),$ and, therefore,
$N_{h}(y,r_{1})\leq 3N$ when $r_{1}<2r$. Thus

$\sup_{B(y,\frac{r}{16})}h^{2}\geq cr^{-d}\int_{B(y,\frac{r}{16})}h^{2}\geq
cr^{-d}e^{-15N}\int_{B(y,2r)}h^{2}\geq cr^{-d}e^{-15N}\int_{B(x,r)}h^{2}.$

Therefore,

$\sup_{B(y,\frac{r}{10})}|H|\geq\sup_{B(y,\frac{r}{16})}|h|\geq
cr^{-d/2}e^{-7.5N}\left(\int_{B(x,r)}h^{2}\right)^{1/2}.$

Combining the inequalities above, we obtain

$\frac{\sup_{B_{\mathbb{C}}(y,2r)}|H|}{\sup_{B(y,r/10)}|H|}\leq\frac{\sup_{B_{\mathbb{C}}(x,3r)}|H|}{\sup_{B(y,r/10)}|H|}\leq
Ce^{7.5N}\left(\frac{\int_{B(x,8r)}h^{2}}{\int_{B(x,r)}h^{2}}\right)^{1/2}\leq
Ce^{9N}.$

Finally an estimate for the size of the zero set of a holomorphic function,
Proposition 6.7 in [6], implies that

$\mathcal{H}^{d-1}(Z(h)\cap B(y,r/20))\leq C(N_{h}(x,4r)+1).$

We sum these inequalities over all balls $B(y,r/20)$ to obtain the required
estimate for $\mathcal{H}^{d-1}(Z(h)\cap B(x,r))$.

###  A.4. Continuity of eigenfunctions in Lipschitz domains

First we prove the following regularity result.

######  Lemma 11.

Let $\Omega$ be a domain in $\mathbb{R}^{d}$ and let $h$ be a harmonic
function in $\Omega$. Suppose that $B$ is a ball centered on $\partial\Omega$
and that there exists a sequence of functions $\\{h_{n}\\}$, $h_{n}\in
C^{\infty}_{0}(\mathbb{R}^{d})$ with the support of $h_{n}$ contained in
$\Omega$, such that $h_{n}\to h$ and $\nabla h_{n}\to\nabla h$ in
$L^{2}(B\cap\Omega)$. Assume also that $\partial\Omega\cap B\in Lip(\tau)$ and
define $h=0$ on $\partial\Omega\cap B$. Then $h\in
C(\overline{\Omega}\cap\frac{1}{2}B)$.

###### Proof.

We define the function

$v=\begin{cases}h^{2}\ {\text{in}}\ \Omega\cap B,\\\ 0\ {\text{in}}\
B\setminus\Omega.\end{cases}$

Then $v\in L^{1}(B)$. Let $\varphi\in C_{0}^{\infty}(B)$. We have

(20)
$\int_{B}v\Delta\varphi=\lim_{n\to\infty}\int_{B}h_{n}^{2}\Delta\varphi\\\
=-2\lim_{n\to\infty}\int_{B}h_{n}\nabla
h_{n}\cdot\nabla\varphi=-2\int_{B\cap\Omega}h\nabla h\cdot\nabla\varphi.$

On the other hand, since $h$ is harmonic in $\Omega$ , we obtain

$0=\int_{\Omega}\nabla h\cdot\nabla(h_{n}\varphi)=\int_{\Omega}h_{n}\nabla
h\cdot\nabla\varphi+\int_{\Omega}\varphi\nabla h\cdot\nabla h_{n}.$

Taking the limit as $n\to\infty$, we get

$\int_{\Omega}h\nabla h\cdot\nabla\varphi=-\int_{\Omega}\varphi|\nabla
h|^{2}.$

Combining the last identity and (20) gives

$\int_{B}v\Delta\varphi=2\int_{B\cap\Omega}|\nabla h|^{2}\varphi.$

In particular, $v$ is subharmonic in $B$ in the weak sense: If $\varphi\geq
0$, $\varphi\in C_{0}^{\infty}(B)$, then $\int_{B}v\Delta\varphi\geq 0$. If
$\alpha$ is a standard mollifier,
$\alpha_{\delta}(x)=\delta^{-d}\alpha(\delta^{-1}x)$, and
$v_{\varepsilon}=v*\alpha_{\varepsilon r}$, where $r$ is the radius of $B$.
Then $v_{\varepsilon}$ is subharmonic in $(1-\varepsilon)B$ and
$v_{\varepsilon}\to v$ in $L^{1}(B)$ and almost everywhere. In particular, $v$
satisfies the mean value inequality at each its Lebesgue point. Clearly any
$y\in\Omega\cap B$ is a Lebesgue point of $v$ as $v=h^{2}$ in $\Omega\cap B$
and $h\in C(\Omega)$. So for any $y\in\Omega\cap B$ and any ball $B_{1}\subset
B$ centered at $y$ we have

$v(y)\leq\frac{1}{|B_{1}|}\int_{B_{1}}v.$

In particular,

$\sup_{\frac{2}{3}B\cap\Omega}h^{2}\leq\frac{3^{d}}{|B|}\int_{B\cap\Omega}h^{2}<\infty.$

Suppose that $x_{1}\in\partial\Omega\cap\frac{1}{2}B$. There exists a cone
$\mathcal{C}$ with the vertex at $x_{1}$ such that
${\mathcal{C}}\cap(\Omega\cap B)=\varnothing$ and the aperture of
$\mathcal{C}$ does not depend on $x_{1}$ (it depends on $\tau$ only). We use
the following simple fact. If $y_{1}\in\mathbb{R}^{d}$ and
$\rho>2\,{\operatorname{dist}}(x_{1},y_{1})$, then

$|B(y_{1},\rho)\cap\mathcal{C}|\geq\alpha|B(y_{1},\rho)|,$

for some $\alpha=\alpha(\tau)\in(0,1)$.

Let $m_{k}=\sup_{B(x_{1},3^{-k}r)\cap\Omega}|h|$ for $k\geq 2$. We know that
$m_{k}<\infty$. Let $y\in B(x_{1},3^{-k}r)\cap\Omega$, $k\geq 3$. By the mean
value theorem applied to $v$, we obtain

$v(y)\leq\frac{1}{|B(y,2\cdot 3^{-k}r)|}\int_{B(y,2\cdot
3^{-k}r)}v\leq(1-\alpha)m_{k-1}^{2}.$

Thus
$\sup_{B(x_{1},3^{-k}r)\cap\Omega}|h|\leq(1-\alpha)^{(k-2)/2}\sup_{\frac{2}{3}B\cap\Omega}|h|$.
We conclude that

$\lim_{y\to x_{1},y\in\Omega}h(y)=0.$

∎

We remark that the argument above implies that $h$ is Hölder continuous in
$\overline{\Omega}\cap B$ and there exist $C>0$ and $\beta\in(0,1)$ such that

$|h(y)|\leq
C{\operatorname{dist}}(y,\partial\Omega)^{\beta}r^{-\beta}\sup_{\Omega\cap\frac{2}{3}B}|h|,\quad
y\in\Omega\cap\frac{1}{2}B.$

######  Corollary 2.

Let $\Omega_{0}\subset\mathbb{R}^{n}$ be a bounded Lipschitz domain. Let
$u_{\lambda}$ be Laplace Dirichlet eigenfunction in $\Omega_{0}$. Then
$u_{\lambda}$ extended by zero to $\partial\Omega_{0}$ is continuous on
$\overline{\Omega}_{0}$.

###### Proof.

We have $u_{\lambda}\in W^{1,2}_{0}(\Omega_{0})\cap C^{\infty}(\Omega_{0})$
and $\Delta u_{\lambda}+\lambda u_{\lambda}=0$ in $\Omega_{0}$. We consider
the harmonic function $h(x,t)=e^{\sqrt{\lambda}t}u_{\lambda}(x)$ in
$\Omega=\Omega_{0}\times\mathbb{R}$. We note that for any $B$ centered on
$\partial\Omega$, $h$ satisfies the assumptions of Lemma 11. Then $h$ is
continuous in $\overline{\Omega}$ and vanishes on $\partial\Omega$. This
implies that $u_{\lambda}\in C(\overline{\Omega}_{0})$ and $u_{\lambda}=0$ on
$\partial\Omega_{0}$. ∎

###  A.5. Quantitative Cauchy uniqueness

We give an elementary proof of Lemma 2 in this section for the convenience of
the reader.

Let $G(x,y)=-c_{d}|x-y|^{2-d}$ be the fundamental solution of the Laplace
equation in $\mathbb{R}^{d}$ when $d\geq 3$ (similar computations can be done
with $G(x,y)=c_{2}\log|x-y|$ for $d=2$). We write $\partial
B_{+}=\Gamma\cup\Sigma$, where $\Gamma$ is the flat part of the boundary and
$\Sigma=\partial B_{+}\setminus\Gamma$. We denote by $n$ the outer normal to
$\partial B_{+}$. Then for $x\in B_{+}$, the Green formula implies

$h(x)=\int_{\partial B_{+}}\left[\frac{\partial G}{\partial
n}(x,y)h(y)-G(x,y)\frac{\partial h}{\partial n}(y)\right]dy\\\
=\int_{\Gamma}\left[\frac{\partial G}{\partial
n}(x,y)h(y)-G(x,y)\frac{\partial h}{\partial n}(y)\right]dy\\\
+\int_{\Sigma}\left[\frac{\partial G}{\partial
n}(x,y)h(y)-G(x,y)\frac{\partial h}{\partial n}(y)\right]dy\\\
=h_{1}(x)+h_{2}(x).$

The functions $h_{1}$ and $h_{2}$ are defined in the complements of $\Gamma$
and $\Sigma$ respectively and are harmonic in the corresponding domains.
Moreover, for $x\not\in\overline{B}_{+}$, applying the Green formula to the
functions $h$ and $G(x,\cdot)$ in $B_{+}$, we obtain $h_{1}(x)+h_{2}(x)=0$.

First, we estimate the value of $h_{1}$ at some point
$x=(x^{\prime},x^{\prime\prime})\in
B\setminus\Gamma\subset\mathbb{R}^{d-1}\times\mathbb{R}$. We divide the
integral into two

$h_{1}(x)=\int_{\Gamma}\frac{\partial G}{\partial
n}(x,y)h(y)dy-\int_{\Gamma}G(x,y)\frac{\partial h}{\partial
n}(y)dy=I_{1}(x)+I_{2}(x).$

Since $|\partial h/\partial n|<\varepsilon$ on $\Gamma$, the second integral
is bounded by

$|I_{2}(x)|\leq
c_{d}\varepsilon\int_{B^{d-1}(x^{\prime},2)}|x^{\prime}-y^{\prime}|^{2-d}dy^{\prime}\leq
C\varepsilon.$

To estimate the first term, we note that for $y\in\Gamma$,

$\frac{\partial G}{\partial n}(x,y)=c_{d}^{\prime}x^{\prime\prime}|x-y|^{-d},$

and thereby

$\int_{\Gamma}\left|\frac{\partial G}{\partial n}(x,y)\right|dy\leq
c_{d}^{\prime}\int_{\mathbb{R}^{d-1}}\frac{|x^{\prime\prime}|}{(x^{\prime\prime
2}+|x^{\prime}-y^{\prime}|^{2})^{d/2}}dy^{\prime}=c^{\prime\prime}_{d}.$

Using that $|h(y)|<\varepsilon$ on $\Gamma$, we conclude that
$|I_{1}(x)|<C\varepsilon$ in $B\setminus\Gamma$. Therefore $|h_{1}(x)|\leq
C\varepsilon$ in $B\setminus\Gamma$. Since $h_{1}(x)+h_{2}(x)=0$ when
$x\in\mathbb{R}^{d}\setminus\overline{B}_{+}$, and $|h_{1}+h_{2}|=|h|\leq 1$
in $B_{+}$, we obtain that $h_{2}(x)$ satisfies

$|h_{2}(x)|<C\varepsilon\ \ {\text{in}}\
B_{-}=B\setminus\overline{B}_{+}\quad{\text{and}}\quad|h_{2}(x)|\leq
1+C\varepsilon\ \ \text{in}\ B_{+}.$

Now we apply the three sphere inequality (6). We note that $h_{2}$ is harmonic
in $B$. First we take $x=(0,-1/5)$ and $r=1/5$ and obtain

$\sup_{B(0,1/10)}|h_{2}|\leq\sup_{B(x,3/10)}|h_{2}|\leq
2^{d}(\sup_{B(x,1/5)}|h_{2}|)^{1/2}(\sup_{B(x,4/5)}|h_{2}|)^{1/2}\leq
C\varepsilon^{1/2}.$

Then we apply the same theorem to the balls centered at the origin three
times. Since $(3/2)^{3}\cdot(1/10)>1/3$ and $(3/2)^{2}\cdot(1/10)<1/4$, we
conclude that

$\sup_{\frac{1}{3}B}|h_{2}(x)|\leq C\varepsilon^{1/16}.$

Finally, combining the last inequality with the bound $|h_{1}|\leq
C\varepsilon$ in $B_{+}$, we get the required estimate $|h|\leq
C\varepsilon^{\gamma}$ in $\frac{1}{3}B_{+}$.

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# Mean Field Effect on $J/\psi$ Production in Heavy Ion Collisions

Baoyi Chen, Kai Zhou, and Pengfei Zhuang  Physics Department, Tsinghua
University, Beijing 100084, China

###### Abstract

The mass shift effect at finite temperature on $J/\psi$ production in
relativistic heavy ion collisions is calculated in a detailed transport
approach, including both mean field and collision terms. While the momentum-
integrated nuclear modification factor $R_{AA}$ is not sensitive to the mass
shift, the reduced threshold for $J/\psi$ regeneration in the quark-gluon
plasma leads to a remarkable enhancement for the differential $R_{AA}$ at low
transverse momentum.

###### pacs:

25.75.-q, 12.38.Mh, 24.85.+p

$J/\psi$ is a tightly bound state of charm quarks $c$ and $\overline{c}$, its
dissociation temperature $T_{d}$ in hot medium is much higher than the
critical temperature $T_{c}$ of the deconfinement phase transition. Therefore,
the measured $J/\psi$s in nuclear collisions at Relativistic Heavy Ion
Collider (RHIC) and Large Hadron Collider (LHC) carry the information of the
hot medium and has long been considered as a signature matsui  of the new
state of matter, the so-called quark-gluon plasma (QGP) qgp .

The $J/\psi$ properties are significantly affected by the deconfinement phase
transition. From the calculations with quantum chromodynamics (QCD) sum rules
megias ; morita  and QCD second-order Stark effect lee , both the $J/\psi$
width and mass are largely changed in a static hot medium. For a dynamically
evolutive QGP phase created in the early stage of heavy ion collisions, the
width is induced by $J/\psi$ decay like the gluon dissociation $g+J/\psi\to
c+\overline{c}$ na50 ; blaizot ; capella ; hufner ; polleri ; bratkovskaya ;
zhuang ; zhu ; wong , and the mass shift comes from the mean field effect of
the medium. Since the mass shift for a heavy quark system is expected to be
weak at low temperature, it is neglected in almost all the model calculations.
However, in the region above and close to the critical temperature, there is a
sudden change in the mass of $J/\psi$. For instance, at temperature
$T/T_{c}=1.1$ the mass shift $\delta m_{J/\psi}=m_{J/\psi}(T)-m_{J/\psi}(0)$
can reach $-(100-200)$ MeV megias ; morita ; lee , which is already comparable
with the mass change for light hadrons leupold  and should have remarkable
consequence in $J/\psi$ production. In this paper, we study the hot nuclear
matter effect on $J/\psi$ production in heavy ion collisions at RHIC and LHC
energies, including not only the gluon dissociation but also the mean field.

Let’s first qualitatively estimate the mean field effect on the $J/\psi$
distribution. The decreased mass reduces the threshold value for the $J/\psi$
production in QGP, and should result in an enhancement for the $J/\psi$ yield.
Secondly, the attractive force between the inhomogeneous medium and $J/\psi$,

${\bf F}({\bf x},{\bf p})=-{m_{J/\psi}\over
E_{J/\psi}}{\bf\nabla}m_{J/\psi}=-{m_{J/\psi}\over E_{J/\psi}}{\partial
m_{J/\psi}\over\partial T}{\bf\nabla}T$ (1)

with $J/\psi$ energy $E_{J/\psi}=\sqrt{m_{J/\psi}^{2}+{\bf p}^{2}}$, pulls
$J/\psi$ to the center of the fireball and kicks $J/\psi$ to a lower momentum
region. This will lead to an enhancement at low momentum and a reduction at
high momentum.

Since charmonia are so heavy and difficult to be thermalized in hot medium, we
use a detailed transport approach yan  to describe their distribution
functions $f_{\Psi}({\bf p},{\bf x},\tau|{\bf b})$ in the phase space at fixed
impact parameter ${\bf b}$ of a nucleus-nucleus collision,

${\partial f_{\Psi}\over\partial\tau}+{{\bf p}\over
E_{\Psi}}\cdot{\bf\nabla}_{x}f_{\Psi}+{\bf
F}\cdot{\bf\nabla}_{p}f_{\Psi}=-\alpha_{\Psi}f_{\Psi}+\beta_{\Psi},$ (2)

where ${\bf\nabla}_{x}$ and ${\bf\nabla}_{p}$ indicate three dimensional
derivatives in coordinate and momentum spaces. Considering the fact that in
proton-proton collisions the contribution from the decay of the excited
charmonium states to the $J/\psi$ yield is about $40\%$ herab , we must take
transport equations not only for the ground state $\Psi=J/\psi$ but also the
excited states $\Psi=\psi^{\prime}$ and $\chi_{c}$. The collision terms on the
right hand side are from the charmonium inelastic interaction with the medium.
The lose term $\alpha$ zhu ; yan  arising from the gluon dissociation process
controls the $J/\psi$ suppression, and the gain term $\beta$ yan  related to
$\alpha$ by detailed balance governs the $J/\psi$ regeneration pbm ;
gorenstein ; thews1 ; rapp ; thews2 ; zhao  in the QGP phase. The gluons and
charm quarks in $\alpha$ and $\beta$ are assumed to be thermalized at RHIC and
LHC energies. The cross section for the gluon dissociation $\sigma_{\Psi}(0)$
in vacuum has been calculated through the method of operator production
expansion peskin ; bhanot , and its value at finite temperature can be
obtained from its classical relation to the charmonium size $r_{\Psi}$,
$\sigma_{\Psi}(T)=\langle r_{\Psi}^{2}(T)\rangle/\langle
r_{\Psi}^{2}(0)\rangle\sigma_{\Psi}(0)$, where the averaged $r$-square can be
derived from the Schrödinger equation satz  with lattice simulated potential
karsch ; shuryak . In Fig.1, we show the $J/\psi$ decay rate
$\Gamma\equiv\alpha_{J/\psi}$ as a function of transverse momentum at fixed
temperature $T/T_{c}=1.1$ and $1.5$. In the calculation here we have chosen a
typical medium velocity $v_{QGP}=0.5$ and assumed that $\vec{v}_{QGP}$ and
$J/\psi$ momentum $\vec{p}$ have the same direction. In the vicinity of the
phase transition, the width is not sensitive to the momentum, and the
amplitude is in qualitative agreement with the result calculated from QCD sum
rules morita . When the temperature increases from $1.1T_{c}$ to $1.5T_{c}$,
the width increases by a factor of about 2.

Figure 1: The $J/\psi$ decay width as a function of transverse momentum
$p_{t}$ at temperature $T/T_{c}=1.1$ and $1.5$. The medium velocity is fixed
as $v_{QGP}=0.5$ and its direction is chosen as the same as the $J/\psi$
momentum.

Different from the previous study, we consider here not only the inelastic
processes, but also a mean field term ${\bf F}\cdot{\bf\nabla}_{p}f_{\Psi}$.
The local temperature $T({\bf x},\tau)$ appeared in the elastic and inelastic
terms are obtained from the hydrodynamic equations. At RHIC and LHC energies,
one can take the $2+1$ dimensional version heinz1 ; hirano1 ; zhu  of the
relativistic hydrodynamic equations

$\partial_{\mu}T^{\mu\nu}=0,\ \ \partial_{\mu}N^{\mu}=0$ (3)

to describe the space-time evolution of the QGP, where $T_{\mu\nu}$ and
$N_{\mu}$ are the energy-momentum tensor and baryon number current of the
system. In our numerical calculation, the QGP formation time is chosen to be
$\tau_{0}=0.6$ fm/c and the initial thermodynamics is determined by the
colliding energy and the nuclear geometry zhu ; hirano2 .

The charm quark mass $m_{c}$ is determined by the charmonium mass $m_{\Psi}$
and its binding energy $\epsilon_{\Psi}$ satz ,
$m_{c}=\left(m_{\Psi}-\epsilon_{\Psi}\right)/2$. The cold nuclear matter
effects on $J/\psi$ production, such as nuclear absorption gerschel , Cronin
effect gavin ; hufner2  and shadowing effect vogt  can be reflected in the
initial charmonium distributions $f_{\Psi}({\bf p},{\bf x},\tau_{0}|{\bf b})$.
In the following we neglect the nuclear absorption at RHIC and LHC energies
due to the small QGP formation time, and consider the Cronin effect with a
Gaussian smearing treatment zhao ; liu .

The transport equation with the mean field force can not be solved
analytically. Supposing that the very small elastic cross section for
charmonia at hadron level is still true at parton level, the mean field term
can be considered as a perturbation, and the transport equation can be solved
perturbatively. Fully neglecting the mass shift, the zeroth-order transport
equation can be strictly solved with the solution yan

$f_{\Psi}^{(0)}({\bf p},{\bf x},\tau|{\bf b})=f_{\Psi}({\bf p},{\bf
x}_{0},\tau_{0}|{\bf b})e^{-\int_{\tau_{0}}^{\tau}d\tau_{1}\alpha_{\Psi}({\bf
p},{\bf x}_{1},\tau_{1}|{\bf
b})}+\int_{\tau_{0}}^{\tau}d\tau_{1}\beta_{\Psi}({\bf p},{\bf
x}_{1},\tau_{1}|{\bf b})e^{-\int_{\tau_{1}}^{\tau}d\tau_{2}\alpha_{\Psi}({\bf
p},{\bf x}_{2},\tau_{2}|{\bf b})}$ (4)

with the coordinate shift ${\bf x}_{n}={\bf x}-{\bf
p}/E_{\Psi}(\tau-\tau_{n})$ which reflects the leakage effect matsui ; hufner3
during the time period $\tau-\tau_{n}$. The first term on the right hand side
is the contribution from the initial production, it suffers from the gluon
dissociation during the whole evolution of QGP. The second term arises from
the regeneration and suffers also the suppression from the production time
$\tau_{1}$ to the end of QGP.

Taking into account the fact that the dissociation temperatures for the
excited states are around $T_{c}$ satz , we do not consider their mass shifts
in QGP. Therefore, for the transport equations for $\psi^{\prime}$ and
$\chi_{c}$, their masses are temperature independent and the mean field forces
disappear, the zeroth-order solutions become the full distributions,
$f_{\psi^{\prime}}=f_{\psi^{\prime}}^{(0)}$ and
$f_{\chi_{c}}=f_{\chi_{c}}^{(0)}$.

We now consider the mean field effect on the $J/\psi$ distribution
$f_{J/\psi}$. We extract the mass shift from the QCD second-order Stark effect
lee ,

$\displaystyle\delta m_{J/\psi}(T)$ $\displaystyle=$ $\displaystyle
m_{J/\psi}(T)-m_{J/\psi}(0)$ $\displaystyle=$ $\displaystyle-{7\pi^{2}\over
18}{a^{2}\over e}\left[{2\over 11}M_{0}(T)+{3\over
4}{\alpha_{s}(T)\over\pi}M_{2}(T)\right],$

which is used as input for the spectral function analysis with QCD sum rules
morita , where the constants are taken as $a=0.271$ fm and $e=640$ MeV, the
coupling constant $\alpha_{s}$ is temperature dependent lee ; morita , and the
functions $M_{0}=\epsilon-3P$ and $M_{2}=\epsilon+P$ determined by the energy
density $\epsilon$ and pressure $P$ of the medium are extracted from the
lattice QCD simulations boyd . Leaving out first the mean field force induced
by the inhomogeneous property of the fireball but keeping the temperature
dependence of the mass, the solution of the transport equation has the same
form with the zeroth-order distribution, the only difference is the
replacement of $m_{J/\psi}(0)\rightarrow m_{J/\psi}(T)$,

$f_{J/\psi}^{(1)}=f_{J/\psi}^{(0)}\big{|}_{m_{J/\psi}(0)\rightarrow
m_{J/\psi}(T)}.$ (6)

If the effect of the mean field force is not very strong, we may infer that it
is not such a bad approximation to replace $f_{J/\psi}$ in the mean field term
by the known $f_{J/\psi}^{(1)}$, the approximate transport equation

${\partial f_{J/\psi}\over\partial\tau}+{{\bf p}\over
E_{J/\psi}}\cdot{\bf\nabla}_{x}f_{J/\psi}=-\alpha_{J/\psi}f_{J/\psi}+\beta_{J/\psi}-{\bf
F}\cdot{\bf\nabla}_{p}f_{J/\psi}^{(1)}$ (7)

is then similar to the equation for $f_{J/\psi}^{(1)}$ but with a replacement
for the regeneration

$\beta_{J/\psi}\to\beta_{J/\psi}-{\bf F}\cdot{\bf\nabla}_{p}f_{J/\psi}^{(1)}.$
(8)

This means that the mean field force can be considered as an effective
regeneration, which is not from the coalescence of heavy quarks but due to the
$J/\psi$ mass shift. The approximate solution so obtained is known as the
second-order $J/\psi$ distribution,

$f_{J/\psi}^{(2)}({\bf p},{\bf x},\tau|{\bf b})=f_{J/\psi}^{(1)}({\bf p},{\bf
x},\tau|{\bf b})-\int_{\tau_{0}}^{\tau}d\tau_{1}\left[{\bf F}({\bf
x}_{1})\cdot{\bf\nabla}_{p}f_{J/\psi}^{(1)}({\bf p},{\bf x}_{1},\tau_{1}|{\bf
b})\right]e^{-\int_{\tau_{1}}^{\tau}d\tau_{2}\alpha({\bf p},{\bf
x}_{2},\tau_{2}|{\bf b})}.$ (9)

Substituting the obtained $f_{J/\psi}^{(2)}$ into the mean field term, and
solving the transport equation again, we can derive the third-order
distribution function $f_{J/\psi}^{(3)}$. With the similar way, the
distribution to the $n$-th order can be generally expressed as a series of the
mean field force,

$\displaystyle f_{\Psi}^{(n)}({\bf p},{\bf x},\tau|{\bf b})$ $\displaystyle=$
$\displaystyle f_{\Psi}^{(1)}({\bf p},{\bf x},\tau|{\bf
b})-\sum_{m=1}^{n-1}(-1)^{m-1}\int_{\tau_{0}}^{\tau}d\tau_{1}\int_{\tau_{0}}^{\tau_{1}}d\tau_{2}\cdots\int_{\tau_{0}}^{\tau_{m-1}}d\tau_{m}$
(10) $\displaystyle\times$ $\displaystyle\left[\left(\prod_{i=1}^{m}{\bf
F}({\bf x}_{i})\cdot{\bf\nabla}_{p}\right)f_{\Psi}^{(1)}({\bf p},{\bf
x}_{m},\tau_{m}|{\bf
b})\right]e^{-\int_{\tau_{m}}^{\tau}d\tau^{\prime}\alpha({\bf p},{\bf
x}^{\prime},\tau^{\prime}|{\bf b})}.$

Note again that the mass shift affects the $J/\psi$ production in two aspects.
The reduced threshold, which is considered already in the first-order
distribution $f_{J/\psi}^{(1)}$, makes the production more easily, and the
attractive force between $J/\psi$ and the matter, which is included only in
the higher-order distributions $f_{J/\psi}^{(n)}\ (n=2,3,\cdots)$, makes a
momentum shift and leads to a low $p_{t}$ enhancement and a corresponding high
$p_{t}$ suppression.

By integrating the charmonium distribution function $f_{\Psi}({\bf p},{\bf
x},\tau|{\bf b})$ over the hyper surface of hadronization determined by
$T({\bf x},\tau)=T_{c}$, and considering the decay of the excited states to
the ground state, we can calculate the number $N_{AA}$ of survived $J/\psi$s
in a heavy ion collision. Let’s first examine the differential nuclear
modification factor
$R_{AA}(p_{t})=N_{AA}(p_{t})/\left(N_{c}N_{pp}(p_{t})\right)$ as a function of
transverse momentum $p_{t}$ in mid rapidity region, where $N_{pp}$ is the
number of $J/\psi$s in a nucleon-nucleon collision and $N_{c}$ the number of
nucleon-nucleon collisions. To see the largest mean field effect, we calculate
first $R_{AA}(p_{t})$ in central collisions with ${\bf b}=0$ where the formed
fireball is hot and large and the surviving time is long. Fig.2 shows our
numerical results for Au+Au collisions at top RHIC energy $\sqrt{s_{NN}}=200$
GeV. The thick and thin lines indicate our results with and without
considering the $J/\psi$ mass shift, by taking $f_{J/\psi}$ and
$f_{J/\psi}^{(0)}$, respectively. The dotted, dashed and solid lines are the
calculations with initial production only, regeneration only and the total.

From our numerical results, the series $f_{J/\psi}^{(n)}\ (n=0,1,2,\cdots)$
converges rapidly, and the deviation
$\left|R_{AA}^{(2)}-R_{AA}^{(1)}\right|/\left|R_{AA}^{(1)}\right|<3\%$ is
valid in any case. This confirms our qualitative estimation that the mean
field effect is mainly reflected in the change in the threshold, the
attractive force is a second order effect and the higher order corrections
with $n>2$ can be safely neglected. For the following numerical calculations
we will take $f_{J/\psi}=f_{J/\psi}^{(2)}$. Since the initial production has
ceased before the formation of the hot medium, the change in the threshold
does not affect it, and the correction is only from the small mean field
force. That is the reason why the initial contribution shown in Fig.2 is
almost independent of the mass shift. However, the regeneration happens in the
hot medium, it is affected by both the reduced threshold and the mean field
force, the overall correction should be much larger in comparison with the
correction to the initial production. Considering the fact that the
regenerated $J/\psi$s in the early stage of the QGP will be eaten up by the
hot medium and only the $J/\psi$s regenerated in the later stage of the QGP
can survive, the enhanced regeneration is mainly in the low $p_{t}$ region, as
shown in Fig.2. At $p_{t}=0$, the total $R_{AA}$ goes up from $0.38$ to
$0.48$, the enhancement is $26\%$. Since some $J/\psi$s are kicked to low
$p_{t}$ region by the mean field force, there is a little reduction of
$R_{AA}$ in the mid $p_{t}(2-3$ GeV) region. At high enough $p_{t}$, there is
almost no effect of the mass shift, and the $J/\psi$ distribution is dominated
by the perturbative QCD in the initial production. In comparison with our
previous calculations liu2 ; zhou , the Gaussian smearing treatment zhao ; liu
for the Cronin effect used here reduces the $R_{AA}$ at high $p_{t}$.

Figure 2: (color online) The differential nuclear modification factor
$R_{AA}(p_{t})$ as a function of transverse momentum $p_{t}$ in Au+Au
collisions at impact parameter $b=0$ and top RHIC energy $\sqrt{s_{NN}}=200$
GeV. The initial production, the regeneration, and the total are shown by
dotted, dashed and solid lines. The thick and thin lines are the calculations
with and without considering the mean field effect. Figure 3: (color online)
The differential nuclear modification factor $R_{AA}(p_{t})$ as a function of
transverse momentum $p_{t}$ in Au+Au collisions at impact parameter $b=4.2$ fm
and top RHIC energy $\sqrt{s_{NN}}=200$ GeV. The data are from the PHENIX
phenix  and STAR star  collaborations, and the thick and thin lines are the
full calculations with and without considering the mean field effect.

To compare with the experimental data, we show in Fig.3 the RHIC data for
Au+Au collisions at centrality $0\%-20\%$ phenix ; star  and our calculation
at $b=4.2$ fm. With decreasing centrality, the fireball temperature drops down
and the mean field effect should be gradually reduced. However, from $b=0$ to
$4.2$ fm, the change in the mean field is small, and the model calculations
are almost the same. In our calculation we used the EKS98 parton distribution
function eks98  for the shadowing evolution and incorporated it with our
transport model through the initial distribution. We show in Fig.3 only the
total calculation, the results with and without the mean field effect can both
describe the data reasonably well. Since the current data are still with large
uncertainty even in low $p_{t}$ region, we need more precise data to
distinguish the mean field effect in the distribution.

Note that in the above calculations the shadowing effect reduces the charm
quark number and in turn the $J/\psi$ regeneration in the medium. Since the
parton momentum fraction $x_{F}$ at RHIC energy does not lie in the remarkable
shadowing region, the shadowing effect on $J/\psi$ production is not
remarkable. However, $x_{F}$ at LHC energy becomes very small and lies in the
strong shadowing region. This would give rise to a considerable reduction of
the $J/\psi$ regeneration. We calculated the $J/\psi$ $R_{AA}(p_{t})$ in Pb+Pb
collisions at LHC energy $\sqrt{s_{NN}}=2.76$ TeV and compared it with the
data from the ALICE collaboration alice , as shown in Fig.4. Besides the
shadowing effect, another important effect one should include in the
comparison with the ALICE data is the B meson decay, which leads to the non-
prompt part in the inclusive $J/\psi$ yield. We use the experimentally
measured p+p data from the CDF cdf , CMS cms  and ATLAS atlas  collaborations
to estimate the decay contribution and take the B meson quench in the medium.
The contribution from the B meson decay is important at high $p_{t}$ but the
influence is small at low $p_{t}$. The charm quark production cross section
$\sigma_{c\bar{c}}=0.38$ mb comes from the combination of the FONLL
calculation fonll  and the p+p data pp . From the numerical calculation, the
shadowing effect at LHC results in a $25\%$ reduction of the $J/\psi$
regeneration, and this makes a better agreement between the calculation
including shadowing effect and the data at low $p_{t}$, see Fig.4.

Figure 4: (color online) The differential nuclear modification factor
$R_{AA}(p_{t})$ as a function of transverse momentum $p_{t}$ in Pb+Pb
collisions at LHC energy $\sqrt{s_{NN}}=2.76$ TeV. The data are from the ALICE
collaboration alice  at centrality $0\%-90\%$ and at forward rapidity
$2.5<y<4$, and the model calculation is for the impact parameter $b=7.2$ fm.
The initial production, the regeneration, and the total are shown by dotted,
dashed and solid lines, and the thick and thin lines are the calculations with
and without considering the mean field effect.

While the mean field effect changes significantly the differential nuclear
modification factor at low transverse momentum, it does not affect the global
yield remarkably. From Figs.2-4, the most important mean field effect is at
$p_{t}=0$ and the effective region is around $p_{t}=0.5$ GeV which is much
less than the averaged $J/\psi$ transverse momentum $\langle
p_{t}\rangle\simeq 2-3$ GeV at RHIC and LHC energies. Therefore, the
$p_{t}$-integrated nuclear modification factor $R_{AA}(N_{p})$ as a function
of the number $N_{p}$ of participant nucleons becomes not sensitive to the
mean field effect. From our numerical calculations, the mass shift induced
enhancement of $R_{AA}(N_{p})$ is very small in peripheral and semi-central
collisions and less than $10\%$ even in most central collisions. Different
from $R_{AA}$ which is a summation of the initial production and regeneration,
the averaged transverse momentum square $\langle p_{t}^{2}\rangle$ is governed
by the fraction of the regeneration liu2 ; zhou . We found that the enhanced
$J/\psi$ yield at low $p_{t}$ leads to a less than $10\%$ suppression of
$\langle p_{t}^{2}\rangle$ in most central collisions.

In summery, we developed a perturbative expansion to study the mean field
effect on $J/\psi$ production in heavy ion collisions. Taking the mean field
term as a perturbation, we analytically solved the $J/\psi$ transport equation
with elastic and inelastic terms to any order and found a rapid convergence of
the perturbative expansion. The initial $J/\psi$ production, which happens
before the formation of QGP, is not sensitive to the mean field force, but the
continuous regeneration, which happens in the evolution of QGP, is
significantly affected by the mean field. As a result, the differential
nuclear modification factor of $J/\psi$ is enhanced at low $p_{t}$ in heavy
ion collisions at RHIC and LHC.

Acknowledgement: We thank useful discussions with Yunpeng Liu, Zhen Qu and Nu
Xu. The work is supported by the NSFC (Grant Nos. 10975084 and 11079024) and
RFDP (Grant No.20100002110080 ).

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