# A basic homogenization problem for the $p$-Laplacian in ${\mathbb{R}}^{d}$
perforated along a sphere: $L^{\infty}$ estimates

Peter V. Gordon Department of Mathematical Sciences, Kent State University,
Kent, OH 44242, USA. E-mail<EMAIL_ADDRESS>Fedor Nazarov Department of
Mathematical Sciences, Kent State University, Kent, OH 44242, USA. E-mail:
<EMAIL_ADDRESS>Yuval Peres Department of Mathematical Sciences, Kent
State University, Kent, OH 44242, USA. E-mail<EMAIL_ADDRESS>
###### Abstract

We consider a boundary value problem for the $p$-Laplacian, posed in the
exterior of small cavities that all have the same $p$-capacity and are
anchored to the unit sphere in $\mathbb{R}^{d}$, where $1<p<d.$ We assume that
the distance between anchoring points is at least $\varepsilon$ and the
characteristic diameter of cavities is $\alpha\varepsilon$, where
$\alpha=\alpha(\varepsilon)$ tends to 0 with $\varepsilon$. We also assume
that anchoring points are asymptotically uniformly distributed as
$\varepsilon\downarrow 0$, and their number is asymptotic to a positive
constant times $\varepsilon^{1-d}$. The solution $u=u^{\varepsilon}$ is
required to be 1 on all cavities and decay to 0 at infinity. Our goal is to
describe the behavior of solutions for small $\varepsilon>0$. We show that the
problem possesses a critical window characterized by
$\tau:=\lim_{\varepsilon\downarrow 0}\alpha/\alpha_{c}\in(0,\infty)$, where
$\alpha_{c}=\varepsilon^{1/\gamma}$ and $\gamma=\frac{d-p}{p-1}.$ We prove
that outside the unit sphere, as $\varepsilon\downarrow 0$, the solution
converges to $A_{*}U$ for some constant $A_{*}$, where
$U(x)=\min\\{1,|x|^{-\gamma}\\}$ is the radial $p$-harmonic function outside
the unit ball. Here the constant $A_{*}$ equals 0 if $\tau=0$, while $A_{*}=1$
if $\tau=\infty$. In the critical window where $\tau$ is positive and finite,
$A_{*}\in(0,1)$ is explicitly computed in terms of the parameters of the
problem. We also evaluate the limiting $p$-capacity in all three cases
mentioned above. Our key new tool is the construction of an explicit ansatz
function $u_{A_{*}}^{\varepsilon}$ that approximates the solution
$u^{\varepsilon}$ in $L^{\infty}(\mathbb{R}^{d})$ and satisfies $\|\nabla
u^{\varepsilon}-\nabla u_{A_{*}}^{\varepsilon}\|_{L^{p}(\mathbb{R}^{d})}\to 0$
as $\varepsilon\downarrow 0$.

Keywords: $p$-Laplacian, $p$-capacity, homogenization, $L^{\infty}$ estimates.

Mathematics Subject Classification MSC 2020: 35J92, 31C45, 35B27, 35B40,
35J20, 35J25.

##  1 Introduction

In this paper we consider a boundary value problem for the $p$-Laplacian in a
domain obtained from ${\mathbb{R}}^{d}$ by perforating it (i.e., removing
small cavities) along the unit sphere. The problem is formulated as follows.
Given $\varepsilon>0$, let $S=S(\varepsilon)$ be a finite set of points on the
unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^{d}$ such that the Euclidean
distance between any two points in $S$ is at least $\varepsilon.$ Points in
$S$ will be referred to as anchors.

Figure 1:  Two examples of sets $\Gamma$ in $\mathbb{R}^{3}$ that are unions
of small cavities anchored to the unit sphere $\mathbb{S}^{2}$: congruent
cones (left) and spheres, cubes and cylinders of the same $p$-capacity
(right).

For each anchor $s$, let $K_{s}$ be a compact subset of the closed unit ball
$\bar{B}(0,1)\subset{\mathbb{R}}^{d}$. Let
$0<\alpha=\alpha(\varepsilon)\leq\frac{1}{80}$ satisfy $\alpha(\varepsilon)\to
0$ as $\varepsilon\to 0$. Define

$\displaystyle\Gamma=\Gamma_{\varepsilon}:=\underset{s\in
S}{\bigcup}(s+\alpha\varepsilon K_{s})\,.$ (1.1)

Two examples of such sets are depicted in Figure 1.

Assume that $1<p<d$ and let $u=u^{\varepsilon}$ be the Perron solution of the
boundary value problem

$\displaystyle\left\\{\begin{array}[]{lll}\Delta_{p}u={\rm div}(|\nabla
u|^{p-2}\nabla u)=0&\mbox{in}&\mathbb{R}^{d}\setminus\Gamma,\\\
u=1&\mbox{on}&\Gamma,\\\ u(x)\to
0&\mbox{as}&|{x}|\to\infty.\end{array}\right.$ (1.5)

(We recall the definition of Perron solutions in Section 2.) We are interested
in the asymptotic behavior of $u^{\varepsilon}$ as $\varepsilon\to 0$ under
the following two key hypotheses:

(${\bf H_{1}}$)

The anchors are asymptotically equidistributed, that is

$\displaystyle\varepsilon^{d-1}\sum_{s\in
S(\varepsilon)}\delta_{s}\overset{\ast}{\rightharpoonup}\sigma\mu\>~{}\mbox{as}~{}\varepsilon\to
0,$ (1.6)

where $\mu$ is the uniform probability measure on the unit sphere
$\mathbb{S}^{d-1}$ and $\sigma>0$.

(${\bf H_{2}}$)

All the sets $K_{s}$ have the same ${\mathbf{p}}$-capacity, i.e., there is
some compact set $K\subseteq\bar{B}(0,1)\subset{\mathbb{R}}^{d}$ such that
${\rm cap}_{p}(K_{s})={\rm cap}_{p}(K)>0$ for all $\varepsilon>0$ and $s\in
S(\varepsilon)$, where

$\displaystyle{\rm
cap}_{p}(K):=\inf\left\\{\int_{\mathbb{R}^{d}}|\nabla\psi|^{p}:~{}\psi\in
C_{0}^{\infty}(\mathbb{R}^{d}),~{}\psi\geq 1~{}\mbox{on}~{}K\right\\},$ (1.7)

see [hkm, Chapter 2]. A special case to keep in mind is when all the $K_{s}$
are rotated copies of $K$, as in the left part of Figure 1.

The Perron solution $u$ of (1.5) is related to the $p$-capacity of $\Gamma.$
Specifically, the following identity holds [hkm, Theorems 9.33 and 9.35]:

$\displaystyle{\rm cap}_{p}(\Gamma)=\int_{\mathbb{R}^{d}}|\nabla u|^{p}.$
(1.8)

In this context, $u$ is called the $p$-equilibrium potential of $\Gamma.$ In
previous works in similar setups [KS14, KS16, GPPS] (summarized at the end of
this section), asymptotics of $u^{\varepsilon}$ in the Sobolev space $H^{1,p}$
were studied. The methods of the present paper enable us to obtain precise
$L^{\infty}$ asymptotics.

To motivate our results, note that since the $p$-capacity is sub-additive
[hkm, Theorem 2.2], we have:

$\displaystyle{\rm cap}_{p}(\Gamma)\leq\underset{s\in S}{\sum}{\rm
cap}_{p}\bigl{(}s+\alpha\varepsilon
K_{s}\bigr{)}=|S|(\alpha\varepsilon)^{d-p}{\rm
cap}_{p}(K)=\frac{(\alpha\varepsilon)^{d-p}}{\varepsilon^{d-1}}(\varepsilon^{d-1}|S|){\rm
cap}_{p}(K)\,,$ (1.9)

where we used the easily checked scaling relation

$\displaystyle{\rm cap}_{p}(aK)=a^{d-p}{\rm cap}_{p}(K),\qquad\forall a>0.$
(1.10)

Taking into account that $\varepsilon^{d-1}|S|\to\sigma$ as $\varepsilon\to
0$, we obtain that

$\displaystyle{\rm cap}_{p}(\Gamma)\leq
C\left(\frac{\alpha}{\alpha_{c}}\right)^{d-p}\,,$ (1.11)

where $C$ is some constant independent of $\varepsilon,$

$\displaystyle\alpha_{c}:=\varepsilon^{\frac{1}{\gamma}}\,,$ (1.12)

and

$\displaystyle\ \gamma:=\frac{d-p}{p-1}\,.$ (1.13)

On the other hand, since $p$-capacity is monotone and
$\Gamma\subset\bar{B}(0,1+\varepsilon)$, we have

$\displaystyle{\rm cap}_{p}(\Gamma)\leq(1+\varepsilon)^{d-p}{\rm
cap}_{p}(\bar{B}(0,1)).$ (1.14)

Thus, when the limiting value (as $\varepsilon\to 0$) of
$\frac{\alpha}{\alpha_{c}}$ increases from $0$ to $\infty$, the natural upper
bounds (1.11) and (1.14) for ${\rm cap}_{p}(\Gamma)$ cross each other. It
turns out that in the same regime, the solution $u$ of (1.5) (outside of
$\mathbb{S}^{d-1}$) gradually transitions from $0$ to the $p$-equilibrium
potential $U$ of the unit ball, defined as the solution of

$\displaystyle\left\\{\begin{array}[]{lll}\Delta_{p}U=0&\mbox{in}&\mathbb{R}^{d}\setminus\bar{B}(0,1),\\\
U=1&\mbox{on}&\bar{B}(0,1),\\\ U(x)\to
0&\mbox{as}&|x|\to\infty\,,\end{array}\right.$ (1.18)

and given by

$\displaystyle U(x):=\left\\{\begin{array}[]{ll}1&0\leq|x|\leq 1\,,\\\
|x|^{-\gamma}&|x|>1\,.\end{array}\right.$ (1.21)

The discussion above motivates our third hypothesis:

(${\bf H_{3}}$)

$\lim_{\varepsilon\to
0}\alpha(\varepsilon)\varepsilon^{-1/\gamma}=\tau\in[0,\infty]$.

Most of the paper will be devoted to the analysis of the critical window,
where $0<\tau<\infty$. Our first result describes the asymptotics of the
capacity ${\rm cap}_{p}(\Gamma_{\varepsilon})$ and of the potential
$u=u^{\varepsilon}$ away from the unit sphere.

######  Theorem 1.1.

Suppose that hypotheses ${\rm(H_{1}),(H_{2}),(H_{3})}$ hold. Then, as
$\varepsilon\to 0$,

$\displaystyle
u^{\varepsilon}(x)\to\left\\{\begin{array}[]{lll}0&\mbox{if}&\tau=0\,,\\\
A_{*}U(x)&\mbox{if}&\ \tau\in(0,\infty)\,,\\\
U(x)&\mbox{if}&\tau=\infty\end{array}\right.$ (1.25)

uniformly on compact subsets of $\mathbb{R}^{d}\setminus\mathbb{S}^{d-1},$
where for $\tau\in[0,\infty)$,

$\displaystyle A_{*}=A_{*}(\tau)=\frac{\left(\sigma\tau^{d-p}{\rm
cap}_{p}(K)\right)^{\frac{1}{p-1}}}{\left(\sigma\tau^{d-p}{\rm
cap}_{p}(K)\right)^{\frac{1}{p-1}}+\left({\rm
cap}_{p}(\bar{B}(0,1))\right)^{\frac{1}{p-1}}}\,.$ (1.26)

Furthermore,

$\displaystyle{\rm
cap}_{p}(\Gamma_{\varepsilon})\to\left\\{\begin{array}[]{lll}0&\mbox{if}&\tau=0\,,\\\
A_{*}^{p}{\rm cap}_{p}(\bar{B}(0,1))+(1-A_{*})^{p}{\rm
cap}_{p}(K)\sigma\tau^{d-p}&\mbox{if}&\ \tau\in(0,\infty)\,,\\\ {\rm
cap}_{p}(\bar{B}(0,1))&\mbox{if}&\tau=\infty\,.\end{array}\right.$ (1.30)

It will be convenient to define $A_{*}=1$ if $\tau=\infty$. If we assume that
$\alpha=C\varepsilon^{\zeta}$ for $\zeta>0$, then this theorem reveals a phase
transition when the exponent $\zeta$ crosses $1/\gamma$.

Example. To illustrate the Theorem, consider the simple special case where
$d=3,p=2$ (so $\gamma=1$ and $\alpha_{c}=\varepsilon$) and
$K_{s}=\bar{B}(0,1)$ for all $s$. Suppose that $\alpha=\tau\alpha_{c}$. In
this case, $U(x)=\min\\{1,|x|^{-1}\\}$ and $u^{\varepsilon}(x)$ can be
interpreted as the probability that a Brownian motion started from $x$ ever
hits $\Gamma=\cup_{s\in S}B(s,\tau\varepsilon^{2})$. The expression for
$A_{*}$ simplifies to $A_{*}=\sigma\tau/(1+\sigma\tau)$.

Figure 2: The ansatz function $u_{A_{*}}$ and its level sets, determined by
$12$ equally spaced anchors on the unit circle ($d=2$). Here $K=\bar{B}(0,1),$
$~{}p=1.5,~{}\gamma=1,~{}\varepsilon\approx 0.52,~{}\tau=1/40$ and
$A_{*}\approx 0.5$.

Figure 3: The phase transition as $\tau$ increases: A single bump for
$\tau=1/4$ ($A_{*}\approx 0.25$), $\tau=1$ ($A_{*}\approx 0.5$), and $\tau=3$
($A_{*}\approx 0.75$). Parameters: $p=1.5,$ $\gamma=1,$ $\varepsilon\approx
5\cdot 10^{-3}.$ The number of anchors is $1250$, only a square of side-length
$20\varepsilon\alpha_{c}$ centered at one of them is depicted.

For $\tau\in(0,\infty)$, the expression (1.25) suggests that ${\rm
cap}_{p}(\Gamma_{\varepsilon})$ should approach $A_{*}^{p}{\rm
cap}_{p}(\bar{B}(0,1))$ as $\varepsilon\to 0$. This, however, contradicts the
actual limiting value of the capacity given by (1.30). Indeed, the limiting
equilibrium potential only captures the first summand in (1.30), which
accounts for the contribution from the bulk. The second term in (1.30), which
accounts for the contribution of the equilibrium potential near the cavities,
completely disappears when looking at (1.25) alone. This is similar to the
“term coming from nowhere” discussed in [CM97].

The preceding observation raises the question: How does the solution of (1.5)
behave near the unit sphere? To answer this question, we will introduce the
$p$-potential of $K_{s}$ in a ball $B(0,R)$, namely, the Perron solution
$V_{R}^{s}:{\mathbb{R}}^{d}\to[0,1]$ of

$\displaystyle\left\\{\begin{array}[]{lll}\Delta_{p}V_{R}^{s}=0&\mbox{in}&B(0,R)\setminus
K_{s},\\\ V_{R}^{s}=1&\mbox{on}&K_{s},\\\ V_{R}^{s}(x)=0&\mbox{for}&|x|\geq
R.\end{array}\right.$ (1.34)

We also define $V_{\infty}^{s}$ by replacing the last requirement in (1.34) by
$V_{\infty}^{s}(x)\to 0$ as $|x|\to\infty$. With $U$ defined by (1.21) and
$U_{1+\varepsilon}(x):=U\Bigl{(}\frac{x}{1+\varepsilon}\Bigr{)}$, we define,
for each $A\in[0,1]$, the ansatz function

$\displaystyle u_{A}(x):=AU_{1+\varepsilon}(x)+(1-A)\sum_{s\in
S}V_{\frac{1}{10\alpha}}^{s}\left(\frac{x-s}{\alpha\varepsilon}\right)\,.$
(1.35)

For $\tau<\infty$, the convergence $u^{\varepsilon}(x)\to A_{*}U(x)$ as
$\varepsilon\to 0$ in Theorem 1.1 does not hold uniformly in
${\mathbb{R}}^{d}\setminus{\mathbb{S}}^{d-1}$ because
$\sup_{x\in{\mathbb{R}}^{d}\setminus{\mathbb{S}}^{d-1}}u^{\varepsilon}(x)=1$.
The following theorem states that $\nabla u_{A_{*}}$ approximates $\nabla u$
in $L^{p}({\mathbb{R}}^{d})$ and $u_{A_{*}}$ approximates $u$ in
$L^{\infty}({\mathbb{R}}^{d})$ as $\varepsilon\to 0$. See Figures 2 and 3 for
a depiction of $u_{A_{*}}$ and of how it changes when $\tau$ increases.

######  Theorem 1.2.

Suppose that hypotheses ${\rm(H_{1}),(H_{2}),(H_{3})}$ hold. Then as
$\varepsilon\to 0$, we have

$\displaystyle\|\nabla u-\nabla u_{A_{*}}\|_{{L^{p}(\mathbb{R}^{d})}}\to 0,$
(1.36)

and

$\displaystyle\|u-u_{A_{*}}\|_{L^{\infty}(\mathbb{R}^{d})}\to 0\,.$ (1.37)

######  Remark 1.1.

Several technical challenges arise because there are no regularity assumptions
on the compact sets $K_{s}$. First, the solution $u$ of (1.5) depends
monotonically on $\alpha(\varepsilon)$ only if $K_{s}$ are star-shaped.
Moreover, in general $u$ need not be continuous on the boundary of $\Gamma$.

Related work. There is an extensive literature devoted to the $p$-Laplacian
and nonlinear potential theory, see, e.g., the books [Mazya, Lindq, hkm,
Adams-Hedberg] and the review paper [pLrev]. Chapter 3 of [MKhomo] discusses
homogenization problems for the $p$-Laplacian. Most works on this topic focus
on the derivation of effective limiting equations in domains that are
perforated in the bulk; see, e.g., [MKhomo, CM97] and references therein. In
the papers [KS14, KS16, GPPS], which are most closely related to our study,
the authors considered homogenization for the $p$-Laplacian in domains where
the perforation takes place only near a $(d-1)$ dimensional surface. In [KS14]
the authors assume that $p<1+d/2$ and the cavities are obtained by
intersecting a periodic structure with a hyperplane; the critical scaling
$\alpha_{c}$ was already identified there under a mild regularity assumption
on the cavities. In [KS16], the hyperplane is replaced by a convex surface and
$p<1+d/4$. In [GPPS] the cavities are obtained by intersecting a periodic
collection of small balls with the $\varepsilon$-neighborhood of a smooth
surface. The latter paper also contains a detailed review of earlier
literature. In the works mentioned above, the authors determined the
asymptotics of the $p$-capacity of the obstacle, and the limiting behavior of
the solution in Sobolev space in the weak topology; the behavior of the
solution near the perforated surface was not described. That is our main goal
here.

Roadmap. While it is not so hard to show that the solution $u^{\varepsilon}$
is generally well-behaved away from the cavities $s+\alpha\varepsilon K_{s}$,
with rare regions of high local energy, the main obstacle to obtaining our
results (especially the $L^{\infty}$ estimates in Theorem 1.2), is ruling out
such exceptional regions.

The rest of the paper is organized as follows. Section 2 presents some
necessary background and preliminaries on $p$-potentials. In Section 3, we
bound the oscillation of $u^{\varepsilon}$ in cones and, using the Besicovich
covering lemma, obtain a lower bound for its energy in cones. In Section 4, we
combine this lower bound with the global energy minimization property of
$u^{\varepsilon}$, to infer a tight upper bound for its energy in cones; we
then use this to deduce that $u^{\varepsilon}$ is approximately constant on
$\partial B(0,1+\delta)$ (where $\varepsilon\ll\delta\ll 1$), and prove
Theorem 1.1. Section 5 shows that if $u^{\varepsilon}$ is approximately
constant on the boundary of a ball $\partial B(0,1+\delta)$, then it is also
approximately constant inside this ball, with the exception of small balls
$B(s,\varepsilon/10)$ around the anchors. The comparison argument in that
section is purely analytic, but is motivated by the connection with noisy tug
of war games described in [PS]. In the final section, we combine the results
from the preceding sections and derive Theorem 1.2.

##  2 Preliminaries

###  2.1 Notation

Given an open set $\Omega\subset\mathbb{R}^{d}$, let $H^{1,p}(\Omega)$ denote
the Sobolev space of $L^{p}(\Omega)$ functions with weak gradient in
$L^{p}(\Omega)$, and let $H^{1,p}_{0}(\Omega)$ denote the closure in
$H^{1,p}(\Omega)$ of $C_{0}^{\infty}(\Omega)$. Write $H_{loc}^{1,p}(\Omega)$
for the space of functions on $\Omega$ which, when restricted to every
subdomain $\Omega_{1}$ compactly contained in $\Omega$, are in
$H^{1,p}(\Omega_{1})$.

######  Definition 2.1.

[hkm, Chapters 6,7] Given an open set $\Omega\subset\mathbb{R}^{d}$, a
continuous function $w\in H_{loc}^{1,p}(\Omega)$ is called $p$-harmonic if
$\Delta_{p}w=0$ weakly, i.e., if

$\int_{\Omega}|\nabla w|^{p-2}\nabla w\cdot\nabla\eta=0\,,\quad\forall\eta\in
C_{0}^{\infty}(\Omega)\,.$

A function $\psi:\Omega\to(-\infty,\infty]$ is $p$-superharmonic if it is
lower semi-continuous, not identically $\infty$ on any connected component of
$\Omega$, and satisfies the following comparison inequality on compactly
contained subdomains $D\subset\Omega$: If $\psi\geq w$ on $\partial D$ and
$w\in C(\bar{D})$ is $p$-harmonic in $D$, then $\psi\geq w$ in $D$.

The following principle, a special case of [hkm, Proposition 7.6], will be
used several times.

######  Lemma 2.1 (Comparison principle).

Let $v,w$ be bounded $p$-harmonic functions in an open set
$\Omega\subset{\mathbb{R}}^{d}$. If

$\limsup_{x\to y}v(x)\leq\liminf_{x\to y}w(x)$

for all $y\in\partial\Omega$ (including $y=\infty$ if $\Omega$ is unbounded)
then $v\leq w$ in $\Omega$.

Next we give the definition of Perron solutions from [hkm, Chapter 9].

######  Definition 2.2.

Given a domain $\Omega\subset\mathbb{R}^{d}$ and boundary values
$f:M\to\mathbb{R}$, where $\partial\Omega\subset M\subset\Omega^{c}$, the
upper class $U_{f}^{\Omega}$ of $f$ consists of $p$-superharmonic functions
$\psi:\Omega\to(-\infty,\infty]$ that are bounded below and satisfy

$\displaystyle\liminf_{x\to y}\psi(x)\geq f(y),\quad\forall
y\in\partial\Omega.$ (2.1)

(Recall that if $\Omega$ is unbounded, then we include $\infty$ in
$\partial\Omega$.) The upper Perron solution

$h=\bar{H}_{f}^{\Omega}:\Omega\cup M\to[-\infty,\infty]$

of the boundary value problem

$\displaystyle\left\\{\begin{array}[]{lll}\Delta_{p}h=0&\mbox{in}&\Omega\,,\\\
h(y)=f(y)&\mbox{for}&y\in M\\\ \end{array}\right.$ (2.4)

is defined in $\Omega$ by

$\displaystyle\bar{H}_{f}^{\Omega}(x):=\inf\\{\psi(x)\,:\,\psi\in
U_{f}^{\Omega}\\}$ (2.5)

and extended to agree with $f$ in $M$. The lower Perron solution of (2.4) is
defined by $\underline{H}_{f}^{\Omega}:=-\bar{H}_{-f}^{\Omega}$.

We say that $f$ is resolutive in $\Omega$ if the upper and lower Perron
solutions of (2.4) coincide; in this case we refer to both of these simply as
the Perron solution.

Theorem 9.25 in [hkm] ensures that if $f$ is continuous on $\partial\Omega$
and $(\mathbb{R}^{d}\setminus\Omega)$ contains a compact set of positive
$p$-capacity, then $f$ is resolutive.

Following [hkm], we define the Dirichlet space

$\displaystyle L^{1,p}(\Omega):=\\{v\in H_{loc}^{1,p}(\Omega):~{}\nabla v\in
L^{p}(\Omega)\\},$ (2.6)

and let $L_{0}^{1,p}(\Omega)$ be the closure of $C_{0}^{\infty}(\Omega)$ in
$L^{1,p}(\Omega)$ with respect to the semi-norm $\left(\int_{\Omega}|\nabla
v|^{p}\right)^{\frac{1}{p}}$, see [hkm, Chapter 1.9, p. 13].

######  Definition 2.3.

For a function $v\in L^{1,p}(\Omega)$, we define its energy in a Borel set
$B\subset\Omega$ by

$\displaystyle E(v,B)=\int_{B}|\nabla v|^{p}\,.$ (2.7)

We write $E(v,{\mathbb{R}}^{d})$ simply as $E(v)$.

We now define the $p$-capacity (with respect to a domain $\Omega$) of a
compact set $K\subset\Omega$ by

$\displaystyle{\rm cap}_{p}(K,\Omega):=\inf\\{E(\psi,\Omega),~{}\psi\geq
1~{}\mbox{on}~{}K,~{}\psi\in C_{0}^{\infty}(\Omega)\\}.$ (2.8)

This definition readily implies that

$\displaystyle{\rm cap}_{p}(K,B(0,R))\downarrow{\rm cap}_{p}(K)\;{\rm
as}\;R\uparrow\infty\,,$ (2.9)

where ${\rm cap}_{p}(K)={\rm cap}_{p}(K,{\mathbb{R}}^{d})$. In [hkm, p. 27-28]
it is shown that

$\displaystyle{\rm cap}_{p}(K,\Omega)=\inf\\{E(\psi,\Omega),~{}\psi\geq
1~{}\mbox{on}~{}K,~{}\psi\in H_{0}^{1,p}(\Omega)\cap C(\Omega)\\}.$ (2.10)

Fix $\phi\in C_{0}^{\infty}(\Omega)$ such that $\phi=1$ on $K$, and define the
admissible class

$\displaystyle{\mathscr{A}}(K,\Omega,\phi):=\\{\psi\in
L_{0}^{1,p}(\Omega)~{}\mbox{\rm such that}~{}\psi-\phi\in
L_{0}^{1,p}(\Omega\setminus K)\\}\,.$ (2.11)

Then we claim that

$\displaystyle{\rm cap}_{p}(K,\Omega)\leq E(\psi,\Omega)\;\;\mbox{\rm
for}\;\psi\in{\mathscr{A}}(K,\Omega,\phi)\,.$ (2.12)

Indeed, suppose that $\psi\in L_{0}^{1,p}(\Omega)$ and $h_{n}\in
C_{0}^{\infty}(\Omega\setminus K)$ satisfy $\nabla(\psi-\phi-h_{n})\to 0$ in
$L^{p}(\Omega\setminus K)$. Then $\nabla(\phi+h_{n})=\nabla\phi=0$ a.e. in
$K$. Therefore, (2.10) implies that

${\rm cap}_{p}(K,\Omega)\leq E(\phi+h_{n},\Omega)=E(\phi+h_{n},\Omega\setminus
K)\to E(\psi,\Omega\setminus K)\,$

as $n\to\infty$.

Next, let $\psi^{K}$ be the Perron solution of

$\displaystyle\left\\{\begin{array}[]{lll}\Delta_{p}\psi^{K}=0&\mbox{in}&\Omega\setminus
K\,,\\\ \psi^{K}=1&\mbox{on}&K\,,\\\
\psi^{K}=0&\mbox{on}&\partial\Omega\,,\end{array}\right.$ (2.16)

where by convention, $\infty\in\partial\Omega$ if $\Omega$ is unbounded. By
[hkm, Theorem 9.33], $\psi^{K}\in{\mathscr{A}}(K,\Omega,\phi)$, and it
coincides with the $p$-potential of $K$ defined in [hkm, Chapter 6]. Moreover,
$\psi^{K}$ is continuous and $p$-harmonic on $\Omega\setminus K$, and it
satisfies $\psi^{K}\equiv 1$ on $K$. It follows from Corollary 1.21 in [hkm]
that $\nabla\psi^{K}(x)=0$ for almost every $x\in K$. By [hkm, Theorem 9.35],
$E(\psi^{K},\Omega)={\rm cap}_{p}(K,\Omega)$. This yields an alternative
definition of $p$-capacity: If $\phi\in C_{0}^{\infty}(\Omega)$ satisfies
$\phi=1$ on $K$, then

$\displaystyle{\rm
cap}_{p}(K,\Omega):=\min\\{E(\psi,\Omega):~{}\psi\in{\mathscr{A}}(K,\Omega,\phi)\\}.$
(2.17)

By the strict convexity of the $L^{p}$ norm, $\psi^{K}$ is the unique (up to
translation by a constant) minimizer of the extremal problem (2.17).

Thus the Perron solutions $u,V_{R}^{s}$ and $U$, defined in (1.5), (1.34) and
(1.18), have the following properties:

$\displaystyle\int_{\mathbb{R}^{d}}|\nabla u|^{p}={\rm
cap}_{p}(\Gamma),\quad\int_{B(0,R)}|\nabla V_{R}^{s}|^{p}={\rm
cap}_{p}(K_{s},B(0,R)),\quad\int_{\mathbb{R}^{d}}|\nabla U|^{p}={\rm
cap}_{p}(\bar{B}(0,1))=\gamma^{p-1}\omega_{d-1}\,,$ (2.18)

where $\omega_{d-1}$ is the surface area of the unit ball in $\mathbb{R}^{d}$,
and the last equality is from [hkm, Section 2.11].

######  Remark 2.1.

Fix $\phi\in C_{0}^{\infty}({\mathbb{R}}^{d})$ such that $\phi=1$ on
$\bar{B}(0,1)$ and $\phi=0$ outside $B(0,2)$, and suppose that $R>2$. Since
$V_{R}^{s}$ coincides with the $p$-potential of $K_{s}$ in $B(0,R)$, [hkm,
Lemma 8.5] yields that $V_{R}^{s}-\phi\in H_{0}^{1,p}(B(0,R)\setminus K_{s})$,
which readily implies that for any domain $\Omega$ that contains $B(0,R)$, we
have $V_{R}^{s}\in H_{0}^{1,p}(\Omega\setminus K_{s}$) and
$E(V_{R}^{s},\Omega)=E(V_{R}^{s},B(0,R))$. While the classical gradient of
$V_{R}^{s}$ on $\partial B(0,R)$ need not exist, the distributional gradient
may be taken to be zero on this boundary, and in any case, does not affect the
energy.

We will need an estimate on the rate of convergence in (2.9).

######  Lemma 2.2.

Suppose that $K\subset\bar{B}(0,1)\subset{\mathbb{R}}^{d}$ is compact. Then

$\displaystyle{\rm cap}_{p}(K)\leq{\rm
cap}_{p}(K,B(0,R))\leq\bigl{(}1-R^{-\gamma}\bigr{)}^{-p}{\rm cap}_{p}(K)\,.$
(2.19)

###### Proof.

The first inequality in (2.19) was already noted in (2.9), so we focus on the
second. Let $K_{n}$ be the union of the closed dyadic cubes of side length
$2^{-n}$ that intersect $K$. Denote by $\psi_{n}$ the $p$-potential of $K_{n}$
in ${\mathbb{R}}^{d}$ and let $\delta>0$. Since $K=\cap_{n=1}^{\infty}K_{n}$,
Theorem 2.2(iv) in [hkm] implies that there exists $n$ such that $K_{n}\subset
B(0,1+\delta)$ and

${\rm cap}_{p}(K_{n})\leq{\rm cap}_{p}(K)+\delta\,.$

Note that $\psi_{n}$ is continuous in ${\mathbb{R}}^{d}$ since
${\mathbb{R}}^{d}\setminus K_{n}$ is regular by the corkscrew condition (See
[hkm, Theorems 6.27 and 6.31].) Recall that $U$ given by (1.21) is the
$p$-potential of the unit ball in ${\mathbb{R}}^{d}$. Then $\psi_{n}(x)\leq
U\Bigl{(}\frac{x}{1+\delta}\Bigr{)}$ for all $x$ by the definition of upper
Perron solutions, so the function

$\psi(x)=\max\Bigl{\\{}0,\frac{\psi_{n}(x)-(1+\delta)^{\gamma}R^{-\gamma}}{1-(1+\delta)^{\gamma}R^{-\gamma}}\Bigr{\\}}$

is in $H_{0}^{1,p}(B(0,R))$ by [hkm, Lemmas 1.23 and 1.26]. Thus, using $\psi$
in (2.10) yields

${\rm cap}_{p}(K,B(0,R))\leq
E(\psi,B(0,R))\leq\bigl{(}1-(1+\delta)^{\gamma}R^{-\gamma}\bigr{)}^{-p}{\rm
cap}_{p}(K_{n})\,,$

which implies the second inequality of (2.19) since $\delta>0$ is arbitrary. ∎

###  2.2 Energy estimate on truncated cones

######  Definition 2.4.

Let $y\in{\mathbb{S}}^{d-1}$ and let
$Q_{\delta}(y):=B(y,\delta)\cap{\mathbb{S}}^{d-1}$ be the open spherical cap
of Euclidean radius $\delta$ centered at $y$. We define the spherical cone

$\displaystyle\Lambda_{\delta}(y):=\\{rq:q\in Q_{\delta}(y)\;\,\text{\rm
and}\;\,r>0\\}.$ (2.20)

######  Lemma 2.3.

Let $Q$ be an open set on the unit sphere, and for $R>1$ set

$\displaystyle\Lambda(Q,R):=\\{rq:q\in Q\;\,\text{\rm and}\;\,r>R\\}\,.$
(2.21)

Suppose that a nonnegative function $v\in L^{1,p}(\Lambda(Q,R))\cap
C(\overline{\Lambda(Q,R)})$ satisfies

$\displaystyle v(x)\to 0\quad\mbox{as}\quad|x|\to\infty\,,$ (2.22)

and

$\displaystyle v(Ry)\geq\tilde{A}>0,\quad\forall y\in Q\,.$ (2.23)

Then,

$\displaystyle E(v,\Lambda(Q,R))\geq E(v_{\tilde{A}},\Lambda(Q,R))\,,$ (2.24)

where $v_{\tilde{A}}(x)=\tilde{A}R^{\gamma}U(x)=\tilde{A}U(x/R)$ for all
$x\in\Lambda(Q,R)$.

###### Proof.

Assume (2.22) and (2.23) hold. Since $v\in L^{1,p}(\Lambda(Q,R))$, for $\mu$
almost every $y\in Q$, the gradient $\nabla v(ry)$ exists and is finite for
a.e. $r\in(R,\infty)$. Furthermore, [EG92, Theorem 4.9.2] implies that for
$\mu$ almost every $y\in Q$, the function $r\mapsto v(ry)\,$ is absolutely
continuous in $\,(R,\infty)$. Thus, for $\mu$ almost every $y\in Q$, Hölder’s
inequality yields that

$\displaystyle\tilde{A}\leq\int_{R}^{\infty}\left|\nabla
v(ry)\right|dr=\int_{R}^{\infty}\left|\nabla
v(ry)\right|r^{\frac{d-1}{p}}r^{\frac{1-d}{p}}\,dr\leq\left(\int_{R}^{\infty}\left|\nabla
v(ry)\right|^{p}r^{d-1}\,dr\right)^{\frac{1}{p}}\left(\int_{R}^{\infty}r^{\frac{1-d}{p-1}}dr\right)^{\frac{p-1}{p}}$
$\displaystyle=\left(\frac{1}{\gamma
R^{\gamma}}\right)^{\frac{p-1}{p}}\left(\int_{R}^{\infty}\left|\nabla
v(ry)\right|^{p}r^{d-1}dr\right)^{\frac{1}{p}}.$ (2.25)

Taking $p$-th power and rearranging terms, we obtain

$\displaystyle\tilde{A}^{p}R^{d-p}\gamma^{p-1}\leq\int_{R}^{\infty}\left|\nabla
v(ry)\right|^{p}r^{d-1}dr.$ (2.26)

Let $\omega_{d-1}$ denote the area of the unit sphere. Integrating (2.26) over
$Q$, we obtain

$\displaystyle\tilde{A}^{p}R^{d-p}\gamma^{p-1}\omega_{d-1}\mu(Q)\leq\omega_{d-1}\int_{Q}\int_{R}^{\infty}\left|\nabla
v(ry)\right|^{p}r^{d-1}\,dr\,d\mu(y)=\int_{\Lambda(Q,R)}\left|\nabla
v(x)\right|^{p}\,dx=E(v,\Lambda(Q,R))\,.$ (2.27)

By the condition for equality in Hölder’s inequality, the second inequality in
(2.2) is an equality if for some constant $C$,

$\displaystyle\Bigl{[}|\nabla
v(ry)|r^{\frac{d-1}{p}}\Bigr{]}^{p-1}=Cr^{\frac{1-d}{p}}\quad\mbox{\rm for
a.e.}\;r\geq R\,.$ (2.28)

The function $v_{\tilde{A}}(x)=\tilde{A}R^{\gamma}U(x)$ satisfies (2.28) and
the boundary condition $v_{\tilde{A}}(Ry)=\tilde{A}$ for all $y\in Q$. Since

$|\nabla v_{\tilde{A}}(ry)|=-\frac{d}{dr}v_{\tilde{A}}(ry)$

for $y\in\partial B(0,1)$, the first inequality in (2.2) is also an equality
if $v=v_{\tilde{A}}$. Therefore,

$\displaystyle
E(v_{\tilde{A}},\Lambda(Q,R))=\tilde{A}^{p}R^{d-p}\gamma^{p-1}\omega_{d-1}\mu(Q)\,.$
(2.29)

Combining (2.27) and (2.29), we obtain (2.24). ∎

###  2.3 Energy of an ansatz function

Recall from (1.35) the definition

$u_{A}(x)=AU_{1+\varepsilon}(x)+(1-A)\sum_{s\in
S}V_{\frac{1}{10\alpha}}^{s}\left(\frac{x-s}{\alpha\varepsilon}\right)\,.$

Since the gradients of the summands on the right-hand side have disjoint
supports, we have

$\displaystyle E(u_{A})=A^{p}E(U_{1+\varepsilon})+(1-A)^{p}\sum_{s\in
S}E\Bigl{(}x\mapsto
V_{\frac{1}{10\alpha}}^{s}\Bigl{(}\frac{x-s}{\alpha\varepsilon}\Bigr{)}\Bigr{)}\,.$
(2.30)

(By Remark 2.1, we can compute the energy over all of ${\mathbb{R}}^{d}$ in
the summands on the right-hand side.) For any function $\psi\in
L^{1,p}({\mathbb{R}}^{d})$, vector $s\in{\mathbb{R}}^{d}$ and scalar $r>0$,
scaling arguments yield that

$\displaystyle
E\Bigl{(}x\mapsto\psi\Bigl{(}\frac{x-s}{r}\Bigr{)}\Bigr{)}=r^{d-p}E(\psi)\,.$
(2.31)

Therefore,

$\displaystyle E(u_{A})=A^{p}(1+\varepsilon)^{d-p}E(U)+(1-A)^{p}\sum_{s\in
S}(\alpha\varepsilon)^{d-p}E\bigl{(}V_{\frac{1}{10\alpha}}^{s}\bigr{)}.$
(2.32)

Using the fact that $E(U)={\rm cap}_{p}(\bar{B}(0,1))$, we obtain

$\displaystyle E(u_{A})=(1+\varepsilon)^{d-p}A^{p}{\rm
cap}_{p}(\bar{B}(0,1))+(1-A)^{p}\sum_{s\in S}\kappa_{s}\,,$ (2.33)

where

$\displaystyle\kappa_{s}:={\rm cap}_{p}\left(\alpha\varepsilon
K_{s},B\left(s,\frac{\varepsilon}{10}\right)\right)\,.$ (2.34)

By Lemma 2.2 applied to $K_{s}$ and hypothesis ${\rm(H_{2})}$ on equality of
capacities, we infer that

$\displaystyle\Bigl{|}{\rm
cap}_{p}\Bigl{(}K_{s},B\Bigl{(}0,\frac{1}{10\alpha}\Bigr{)}\Bigr{)}-{\rm
cap}_{p}(K)\Bigr{|}\leq J(\alpha){\rm cap}_{p}(K)\,,$ (2.35)

where

$\displaystyle J(\alpha):=\bigl{(}1-(10\alpha)^{\gamma}\bigr{)}^{-p}-1\to
0\;\;\mbox{\rm as}\;\;\varepsilon\to 0\,.$ (2.36)

Therefore,

$\displaystyle\Bigl{|}E(u_{A})-(1+\varepsilon)^{d-p}A^{p}{\rm
cap}_{p}(\bar{B}(0,1))-(1-A)^{p}(\alpha\varepsilon)^{d-p}|S|{\rm
cap}_{p}(K)\Bigr{|}\leq(1-A)^{p}(\alpha\varepsilon)^{d-p}|S|J(\alpha){\rm
cap}_{p}(K)\,.$ (2.37)

Next, as $\varepsilon\to 0$, we have by (1.12) and the definition of $\tau$ in
${\rm(H_{3})}$ that

$\displaystyle(\alpha\varepsilon)^{d-p}|S|=\bigl{(}\alpha\varepsilon^{-1/\gamma}\bigr{)}^{d-p}\varepsilon^{d-1}|S|\to\tau^{d-p}\sigma\,.$
(2.38)

In conjunction with (2.37), this gives

$\displaystyle E(u_{A})\to\varphi_{\tau}(A)\;\mbox{\rm as}\;\varepsilon\to
0\,,$ (2.39)

where

$\displaystyle\varphi_{\tau}(A)=A^{p}{\rm cap}_{p}(\bar{B}(0,1))+(1-A)^{p}{\rm
cap}_{p}(K)\sigma\tau^{d-p}.$ (2.40)

If $\tau\in[0,\infty)$, then the convergence in (2.39) is uniform in
$A\in[0,1]$. Note that $\varphi_{\infty}(1)={\rm cap}_{p}(\bar{B}(0,1))$ and
$\varphi_{\infty}(A)=\infty$ for all $A\in[0,1)$, and in the latter case, the
convergence in (2.39) means that the left-hand side tends to $\infty$. For
$\tau\in(0,\infty)$, the function $\varphi_{\tau}$ is a continuous strictly
convex function on $[0,1]$. Differentiation shows that it attains its minimum
at $A_{*}=A_{*}(\tau)$ given by (1.26), with

$\displaystyle\varphi_{\tau}(A_{*})=\left(\frac{\left(\sigma\tau^{d-p}{\rm
cap}_{p}(K){\rm
cap}_{p}(\bar{B}(0,1))\right)^{\frac{1}{p-1}}}{\left(\sigma\tau^{d-p}{\rm
cap}_{p}(K)\right)^{\frac{1}{p-1}}+\left({\rm
cap}_{p}(\bar{B}(0,1))\right)^{\frac{1}{p-1}}}\right)^{p-1}.$ (2.41)

Clearly, $\varphi_{\tau}(A_{*})\leq\varphi_{\tau}(1)={\rm
cap}_{p}(\bar{B}(0,1))$ for all $\tau\in[0,\infty]$.

######  Remark 2.2.

Let $\eta:{\mathbb{R}}^{d}\to[0,1]$ be a $C^{\infty}$ cutoff function such
that $\eta\equiv 1$ on $B(0,1)$ and $\eta\equiv 0$ outside $B(0,2)$. Then for
$R>2$, we have that $V_{R}^{s}-\eta\in L_{0}^{1,p}(B(0,R)\setminus K_{s})$ for
each $s\in S$. Thus, for every anchor $s$,

$V_{\frac{1}{10\alpha}}^{s}\left(\frac{x-s}{\alpha\varepsilon}\right)-\eta\
\left(\frac{x-s}{\alpha\varepsilon}\right)\in
L_{0}^{1,p}\Bigl{(}B(s,\varepsilon/10)\setminus(s+\alpha\varepsilon
K_{s})\Bigr{)}\,.$

Consequently, for all $A\in[0,1]$, the ansatz function $u_{A}$ given by (1.35)
belongs to the admissible class ${\cal A}(\Gamma,\mathbb{R}^{d},\phi)$ with

$\phi(x):=A\eta\Bigl{(}\frac{x}{1+\varepsilon}\Bigr{)}+(1-A)\sum_{s\in
S}\eta\Bigl{(}\frac{x-s}{\alpha\varepsilon}\Bigr{)}\,.$

By (2.12), this implies that $E(u)\leq E(u_{A}).$

######  Corollary 2.1.

For every $y$ on the unit sphere, $\delta>0$ and $\tau\in[0,\infty]$, we have

$\displaystyle
E(u_{A},\Lambda_{\delta}(y))\to\mu(Q_{\delta})\varphi_{\tau}(A)\quad\mbox{as}\quad\varepsilon\to
0\,,$ (2.42)

and the convergence is uniform in $A\in[0,1]$ and in $y\in{\mathbb{S}}^{d-1}$,
provided that $\tau<\infty$. Moreover, for every $\delta\in(0,1/2)$ and
sufficiently small $\varepsilon$ (that may depend on $\delta$),

$\displaystyle\forall A\in[0,1],\qquad
E(u_{A},\Lambda_{\delta}(y))\geq(1-\delta)^{p}E(u_{A^{*}},\Lambda_{\delta}(y))\;\;\mbox{\rm
for}\;\;y\in{\mathbb{S}}^{d-1}\,.$ (2.43)

###### Proof.

By the definition of $u_{A}$ and (2.18), we have

$\displaystyle(1-A)^{p}\sum_{s\in S\cap
Q_{\delta-\varepsilon}(y)}\kappa_{s}\leq
E(u_{A},\Lambda_{\delta}(y))-A^{p}(1+\varepsilon)^{d-p}E(U,\Lambda_{\delta}(y))\leq(1-A)^{p}\sum_{s\in
S\cap Q_{\delta+\varepsilon}(y)}\kappa_{s}\,,$ (2.44)

where $\kappa_{s}$ was defined in (2.34). Since $U$ is radial,

$E(U,\Lambda_{\delta}(y))=\mu(Q_{\delta}){\rm
cap}_{p}\bigl{(}\bar{B}(0,1)\bigr{)}\,.$

The rest of the proof of (2.42) proceeds exactly like the proof of (2.39),
since the left-hand and right-hand sides of (2.44) have the same asymptotics:

$\varepsilon^{d-1}|S\cap
Q_{\delta\pm\varepsilon}(y)|\to\sigma\mu(Q_{\delta})\quad\mbox{as}\quad\varepsilon\to
0\,.$

It remains to verify (2.43). If $0<\tau<\infty$, then this follows from the
uniform convergence (in the parameter $A$) in (2.42). If $\tau=0$, then
$A_{*}=0$. In this case, (2.43) is obvious if $A\leq\delta$. On the other
hand, if $\tau=0$ and $A>\delta$, then the left-hand side of (2.43) is at
least $\delta^{p}E(U,\Lambda_{\delta}(y))>0$ which does not depend on
$\varepsilon$, while the right-hand side of (2.43) tends to $0$ as
$\varepsilon\downarrow 0$; Thus, (2.43) holds in this case as well provided
$\varepsilon$ is small enough.

Finally, if $\tau=\infty$, then $A_{*}=1$ and $u_{A^{*}}=U_{1+\varepsilon}$;
in this case, (2.43) is obvious if $A>1-\delta$, while the lower bound

$E(u_{A},\Lambda_{\delta}(y))\geq(1-A)^{p}\sum_{s\in S\cap
Q_{\delta-\varepsilon}(y)}\kappa_{s}\,,$

implies that

$\lim_{\varepsilon\to
0}\min_{A\in[0,1-\delta]}E(u_{A},\Lambda_{\delta}(y))=\infty\,,$

so (2.43) holds if $\tau=\infty$ and $A\leq 1-\delta$ as well. ∎

##  3 Bounding oscillation and energy of $u$ in cones

In this section, the positive constants $C,C_{1},C_{2},\ldots$ depend only on
$d,p$.

######  Lemma 3.1.

Suppose that for some $r>0$ and $z\in{\mathbb{R}}^{d}$ we have

$\displaystyle B(z,r)\subset B(z,5r/4)\subset{\mathbb{R}}^{d}\setminus\Gamma$
(3.1)

and for some $\lambda>0,\beta\geq 0$, the solution $u$ of (1.5) satisfies

$\displaystyle E(u,B(z,5r/4))\leq\lambda^{p}r^{d-1-\beta}\,.$ (3.2)

Then,

$\displaystyle\underset{B(z,r)}{\rm osc}~{}u\leq C_{1}\lambda
r^{\frac{p-1-\beta}{p}}\,,$ (3.3)

where

$\displaystyle\underset{D}{\rm osc}~{}u:=\sup_{D}u-\inf_{D}u$ (3.4)

stands for the oscillation over the set $D$.

###### Proof.

By Poincaré’s inequality, there exists a real $t$ such that

$\displaystyle J_{p}=\int_{B(z,5r/4)}\left|u-t\right|^{p}\leq
C_{2}r^{p}\int_{B(z,5r/4)}|\nabla u|^{p}=C_{2}r^{p}E(u,B(z,5r/4))\,.$ (3.5)

The hypothesis (3.2) and the inequality (3.5) give

$\displaystyle J_{p}\leq C_{2}\lambda^{p}r^{p+d-1-\beta}\,.$ (3.6)

Let

$\displaystyle u_{1}(x)=\max\\{u(x)-t,0\\},\qquad
u_{2}(x)=\max\\{t-u(x),0\\}.$ (3.7)

By [Lindq, Lemma 3.6] we have

$\displaystyle\left\|u_{i}\right\|_{L^{\infty}(B(z,r))}\leq
C_{3}\left(\frac{J_{p}}{r^{d}}\right)^{\frac{1}{p}}\leq C_{4}\lambda
r^{\frac{p-1-\beta}{p}},\qquad i=1,2,$ (3.8)

which implies (3.3) with $C_{1}=2C_{4}$. ∎

######  Definition 3.1.

Fix $\beta=\frac{p-1}{2p}$. Given $\delta<1/20$ and $\varepsilon<\delta/20$,
an anchor $s\in S$ will be called a good anchor if

$\displaystyle
E(u,\Lambda_{\zeta}(s))\leq\zeta^{d-1-\beta},\qquad\forall\zeta\in[\varepsilon,\delta]\,.$
(3.9)

Otherwise, $s$ will be called a bad anchor.

Figure 4: Geometric objects used in the proof of Lemma 3.2: The top figure
indicates the point $x_{0}$ and the line segment $L(x_{0})$ in the cored wedge
$\Lambda_{\varepsilon}^{*}(s)$ defined by (3.10). The bottom figure shows the
points $x_{k}$ and $\xi$ on $L(x_{0})$ and the corresponding overlapping balls
centered at these points.

######  Lemma 3.2.

Fix $\varepsilon,\delta,\beta$ as in the definition above. Suppose $s$ is a
good anchor and let

$\displaystyle\Lambda_{\varepsilon}^{*}(s):=\Bigl{\\{}x\in\Lambda_{\varepsilon/2}(s)\setminus
B\left(s,\frac{\varepsilon}{10}\right):1-\delta\leq|x|\leq
1+\delta\Bigr{\\}}\,.$ (3.10)

Then there exists $C_{5}$, such that

$\displaystyle\underset{\Lambda_{\varepsilon}^{*}(s)}{\rm osc}~{}u\leq
C_{5}\delta^{\frac{p-1-\beta}{p}}\,.$ (3.11)

###### Proof.

Take $x_{0}\in\Lambda_{\varepsilon}^{*}(s)$ and connect $x_{0}$ radially by a
line segment $L(x_{0})$ to the cap
$\Lambda_{\varepsilon}^{*}(s)\cap\\{|x|=1+\delta\\}$ or the cap
$\Lambda_{\varepsilon}^{*}(s)\cap\\{|x|=1-\delta\\}$, depending on whether
$|x_{0}|\geq 1$ or not. (Observe that the oscillation of $u$ on each of these
caps is at most $C_{1}\delta^{\frac{p-1-\beta}{p}}$ by Lemma 3.1.) Assume
first that $1\leq|x_{0}|\leq 1+\delta/4.$ We then define a sequence of points
$\\{x_{k}\\}_{k\geq 0}$ along $L(x_{0})$ and corresponding radii $r_{k}$ as
follows (See Figure 4):

For $k=0,1,\ldots,20$, let $|x_{k}|=|x_{0}|+k\varepsilon/20$ and
$r_{k}=\varepsilon/20$. For $k>20$, let $r_{k}=2^{k-22}\varepsilon/5$ and
$|x_{k}|=|x_{k-1}|+r_{k}$. Denote by $k_{*}$ the last $k$ such that
$|x_{k}|\leq 1+3\delta/4.$ Then for all $k\in[0,k_{*}]$, we have
$|x_{k}-s|\geq 3r_{k}/2$ and $3r_{k}/2\leq|x_{k_{*}}|-1\leq 3\delta/4$, whence
$r_{k}\leq\delta/2$. Note that for $k\leq k_{*}$, we have

$B(x_{k},5r_{k}/4)\subset\Lambda_{\frac{3r_{k}}{2}+2\varepsilon}(s)\setminus\Gamma\,.$

We also have $|x_{k_{*}}|>1+r_{k_{*}}$ and
$1+3\delta/4\leq|x_{k_{*}+1}|=|x_{k_{*}}|+2r_{k_{*}}$, so

$|x_{k_{*}}|-1+r_{k_{*}}>\bigl{(}3|x_{k_{*}}|-3+3r_{k_{*}}-(|x_{k_{*}}|-1-r_{k_{*}})\bigr{)}/3=2(|x_{k_{*}}|+2r_{k_{*}}-1)/3\geq\delta/2\,.$

Let $\xi$ be the point on $L(x_{0})$ that satisfies $|\xi|=1+3\delta/4$.
Observe that $B(\xi,5\delta/{16})\subset\Lambda_{\delta}(s)\setminus\Gamma$.
Applying the preceding lemma to the overlapping balls
$\\{B(x_{k},r_{k})\\}_{k\leq k_{*}}$ and $B(\xi,\delta/4)$ yields, for some
$C_{6}$, that

$\displaystyle\forall k\leq k_{*}\,,\qquad\underset{B(x_{k},r_{k})}{\rm
osc}~{}u\leq
C_{6}r_{k}^{\frac{p-1-\beta}{p}}\,,\qquad\underset{B(\xi,\delta/4)}{\rm
osc}~{}u\leq C_{6}\delta^{\frac{p-1-\beta}{p}}\,.$ (3.12)

Summing the resulting (almost geometric) series concludes the proof for the
oscillation over the line segment $L(x_{0})$. Considering the union of these
line segments over all
$\\{x_{0}\in\Lambda_{\varepsilon}^{*}(s):1\leq|x_{0}|\leq 1+\delta/4\\}$
allows us to bound the oscillation over
$\\{x\in\Lambda_{\varepsilon}^{*}(s):|x|\geq 1\\}$. A similar argument applies
to $\\{x\in\Lambda_{\varepsilon}^{*}(s):|x|\leq 1\\}$. Since these two sets
intersect on the unit sphere, the proof is complete. ∎

######  Lemma 3.3.

Fix $\varepsilon$, $\delta,$ $\beta$ as in Definition 3.1, and let $y$ be a
point on the unit sphere ${\mathbb{S}}^{d-1}$. Suppose that

$E(u,\Lambda_{2\delta}(y))\leq M\delta^{d-1},$ (3.13)

for some $M>0$. Then the set $S_{\beta}$ of bad anchors satisfies
$|S_{\beta}\cap Q_{\delta}(y)|\leq
C_{7}M\delta^{\beta}(\delta/\varepsilon)^{d-1}$, where $C_{7}=C_{7}(d,p)$.

The proof is a variant of the classical proof of the Hardy-Littlewood maximal
inequality.

###### Proof.

By the definition of bad anchors, for each $s\in S_{\beta}\cap Q_{\delta}(y)$
there is a $\zeta=\zeta(s)\in[\varepsilon,\delta]$ such that

$\displaystyle
E(u,\Lambda_{\zeta}(s))>\zeta^{d-1-\beta}\geq\delta^{-\beta}\zeta^{d-1}.$
(3.14)

The caps $Q_{\zeta(s)}(s)$ with $s\in S_{\beta}\cap Q_{\delta}(y)$ form a
Besicovitch covering of $S_{\beta}\cap Q_{\delta}(y)$. By the Besicovitch
Covering Theorem, we can extract a finite subcover
$\\{Q_{\zeta_{j}}(s_{j})\\}_{j=1}^{N}$, where $\zeta_{j}=\zeta(s_{j})$ for
each $j$, so that every point on the unit sphere belongs to at most
$C_{8}=C_{8}(d)$ elements of this subcover. Therefore,

$\displaystyle\sum_{j=1}^{N}{\bf 1}_{\Lambda_{\zeta_{j}}(s_{j})}\leq C_{8}{\bf
1}_{\Lambda_{2\delta}(y)}\,.$ (3.15)

For each $s\in S_{\beta}\cap Q_{\delta}(y)$, there is some $j\leq N$ such that
$s\in Q_{\zeta_{j}}(s_{j})$, whence the cap $Q_{\varepsilon/2}(s)$ is
contained in $Q_{2\zeta_{j}}(s_{j})$. Since the $\varepsilon/2$ neighborhoods
$Q_{\varepsilon/2}(s)$ of anchors are disjoint, for some $C_{9}>0$ we have

$\displaystyle C_{9}|S_{\beta}\cap
Q_{\delta}(y)|\varepsilon^{d-1}\leq\mu\Bigl{[}\bigcup\\{Q_{\varepsilon/2}(s)\,:\,s\in
S_{\beta}\cap
Q_{\delta}(y)\\}\Bigr{]}\leq\mu\Bigl{[}\bigcup_{j=1}^{N}Q_{2\zeta_{j}}(s_{j})\Bigr{]}\leq\sum_{j=1}^{N}\mu[Q_{2\zeta_{j}}(s_{j})]\,.$
(3.16)

Now by (3.14), for each $j$ we have

$\displaystyle\mu[Q_{2\zeta_{j}}(s_{j})]\leq C_{10}\zeta_{j}^{d-1}\leq
C_{10}\delta^{\beta}E(u,\Lambda_{\zeta_{j}}(s_{j}))\,.$ (3.17)

Thus by (3.16) and (3.17),

$\displaystyle|S_{\beta}\cap
Q_{\delta}(y)|\varepsilon^{d-1}\leq\frac{C_{10}}{C_{9}}\delta^{\beta}\sum_{j=1}^{N}E(u,\Lambda_{\zeta_{j}}(s_{j}))=C_{11}\delta^{\beta}\int_{{\mathbb{R}}^{d}}\Bigl{(}|\nabla
u|^{p}\sum_{j=1}^{N}{\bf 1}_{\Lambda_{\zeta_{j}}(s_{j})}\Bigr{)}\leq
C_{11}\delta^{\beta}C_{8}E(u,\Lambda_{2\delta}(y))\,,$ (3.18)

where we have used (3.15) in the last step. Combining (3.13) and (3.18)
concludes the proof. ∎

######  Lemma 3.4.

Let $\beta:=\frac{p-1}{2p}$. Then there exist $\delta_{0}>0$ (which may depend
on $d,p,\sigma$) and $C_{12},C_{13}>0$ (which depend on $d,p$) such that for
$\delta\in(0,\delta_{0})$, there exists $\varepsilon_{0}>0$ with the following
property. For each $\varepsilon\in(0,\varepsilon_{0})$ and
$y\in\mathbb{S}^{d-1}$ such that $u=u^{\varepsilon}$ satisfies

$\displaystyle E(u,\Lambda_{2\delta}(y))\leq\delta^{-\beta/2}\delta^{d-1}\,,$
(3.19)

there exists some $A=A(y,\varepsilon)$, such that

$\displaystyle|u(x)-A|\leq C_{12}\delta^{\beta},\quad\forall
x\in\Lambda_{\delta}(y):|x|=1\pm\delta\,,$ (3.20)

and

$\displaystyle
E(u,\Lambda_{\delta}(y))\geq(1-2\delta^{\beta/3})E(u_{A},\Lambda_{\delta}(y))\,.$
(3.21)

###### Proof.

Since $\beta<(p-1)/2$, we have $(p-1-\beta)/p>(p-1)/2p=\beta$. We will use
this repeatedly below. Also, recall the following consequence of the
definition (1.35) of $u_{A}$ and (2.18), that was already noted in (2.44):

$\displaystyle E(u_{A},\Lambda_{\delta}(y))\leq
A^{p}(1+\varepsilon)^{d-p}E(U,\Lambda_{\delta}(y))+(1-A)^{p}\sum_{s\in S\cap
Q_{\delta+\varepsilon}(y)}\kappa_{s}\,,$ (3.22)

where

$\displaystyle\kappa_{s}:={\rm cap}_{p}\left(\alpha\varepsilon
K_{s},B\left(s,\frac{\varepsilon}{10}\right)\right)\,.$ (3.23)

First observe that under the assumption (3.19), Lemma 3.1 gives that

$\displaystyle\underset{\Lambda_{\delta}(y)\cap\partial B(0,1+\delta)}{\rm
osc}~{}u\leq C_{1}\delta^{\frac{p-1-\beta}{p}}\leq C_{1}\delta^{\beta}\,.$
(3.24)

Similarly,

$\displaystyle\underset{\Lambda_{\delta}(y)\cap\partial B(0,1-\delta)}{\rm
osc}~{}u\leq C_{1}\delta^{\beta}\,.$ (3.25)

Hence,

$\displaystyle u(x)=A_{\pm}+O(\delta^{\beta}),\quad\forall
x\in\Lambda_{\delta}(y):|x|=1\pm\delta,$ (3.26)

where $A_{+}=A_{+}(y)$ and $A_{-}=A_{-}(y)$ are two constants. In view of
Lemma 3.3, if $\delta_{0}>0$ is small enough and $\delta\in(0,\delta_{0})$,
then for sufficiently small $\varepsilon>0$, there exists a good anchor in
$Q_{\delta}(y)$, and therefore by Lemma 3.2 and (3.26), we have
$|A_{+}-A_{-}|\leq C_{13}\delta^{\beta}.$ Thus

$\displaystyle|u(x)-A_{+}|\leq C_{14}\delta^{\beta},\quad\forall
x\in\Lambda_{\delta}(y):|x|=1\pm\delta\,.$ (3.27)

Denote by $Q_{\delta-\varepsilon}^{\sharp}(y)=(S\setminus S_{\beta})\cap
Q_{\delta-\varepsilon}(y)$ the set of good anchors in
$Q_{\delta-\varepsilon}(y).$ Lemma 3.2 and (3.20) imply that there exists
$C_{15}$ such that for each $s\in Q_{\delta-\varepsilon}^{\sharp}(y)$,

$\displaystyle u(x)\leq
A_{1}:=\min\bigl{\\{}1,A_{+}+C_{15}\delta^{\beta}\bigr{\\}}\,,\quad\forall
x\in\partial B\left(s,\frac{\varepsilon}{10}\right)\,.$ (3.28)

Define

$A:=A_{1}\;\mbox{ \rm if }\;A_{+}>1/2\;\mbox{ \rm and
}\;A:=\max\\{0,A_{+}-C_{14}\delta^{\beta}\\}\;\mbox{ \rm if }\;A_{+}\leq
1/2\,.$

We assume that $\delta$ is small enough to ensure that
$C_{15}\delta^{\beta}<1/4$, so $A=1$ iff $A_{1}=1$. Also, note that $A_{1}\leq
A+(C_{14}+C_{15})\delta^{\beta}$, so
$1-A_{1}\geq(1-A)\bigl{(}1-2(C_{14}+C_{15})\delta^{\beta}\bigr{)}$ if
$A_{+}\leq 1/2$. Thus the inequality

$\displaystyle(1-A_{1})^{p}\geq(1-\delta^{\beta/3})(1-A)^{p}$ (3.29)

holds for all values of $A_{+}$, provided that $\delta$ is small enough.

Now that $A$ has been chosen, our next goal will be to prove that if $\delta$
is small enough, then the inequality

$\displaystyle E\left(u,\Lambda_{\delta}(y)\cap
B(0,1+\delta)\right)\geq(1-2\delta^{\beta/3})(1-A)^{p}\sum_{s\in S\cap
Q_{\delta+\varepsilon}(y)}\kappa_{s}$ (3.30)

holds for sufficiently small $\varepsilon$. Note that if $A_{1}=1$ then also
$A=1$, so the right-hand side of (3.30) vanishes, and the inequality certainly
holds; thus we need only prove (3.30) when $A_{1}<1$.

Given $s\in Q_{\delta-\varepsilon}^{\sharp}(y)$, fix a $C^{\infty}$ cutoff
function $\phi_{s}:B\left(s,\frac{\varepsilon}{10}\right)\to[0,1]$ that is
identically 1 in $B(s,\alpha\varepsilon)$ and vanishes outside
$B(s,2\alpha\varepsilon)$. Since $A_{1}<1$ and (3.28) holds, the function
$u_{s}:B(s,\varepsilon/{10})\to[0,1]$ given by

$u_{s}(x):=\max\Bigl{\\{}0,\frac{u(x)-A_{1}}{1-A_{1}}\Bigr{\\}}\,,$

is in the admissible class

${\mathscr{A}}(s+\alpha\varepsilon K_{s},\,B(s,\varepsilon/{10}),\,\phi_{s})$

defined in (2.11), by basic properties of $H_{0}^{1,p}$ (See [hkm, Lemmas 1.23
and 1.26]). Thus, by (2.12),

$\forall s\in Q_{\delta-\varepsilon}^{\sharp}(y),\qquad\kappa_{s}\leq
E\Bigl{(}u_{s},\,B\left(s,\frac{\varepsilon}{10}\right)\Bigr{)}\leq(1-A_{1})^{-p}E\Bigl{(}u,B\left(s,\frac{\varepsilon}{10}\right)\Bigr{)}\,.$

Therefore,

$\displaystyle E\left(u,\Lambda_{\delta}(y)\cap
B(0,1+\delta)\right)\geq\sum_{s\in
Q_{\delta-\varepsilon}^{\sharp}(y)}E\left(u,B\left(s,\frac{\varepsilon}{10}\right)\right)\geq(1-A_{1})^{p}\sum_{s\in
Q_{\delta-\varepsilon}^{\sharp}(y)}\kappa_{s}\,.$ (3.31)

Next, we will compare the right-hand-sides of (3.31) and (3.30). The
equidistribution hypothesis $({\rm H_{1}})$ implies that for $\varepsilon$
sufficiently small,

$\displaystyle|S\cap
Q_{\delta}(y)|\geq(1-\delta)\sigma\varepsilon^{1-d}\mu(Q_{\delta})\geq
C_{17}\sigma(\delta/\varepsilon)^{d-1}\,.$ (3.32)

Invoking Lemma 3.3 with $M=\delta^{-\beta/2}$ yields that

$\displaystyle|S_{\beta}\cap Q_{\delta}(y)|\leq
C_{7}\delta^{\beta/2}(\delta/\varepsilon)^{d-1}\,.$ (3.33)

The $\mu$-measure of the shell $Q_{\delta+2\varepsilon}(y)\setminus
Q_{\delta-2\varepsilon}(y)$ is $O(\varepsilon\delta^{d-2})$, so the number of
anchors in the smaller shell $Q_{\delta+\varepsilon}(y)\setminus
Q_{\delta-\varepsilon}(y)$ is

$O(\delta/\varepsilon)^{d-2}\leq
C_{7}\delta^{\beta/2}(\delta/\varepsilon)^{d-1}$

(since caps of radius $\varepsilon/2$ around these anchors are pairwise
disjoint, and contained in the larger shell.) Thus for small enough
$\varepsilon$,

$\displaystyle\Bigl{|}\Bigl{(}S\cap Q_{\delta+\varepsilon}(y)\Bigr{)}\setminus
Q_{\delta-\varepsilon}^{\sharp}(y)\Bigr{|}$ $\displaystyle\leq$
$\displaystyle\Bigl{|}S_{\beta}\cap
Q_{\delta}(y)\Bigr{|}+\Bigl{|}S\cap\bigl{(}Q_{\delta+\varepsilon}(y)\setminus
Q_{\delta-\varepsilon}(y)\bigr{)}\Bigr{|}$ $\displaystyle\leq$ $\displaystyle
2C_{7}\delta^{\beta/2}(\delta/\varepsilon)^{d-1}\leq\frac{C_{17}}{2}\sigma\delta^{\beta/3}(\delta/\varepsilon)^{d-1}\,,$

where the rightmost inequality assumes that $\delta$ is small enough so that
$2C_{7}\delta^{\beta/6}\leq\frac{C_{17}}{2}\sigma$.

Lemma 2.2 (see also (2.35)) implies that if $\varepsilon>0$ is small enough,
then for every anchor $s$,

$\displaystyle{\rm cap}_{p}(\alpha\varepsilon K)\leq\kappa_{s}\leq 2{\rm
cap}_{p}(\alpha\varepsilon K)\,.$ (3.35)

By comparing (3) to (3.32), and using (3.35), we obtain that

$\displaystyle\sum\Bigl{\\{}\kappa_{s}\,:\,s\in\Bigl{(}S\cap
Q_{\delta+\varepsilon}(y)\Bigr{)}\setminus
Q_{\delta-\varepsilon}^{\sharp}(y)\Bigr{\\}}\leq\delta^{\beta/3}\sum\Bigl{\\{}\kappa_{s}\,:\,s\in
S\cap Q_{\delta+\varepsilon}(y)\Bigr{\\}}\,,$ (3.36)

or equivalently,

$\displaystyle\sum\Bigl{\\{}\kappa_{s}\,:\,s\in
Q_{\delta-\varepsilon}^{\sharp}(y)\Bigr{\\}}\geq\Bigl{(}1-\delta^{\beta/3}\Bigr{)}\sum\Bigl{\\{}\kappa_{s}\,:\,s\in
S\cap Q_{\delta+\varepsilon}(y)\Bigr{\\}}\,.$ (3.37)

Combining this inequality with (3.31) and (3.29), we have established that
(3.30) holds.

Recall from (1.5) that $u(x)\to 0$ as $|x|\to\infty$ and recall that
$U_{R}(x)=U(x/R)$ is identically 1 for $x\in B(0,R)$. By considering
separately the two cases $A_{+}>1/2$ and $A_{+}\leq 1/2$, we infer from (3.27)
that

$\forall x\in\Lambda_{\delta}(y)\cap\partial B(0,1+\delta),\qquad
u(x)\geq\tilde{A}:=(1-C_{20}\delta^{\beta})A\,.$

Thus, Lemma 2.3 implies that

$\displaystyle E\left(u,\\{x\in\Lambda_{\delta}(y):~{}|x|\geq
1+\delta\\}\right)\geq\tilde{A}^{p}E(U_{1+\delta},\Lambda_{\delta}(y))=(1+\delta)^{d-p}\tilde{A}^{p}E(U,\Lambda_{\delta}(y))\,.$
(3.38)

For sufficiently small $\delta$, we have $(1-C_{20}\delta^{\beta})^{p}\geq
1-2\delta^{\beta/3}$, so if $\epsilon$ is small enough, then

$\displaystyle E\left(u,\\{x\in\Lambda_{\delta}(y):~{}|x|\geq
1+\delta\\}\right)\geq(1-\delta^{\beta/3})A^{p}E(U,\Lambda_{\delta}(y))\,.$
(3.39)

Combining (3.30) and (3.39), we obtain by (3.22) that (3.21) holds. This
completes the proof.

∎

##  4 Asymptotics for $u^{\varepsilon}$ in the bulk: Proof of Theorem 1.1

In this section, we first use the lower bound on the energy of $u$ in cones,
obtained in Lemma 3.4, in conjunction with $u$ minimizing energy globally, to
deduce an upper bound for the energy of $u$ in all cones. This will imply that
$u$ is close to $u_{A^{*}}$ on $\partial B(0,1+\delta)$, from which Theorem
1.1 will follow easily.

######  Lemma 4.1.

Let $\beta=\frac{p-1}{2p}$ as in Definition 3.1. There exists $\delta_{0}>0$
(that may depend on $d,p,\sigma$) and $C_{21}>0$ (that may depend on $d,p$)
such that for all $\delta\in(0,\delta_{0}/2)$, points $z\in\mathbb{S}^{d-1}$
and $m>1$, we have

$\displaystyle E(u,\Lambda_{\delta}(z))\leq
E(u_{A_{*}},\Lambda_{\delta}(z))+C_{21}{\delta^{m\beta/3}}\,,$ (4.1)

provided that $\varepsilon$ is sufficiently small. Consequently, there exists
$\theta=\theta(d,p)>0$ such that for $\delta\in(0,\delta_{0}/2)$ and
$\varepsilon$ sufficiently small,

$\displaystyle|u(x)-A_{*}|\leq C_{\\#}\delta^{\theta}\;\;\mbox{\rm for
all}\;\;x\in{\mathbb{R}}^{d}\;\;\mbox{\rm such that}\;\;|x|=1\pm\delta\,,$
(4.2)

where $C_{\\#}$ does not depend on $\varepsilon,\delta$ (but may depend on
$d,p,\tau,\sigma,{\rm cap}_{p}(K)$).

As in the previous section, the constants $C_{i}$ in the proof only depend on
$d,p$.

###### Proof.

Let $\Omega={\mathbb{R}}^{d}\setminus\Gamma$. Observe that for
$x\in{\mathbb{R}}^{d}$ and $y\in{\mathbb{S}}^{d-1}$, we have

$x\in\Lambda_{2\delta}(y)\Leftrightarrow y\in Q_{2\delta}(x/|x|)\,.$

Therefore, by Fubini,

$\displaystyle\underset{{\mathbb{S}}^{d-1}}{\int}E(u,\Lambda_{2\delta}(y))\,d\mu(y)=\underset{{\mathbb{S}}^{d-1}}{\int}\,\,\underset{\Omega\cap\Lambda_{2\delta}(y)}{\int}|\nabla
u(x)|^{p}\,dx\,d\mu(y)=\underset{\Omega}{\int}\underset{Q_{2\delta}(x/|x|)}{\int}\\!\\!\\!|\nabla
u(x)|^{p}\,d\mu(y)\,dx=\mu(Q_{2\delta})E(u)\,.$ (4.3)

Define

$\displaystyle
Y_{\delta}=\Bigl{\\{}y\in{\mathbb{S}}^{d-1}\,:\,E(u,\Lambda_{2\delta}(y))>\delta^{-\beta/2}\delta^{d-1}\Bigr{\\}}\,.$
(4.4)

Since $E(u)={\rm cap}_{p}(\Gamma)\leq{\rm cap}_{p}(\bar{B}(0,2))=C_{22}$ and
$\mu(Q_{2\delta})=O(\delta^{d-1})$, equation (4.3) implies that

$\displaystyle\mu(Y_{\delta})\leq C_{23}\delta^{\beta/2}\,.$ (4.5)

Denote $\chi_{m}(y)=E(u_{A_{*}},\Lambda_{\delta^{m}}(y))$ and observe that if
$\delta^{m}<\delta_{0}(d,p,\sigma)$, then Lemma 3.4 and (2.43) (applied to
$\delta^{m}$ in place of $\delta$) imply that for sufficiently small
$\varepsilon$,

$\displaystyle
E(u,\Lambda_{\delta^{m}}(y))\geq(1-2\delta^{m\beta/3})\chi_{m}(y)\;\;\mbox{\rm
for}\;\;y\in{\mathbb{S}}^{d-1}\setminus Y_{\delta^{m}}\,.$ (4.6)

Next, note that for $z\in{\mathbb{S}}^{d-1}$ and
$y\in{\mathbb{S}}^{d-1}\setminus Q_{\delta+\delta^{m}}(z)$, we have

$x\in\Lambda_{\delta^{m}}(y)\,\Rightarrow\,\Bigl{\\{}x\in{\mathbb{R}}^{d}\setminus\Lambda_{\delta}(z)\;\;{\rm
and}\;\;y\in Q_{\delta^{m}}(x/|x|)\Bigr{\\}}\,.$

Therefore, by Fubini

$\displaystyle\\!\\!\\!\\!\underset{{\mathbb{S}}^{d-1}\setminus
Q_{\delta+\delta^{m}}(z)}{\int}\\!\\!\\!\\!\\!\\!\\!\\!\\!E(u,\Lambda_{\delta^{m}}(y))\,d\mu(y)=\\!\\!\\!\\!\\!\underset{{\mathbb{S}}^{d-1}\setminus
Q_{\delta+\delta^{m}}(z)}{\int}\\!\\!\\!\\!\int_{\Omega\cap\Lambda_{\delta^{m}}(y)}\\!\\!|\nabla
u(x)|^{p}\,dx\,d\mu(y)\leq\\!\\!\\!\\!\\!\\!\\!\\!\underset{\Omega\setminus\Lambda_{\delta}(z)}{\int}\\!\\!\\!\\!\int_{Q_{\delta^{m}}(x/|x|)}\\!\\!\\!|\nabla
u(x)|^{p}\,d\mu(y)\,dx\,.$ (4.7)

The right-hand side factors as
$\mu(Q_{\delta^{m}})E(u,\Omega\setminus\Lambda_{\delta}(z))$, so reversing the
order of expressions gives

$\displaystyle\mu(Q_{\delta^{m}})E(u,\Omega\setminus\Lambda_{\delta}(z))\geq\\!\\!\underset{{\mathbb{S}}^{d-1}\setminus
Q_{\delta+\delta^{m}}(z)}{\int}E(u,\Lambda_{\delta^{m}}(y))\,d\mu(y)\geq\\!\\!\underset{{\mathbb{S}}^{d-1}\setminus[Q_{\delta+\delta^{m}}(z)\cup
Y_{\delta^{m}}]}{\int}(1-2\delta^{m\beta/3})\chi_{m}(y)\,d\mu(y)\,,$ (4.8)

where we have used (2.43) in the last step. By Fubini,

$\displaystyle\int_{{\mathbb{S}}^{d-1}}\chi_{m}(y)\,d\mu(y)=\mu(Q_{\delta^{m}})E(u_{A_{*}})\,$
(4.9)

and

$\displaystyle\int_{Q_{\delta-\delta^{m}}(z)}\chi_{m}(y)\,d\mu(y)\leq\mu(Q_{\delta^{m}})E(u_{A_{*}},\Lambda_{\delta}(z))\,.$
(4.10)

Write $Y^{+}=Y_{\delta^{m}}\cup\Bigl{(}Q_{\delta+\delta^{m}}(z)\setminus
Q_{\delta-\delta^{m}}(z)\Bigr{)}$, so that for small $\delta$,

$\displaystyle\mu(Y^{+})\leq C_{23}\delta^{m\beta/2}+C_{24}\delta^{d-2+m}\leq
2C_{23}\delta^{m\beta/2}\,.$ (4.11)

Now (2.42) implies the bound $\chi_{m}(y)\leq C_{25}\mu(Q_{\delta^{m}})$ for
$\varepsilon$ small enough; together with (4.11), it gives

$\displaystyle\int_{Y^{+}}\chi_{m}(y)\,d\mu(y)\leq
C_{26}\mu(Q_{\delta^{m}})\delta^{m\beta/2}\,.$ (4.12)

Subtracting (4.10) and (4.12) from (4.9) yields

$\displaystyle\underset{{\mathbb{S}}^{d-1}\setminus[Q_{\delta+\delta^{m}}(z)\cup
Y_{\delta^{m}}]}{\int}\chi_{m}(y)\,d\mu(y)\geq\mu(Q_{\delta^{m}})\Bigl{[}E(u_{A_{*}},{\mathbb{R}}^{d}\setminus\Lambda_{\delta}(z))-C_{26}\delta^{m\beta/2}\Bigr{]}\,,$
(4.13)

whence by (4.8),

$\displaystyle
E(u,\Omega\setminus\Lambda_{\delta}(z))\geq(1-2\delta^{m\beta/3})\Bigl{[}E(u_{A_{*}},{\mathbb{R}}^{d}\setminus\Lambda_{\delta}(z))-C_{26}\delta^{m\beta/2}\Bigr{]}\,.$
(4.14)

Thus,

$\displaystyle E(u_{A_{*}})\geq E(u)$ $\displaystyle=$ $\displaystyle
E(u,\Lambda_{\delta}(z))+E(u,\Omega\setminus\Lambda_{\delta}(z))$
$\displaystyle\geq$ $\displaystyle
E(u,\Lambda_{\delta}(z))+(1-2\delta^{m\beta/3})\Bigl{[}E(u_{A_{*}},\Omega\setminus\Lambda_{\delta}(z))-C_{26}\delta^{m\beta/2}\Bigr{]}\,.$

Rearranging terms and using that there exists $C_{27}$ such that if
$\varepsilon$ is small, then $E(u_{A_{*}})\leq C_{27}$, we conclude that

$\displaystyle E(u,\Lambda_{\delta}(z))\leq
E(u_{A_{*}},\Lambda_{\delta}(z))+2C_{27}\delta^{m\beta/3}\,,$ (4.16)

provided $\delta<\delta_{0}$ and $\varepsilon<\varepsilon_{0}(\delta)$ are
small enough. Applying this inequality with $2\delta$ in place of $\delta$,
the hypothesis of Lemma 3.4 is satisfied, provided $m$ is chosen to satisfy
$m\beta/3>d$. Therefore, by (3.20), we have that for sufficiently small
$\varepsilon$,

$\displaystyle|u(x)-A|\leq C_{12}\delta^{\beta}\;\;\mbox{ \rm for
all}\;\;x\in\Lambda_{\delta}(z)\;\;\mbox{\rm such that}\;\;|x|=1\pm\delta\,,$
(4.17)

where $A=A(z,\varepsilon)$. By (3.21) and (4.16),

$\displaystyle(1-2\delta^{\beta/3})E(u_{A},\Lambda_{\delta}(z))\,\leq\,E(u_{A_{*}},\Lambda_{\delta}(z))+2C_{27}\delta^{d}\,,$
(4.18)

since $m\beta/3>d$. Now we separate cases.

Case 1. If $\tau\in(0,\infty)$, then (4.18) and Corollary 2.1 yield, for
sufficiently small $\varepsilon>0$, that

$\displaystyle(1-3\delta^{\beta/3})\mu(Q_{\delta})\varphi_{\tau}(A)\leq\mu(Q_{\delta})\varphi_{\tau}(A_{*})+2C_{27}\delta^{d}\,.$
(4.19)

Thus, since $\varphi_{\tau}(A_{*})\leq C$, we infer that

$C_{28}\delta^{\beta/3}\geq\varphi_{\tau}(A)-\varphi_{\tau}(A_{*})\geq
c_{\\#}|A-A_{*}|^{2p}\,,$

where in the right-hand inequality, $c_{\\#}=c_{\\#}(d,p,\tau,\sigma,{\rm
cap}_{p}(K))>0$. Therefore,

$|A-A_{*}|\leq(C_{28}/c_{\\#})^{1/(2p)}\cdot\delta^{\beta/(6p)}\,.$

In conjunction with (4.17), this yields the final claim of the lemma.

Case 2. If $\tau=0$, then $A_{*}=0$, so $E(u_{A_{*}})\to 0$ as $\varepsilon\to
0$ by (2.39). Since

$E(u_{A},\Lambda_{\delta}(z))\geq C_{29}A^{p}\delta^{d-1}$

by (2.29), we infer from (4.18) that $|A-A_{*}|=A=O(\delta^{1/p})$ for small
$\varepsilon$.

Case 3. If $\tau=\infty$, then $A_{*}=1$ and

$\displaystyle
E(u_{A_{*}},\Lambda_{\delta}(z))=E((1+\varepsilon)^{\gamma}U,\Lambda_{\delta}(z))\leq
C_{30}\delta^{d-1}$ (4.20)

by (2.29). On the other hand, for $A<1$, by (2.44) and the definition of
$\kappa_{s}$, for any $\varepsilon>0$ small enough we have

$\displaystyle\frac{E(u_{A},\Lambda_{\delta}(z))}{(1-A)^{p}}\geq\sum_{s\in
S\cap Q_{\delta-\varepsilon}(y)}\kappa_{s}\geq
C_{31}\sigma(\delta/\varepsilon)^{d-1}(\alpha\varepsilon)^{d-p}{\rm
cap}_{p}(K)\geq c_{0}\delta^{d-1}(\alpha/\alpha_{c})^{d-p}$ (4.21)

for some $c_{0}=c_{0}(d,p,K,\sigma)>0$. The right-hand side of (4.21) tends to
$\infty$ as $\varepsilon\downarrow 0$, so by (4.18), we have
$|A-A_{*}|=1-A\leq\delta$ for small enough $\varepsilon$. ∎

###### Proof of Theorem 1.1.

By Lemma 4.1, if $\delta<\delta_{0}(d,p,\sigma)$, then for sufficiently small
$\varepsilon$, we have

$\forall x\in\partial B(0,1+\delta)\cup\partial B(0,1-\delta),\qquad
A_{*}-C_{\\#}\delta^{\theta}\leq u(x)\leq A_{*}+C_{\\#}\delta^{\theta}\,,$

where $\theta,C_{\\#}$ do not depend on $\varepsilon,\delta$. The comparison
principle (Lemma 2.1) then implies that for sufficiently small $\varepsilon$,

$\displaystyle\forall x\in{\mathbb{R}}^{d}\setminus
B(0,1+\delta),\qquad(A_{*}-C_{\\#}\delta^{\theta})U_{1+\delta}(x)\leq
u(x)\leq(A_{*}+C_{\\#}\delta^{\theta})U_{1+\delta}(x)\,.$ (4.22)

and

$\displaystyle\forall x\in B(0,1-\delta),\qquad
A_{*}-C_{\\#}\delta^{\theta}\leq u(x)\leq A_{*}+C_{\\#}\delta^{\theta}\,.$
(4.23)

(For the latter conclusion, the maximum principle suffices.) Since $\delta>0$
can be chosen arbitrarily small, this concludes the proof of uniform
convergence on compact subsets of ${\mathbb{R}}^{d}\setminus\partial B(0,1)$.
∎

##  5 Separation theorem

The oscillation and energy estimates we used in the previous section to prove
Theorem 1.1 are not sufficient to establish Theorem 1.2. For this purpose, we
will need to bound $u$ closer to the cavities, which is the goal of this
section.

Recall that $\gamma=\frac{d-p}{p-1}$. In this section we prove some estimates
that are valid for all parameters $\delta<\min\\{1/2,1/(2\gamma)\\}$ and
$0<\varepsilon<\delta/10$. In particular, we do not assume the asymptotic
equidistribution hypothesis $(H_{1})$ in (1.6), and instead of the asymptotic
relation $(H_{3})$ $\alpha(\varepsilon)\varepsilon^{-1/\gamma}\to\tau$ we just
assume that for some $\tau_{1}\in(0,\infty)$, we have
$\alpha<\min\\{1/80,\tau_{1}\varepsilon^{1/\gamma}\\}\,.$

Let $S\subset\partial B(0,1)$ be a set of anchors, with Euclidean distance at
least $\varepsilon$ between any two anchors. The next theorem ensures that for
$\alpha$ in the critical window and below it, the bumps in the equilibrium
potential are separated.

######  Theorem 5.1.

For $\zeta>0$, write $\Omega_{\zeta}=B(0,1+\delta)\setminus\bigl{[}\cup_{s\in
S}\bar{B}(s,\zeta\varepsilon)\bigr{]}$. Let $w:\bar{B}(0,1+\delta)\to[0,1]$ be
the Perron solution of the boundary value problem

$\displaystyle\left\\{\begin{array}[]{lll}\Delta_{p}w=0&\mbox{in}&\Omega_{\alpha},\\\
w=0&\mbox{on}&\partial B(0,1+\delta),\\\ w=1&\mbox{on}&\cup_{s\in S}\partial
B(s,\alpha\varepsilon)\,.\end{array}\right.$ (5.4)

Then for some $C=C(d,p)$, we have

$\sup_{z\in\Omega_{1/10}}|w(z)|\leq C\tau_{1}^{\gamma}\delta\,.$ (5.5)

Figure 5: The inner balls are centered at anchors on the unit sphere and have
radius $\alpha\varepsilon$; on their boundary, $w=1$. Each of them is
surrounded by two concentric spheres of radii $\varepsilon/20$ and
$\varepsilon/10$, respectively. All these spheres are contained in larger
spheres of radii $1+\varepsilon/5$ and $1+\delta$ centered at the origin.

The proof of Theorem 5.1 is based on the lemma below which requires the
following notation:

$\displaystyle D$ $\displaystyle:=\sup\Bigl{\\{}w(x)\mid x\in\cup_{s\in
S}\;\partial B\left(s,\frac{\varepsilon}{20}\right)\Bigr{\\}}\,,$
$\displaystyle F$ $\displaystyle:=\sup\Bigl{\\{}w(x)\mid x\in\cup_{s\in
S}\;\partial B\left(s,\frac{\varepsilon}{10}\right)\Bigr{\\}}\,,$
$\displaystyle G$ $\displaystyle:=\sup\Bigl{\\{}w(x)\mid x\in\partial
B\left(0,1+\frac{\varepsilon}{5}\right)\Bigl{\\}}\,.$

We note here that by the maximum principle,

$\sup_{z\in\Omega_{1/10}}w(z)\leq F\,.$ (5.6)

######  Lemma 5.1.

There exist constants $c_{1},c_{2}>0$ and $c_{3}\in(0,1)$ that only depend on
$d,p$, such that

$\displaystyle(a)\qquad D$ $\displaystyle\leq
F+c_{1}\tau_{1}^{\gamma}\varepsilon(1-F)\,,$ (5.7) $\displaystyle(b)\qquad G$
$\displaystyle\leq(1-c_{2}\varepsilon/\delta)F\,,$ (5.8)
$\displaystyle(c)\qquad F$ $\displaystyle\leq(1-c_{3})D+c_{3}G\,.$ (5.9)

###### Proof.

Let $0<r<R$ and set

$h_{r,R}(x)=\frac{|x|^{-\gamma}-R^{-\gamma}}{r^{-\gamma}-R^{-\gamma}}\;\mbox{
\rm for }\;r<|x|<R\,.$ (5.10)

Observe that $h_{r,R}$ is $p$-harmonic in $B(0,R)\setminus\bar{B}(0,r)$ and
takes values $0$ and $1$ on $\partial B(0,R)$ and $\partial B(0,r)$
respectively.

(a) Given the definition of $F,$ the comparison principle (Lemma 2.1) implies
that for each $s\in S$, we have

$\displaystyle w(x)\leq F+(1-F)\nu_{1}(x-s),\qquad\forall x\in
B\left(s,\frac{\varepsilon}{10}\right)\setminus B(s,\alpha\varepsilon),$
(5.11)

where $\nu_{1}(x)=h_{r,R}(x)$ with $r=\alpha\varepsilon$,
$R={\varepsilon}/{10}.$

Since

$\displaystyle|x|=\varepsilon/20\;\Longrightarrow\;\nu_{1}(x)=\frac{(\varepsilon/20)^{-\gamma}-(\varepsilon/10)^{-\gamma}}{(\alpha\varepsilon)^{-\gamma}-(\varepsilon/10)^{-\gamma}}\leq\frac{(\varepsilon/20)^{-\gamma}}{(\alpha\varepsilon)^{-\gamma}}\leq
c_{1}\tau_{1}^{\gamma}\varepsilon\,,$ (5.12)

we infer from (5.11) that

$\displaystyle w(x)\leq F+(1-F)c_{1}\tau_{1}^{\gamma}\varepsilon\qquad\forall
x\in\partial B\left(s,\frac{\varepsilon}{20}\right)\,.$ (5.13)

This gives (5.7).

(b) Observe that by (5.6) we have $w\leq F$ on $\partial
B(0,1+\varepsilon/10).$ Hence, by the comparison principle, for all $s\in S$
we have

$\displaystyle w(x)\leq F\nu_{2}(x),\qquad\forall x\in
B\left(0,1+\delta\right)\setminus B\left(0,1+\frac{\varepsilon}{10}\right)\,,$
(5.14)

where $\nu_{2}(x)=h_{r,R}(x)$ with $r=1+\varepsilon/10$ and $R=1+\delta.$

Since the derivative of $t\mapsto t^{-\gamma}$ is bounded above and below by
positive constants for $t\in[1,2]$, the mean value theorem gives

$1-\nu_{2}\left(1+\frac{\varepsilon}{5}\right)=\frac{(1+\varepsilon/10)^{-\gamma}-(1+\varepsilon/5)^{-\gamma}}{(1+\varepsilon/10)^{-\gamma}-(1+\delta)^{-\gamma}}\geq
c_{2}\frac{\varepsilon}{\delta}\,.$ (5.15)

Combining (5.14) and (5.15), we have that

$\displaystyle
w(x)\leq\left(1-c_{2}\frac{\varepsilon}{\delta}\right)F,\qquad\forall
x\in\partial B\left(0,1+\frac{\varepsilon}{5}\right).$ (5.16)

This gives (5.8).

(c) Denote by $\xi$ the standard basis vector $(1,0,\ldots,0)$ in
${\mathbb{R}}^{d},$ and consider the open cored half-ball

$H:=\\{x=(x_{1},\ldots,x_{d})\in B(0,8)\setminus\bar{B}(4\xi,1)\;:x_{1}>0\\}.$
(5.17)

Figure 6: The cored half-ball $H$ obtained by removing a unit ball from the
middle of a half-ball of radius 8.

Let $\psi$ be the Perron solution of $\Delta_{p}\psi=0$ in $H$, with boundary
values $\psi=0$ on the flat part $\\{x\in\partial H\,:\,x_{1}=0\\}$ of
$\partial H$, and $\psi=1$ on the curved part $\\{x\in\partial
B(0,8)\,:x_{1}>0\\}\cup\partial B(4\xi,1)$. (By Proposition 9.31 in [hkm], the
upper and lower Perron solutions coincide since the boundary conditions are
lower semi-continuous and $H$ is regular.) By the strong maximum principle
(see [Lindq], Corollary 2.22), we have $\psi(x)<1$ for all $x\in H.$ By the
continuity of $\psi$ in $H$, for some $c_{3}>0$, we have

$\sup\\{\psi(x)\,:\,x\in\partial B(4\xi,2)\\}=1-c_{3}\,.$ (5.18)

Fix $s\in S$ and let $s_{*}=(1+\varepsilon/5)s$. The cored half-ball

$H_{s}:=\\{x\in
B(s_{*},2\varepsilon/5)\setminus\bar{B}(s,\varepsilon/20)\;:\langle
x,s\rangle<1+\varepsilon/5\\}\,$ (5.19)

is a rotated and translated copy of $H$, scaled by $\varepsilon/20$. The
function $w$ is continuous on $\bar{H_{s}}$, and $p$-harmonic in $H_{s}$, with
boundary values $w\leq G$ on the flat part of $\partial H_{s}$ and $w\leq D$
on the curved part. Therefore, by the comparison principle and (5.18), we have

$\sup\\{w(x)\,:\,x\in\partial B(s,\varepsilon/10)\\}\leq
G+(1-c_{3})(D-G)=(1-c_{3})D+c_{3}G\,.$ (5.20)

This proves (5.9). ∎

###### Proof of Theorem 5.1.

Combining (5.7)-(5.9), we have

$\displaystyle
F\leq(1-c_{3})\bigl{[}F+c_{1}\tau_{1}^{\gamma}\varepsilon(1-F)\bigr{]}+c_{3}\bigl{[}1-c_{2}\varepsilon/\delta\bigr{]}F.$
(5.21)

This gives

$\displaystyle
0\leq(1-c_{3})c_{1}\tau_{1}^{\gamma}\varepsilon(1-F)-c_{3}c_{2}(\varepsilon/\delta)F\,,$
(5.22)

which in turn implies that

$\displaystyle c_{3}c_{2}(\varepsilon/\delta)F\leq
c_{1}\tau_{1}^{\gamma}\varepsilon\,,$ (5.23)

whence

$\displaystyle
F\leq\left(\frac{c_{1}}{c_{3}c_{2}}\right)\tau_{1}^{\gamma}\delta\,.$ (5.24)

Thus there exists a constant $C_{32}>0$ such that $F\leq
C_{32}\tau_{1}^{\gamma}\delta$. By (5.6), this completes the proof. ∎

##  6 Asymptotics for $u^{\varepsilon}$ near the unit sphere: Proof of Theorem
1.2

We will need Clarkson’s inequalities, in the slightly more general form given
in [Boas] for functions $f,g$ taking values in ${\mathbb{R}}^{d}$:

$\displaystyle p\geq
2\Rightarrow\quad\left\|\frac{f+g}{2}\right\|_{p}^{p}+\left\|\frac{f-g}{2}\right\|_{p}^{p}\leq\frac{1}{2}\left(\|f\|_{p}^{p}+\|g\|_{p}^{p}\right)\,,$
(6.1)

and

$\displaystyle 1<p\leq
2\Rightarrow\quad\left\|\frac{f+g}{2}\right\|_{p}^{q}+\left\|\frac{f-g}{2}\right\|_{p}^{q}\leq\left(\frac{1}{2}\|f\|_{p}^{p}+\frac{1}{2}\|g\|_{p}^{p}\right)^{\frac{q}{p}}\,,$
(6.2)

where $\frac{1}{p}+\frac{1}{q}=1$.

###### Proof of Theorem 1.2.

Recall that $\beta=(p-1)/(2p)$. As before, we let $C_{i}$ denote constants
that depend only on $d,p$. Pick a small $\delta>0$. We invoke Lemma 4.1 with
$m$ chosen to satisfy $m\beta/3\geq d$, and infer from (2.42) that

$\forall y\in{\mathbb{S}}^{d-1},\quad E(u,\Lambda_{2\delta}(y))\leq
E(u_{A_{*}},\Lambda_{2\delta}(y))+2^{d}C_{21}\delta^{d}\leq
C_{33}\delta^{d-1}\,,$

provided $\varepsilon$ is small enough. Thus the hypothesis of Lemma 3.4 holds
for every $y\in{\mathbb{S}}^{d-1}$, so that lemma yields

$\displaystyle
E(u,\Lambda_{\delta}(y))\geq(1-2\delta^{\beta/3})E(u_{A},\Lambda_{\delta}(y))\,.$
(6.3)

for some $A$ that may depend on all parameters of the problem, including
$\delta$ and $\varepsilon$. In conjunction with (2.43), this yields that for
small enough $\delta$,

$\displaystyle
E(u,\Lambda_{\delta}(y))\geq(1-3\delta^{\beta/3})E(u_{A_{*}},\Lambda_{\delta}(y))\,.$
(6.4)

By averaging this inequality over $y\in{\mathbb{S}}^{d-1}$, Fubini’s Theorem
yields (as in the proof of (4.3)) that for sufficiently small $\varepsilon$,

$\displaystyle
E(u)\mu(Q_{\delta})\geq(1-3\delta^{\beta/3})E(u_{A_{*}})\mu(Q_{\delta})\,.$
(6.5)

We deduce that for $\varepsilon$ small enough,

$\displaystyle 0\leq E(u_{A_{*}})-E(u)\leq C\delta^{\beta/3}\,.$ (6.6)

Let $\phi$ be the cutoff function defined in Remark 2.2. Then $u,u_{A_{*}}$
are both in the convex admissible class
${\mathscr{A}}(\Gamma,{\mathbb{R}}^{d},\phi)$ defined in (2.11), and $u$
minimizes the energy in this class, so the inequality

$E\Bigl{(}\frac{u+u_{A_{*}}}{2}\Bigr{)}\geq E(u)$

holds. In conjunction with (6.6) and Clarkson’s inequalities, this implies
that

$\|\nabla u-\nabla u_{A_{*}}\|_{p}=O(\delta^{c})$

for some constant $c>0$ and all sufficiently small $\varepsilon$. Since
$\delta$ can be arbitrarily small, this proves the first statement of the
theorem.

It only remains to prove the $L^{\infty}$ convergence.

Recall that we have already established in (4.22) that for small $\delta$ and
(given $\delta$) sufficiently small $\varepsilon$, we have

$\displaystyle\forall x\in{\mathbb{R}}^{d}\setminus
B(0,1+\delta),\qquad(A_{*}-C_{\\#}\delta^{\theta})U_{1+\delta}(x)\leq
u(x)\leq(A_{*}+C_{\\#}\delta^{\theta})U_{1+\delta}(x)\,.$ (6.7)

for some $0<\theta<1$.

If $\tau=\infty$, then $A^{*}=1$ and the first inequality in (6.7), together
with the minimum principle, imply that $u\geq 1-C_{\\#}\delta^{\theta}$ in
$B(0,1+\delta)$, completing the proof in this case. Thus, we may assume that
$\tau<\infty$.

Next, recall the continuous $p$-harmonic function $w$ from Theorem 5.1 defined
in the regular domain

$\Omega_{\alpha}=B(0,1+\delta)\setminus\bigl{[}\cup_{s\in
S}\bar{B}\bigl{(}s,\alpha\varepsilon\bigr{)}\bigr{]}$

considered there. Theorem 5.1 implies that $w\leq
C_{34}(2\tau)^{\gamma}\delta$ in the closure of the domain

$\Omega_{1/10}=B(0,1+\delta)\setminus\bigl{[}\cup_{s\in
S}\bar{B}\bigl{(}s,\varepsilon/10\bigr{)}\bigr{]}\,.$

The inequality $u(x)-A_{*}-C_{\\#}\delta^{\theta}\leq w$ holds on
$\partial\Omega_{\alpha}$, so it must holds in $\Omega_{\alpha}$ as well, by
the comparison principle (Lemma 2.1). In particular, for $\delta<\delta_{0}$
small enough (that may depend on $d,p,\sigma,\tau$), and $\epsilon$ small
enough given $\delta$, we have

$\forall x\in\partial B(s,\varepsilon/10),\quad u\leq
A_{*}+C_{\\#}\delta^{\theta}+C_{34}(2\tau)^{\gamma}\delta\leq
A_{*}+2C_{\\#}\delta^{\theta}\,.$

On the other hand, $u\geq A_{*}-C_{\\#}\delta^{\theta}$ in $B(0,1+\delta)$ by
(6.7) and the minimum principle (or by Lemma 2.1.) Thus

$\displaystyle|u-u_{A_{*}}|\leq
2C_{\\#}\delta^{\theta}\quad\text{in}\quad\Omega_{1/10}\,.$ (6.8)

Finally, fix an anchor $s\in S$ and let
$D_{s}:=B(s,\varepsilon/10)\setminus(s+\alpha\varepsilon K_{s})$. Observe that
$u_{A^{*}}$, restricted to $D_{s}$, is the Perron solution of the $p$-Laplace
equation in $D_{s}$, with the boundary condition $f_{s}$ which is identically
1 on $s+\alpha\varepsilon K_{s}$ and equals $A_{*}$ on $\partial
B(s,\varepsilon/10)$. The inequality

$\displaystyle u_{A_{*}}(x)-2C_{\\#}\delta^{\theta}\leq u(x)\leq
u_{A_{*}}(x)+2C_{\\#}\delta^{\theta}$ (6.9)

holds for $x\in\partial D_{s}$.

Let $\psi$ be a superharmonic function in the upper class $U_{f}^{\Omega}$
(see Definition 2.2) corresponding to the boundary conditions

$f\equiv 1\;\text{on}\;\Gamma,\quad f(\infty)=0$

in the definition (1.5) of $u$ as a Perron solution in
$\Omega={\mathbb{R}}^{d}\setminus\Gamma$. Then $\psi+2C_{\\#}\delta^{\theta}$,
restricted to $D_{s}$, is in the upper class $U_{f_{s}}^{D_{s}}$ by (6.9), so
$\psi+2C_{\\#}\delta^{\theta}\geq u_{A_{*}}$ in $D_{s}$. Taking an infimum
over all such $\psi\in U_{f}^{\Omega}$, we infer that

$\displaystyle u+2C_{\\#}\delta^{\theta}\geq
u_{A_{*}}\quad\text{in}\;D_{s}\,.$ (6.10)

Similarly, let $\psi_{1}$ be a superharmonic function in the upper class
$U_{-f}^{\Omega}$. Then $\psi_{1}+2C_{\\#}\delta^{\theta}$, restricted to
$D_{s}$, is in the upper class $U_{-f_{s}}^{D_{s}}$ by (6.9), so
$\psi_{1}+2C_{\\#}\delta^{\theta}\geq-u_{A_{*}}$ in $D_{s}$. Taking an infimum
over all such $\psi_{1}\in U_{-f}^{\Omega}$, we infer that

$\displaystyle-u+2C_{\\#}\delta^{\theta}\geq-
u_{A_{*}}\quad\text{in}\;D_{s}\,.$ (6.11)

Combining this with (6.10) concludes the proof, since $\delta$ can be
arbitrarily small. ∎

##  7 Concluding remarks.

1. One method to obtain an asymptotically equidistributed set of anchors is to choose $S(\varepsilon)$ as any $\epsilon$-separated set on the sphere of maximal cardinality. The asymptotic equidistribution

$\displaystyle\frac{1}{|S(\varepsilon)|}\sum_{s\in
S(\varepsilon)}\delta_{s}\overset{\ast}{\rightharpoonup}\mu\>~{}\mbox{as}~{}\varepsilon\downarrow
0,$ (7.1)

then follows from a classical argument of Maak, as presented, e.g., in [Don,
Chapter 12], while the existence of the limit
$\sigma=\lim_{\varepsilon\downarrow 0}\varepsilon^{d-1}|S(\varepsilon)|$ is
established in [Ham].

2. The main results of this paper directly extend to problem (1.5) considered in a ball $B(0,R)\subset\mathbb{R}^{d}$ with $R>1.$ Indeed, let $1<p<d,$ $R>1$ and let $w=w^{\varepsilon}$ be the Perron solution of the following problem

$\displaystyle\left\\{\begin{array}[]{lll}\Delta_{p}w=0&\mbox{in}&B(0,R)\setminus\Gamma,\\\
w=1&\mbox{on}&\Gamma,\\\ w=0&\mbox{on}&\partial B(0,R).\end{array}\right.$
(7.5)

Define an ansatz

$\displaystyle w^{\varepsilon}_{A}(x):=AW_{\varepsilon,R}(x)+(1-A)\sum_{s\in
S}V_{\frac{1}{10\alpha}}^{s}\left(\frac{x-s}{\alpha\varepsilon}\right)\,,$
(7.6)

where $W_{\varepsilon,R}$ is the $p$-equilibrium potential of the ball
$\bar{B}(0,1+\varepsilon)$ relative to the ball $B(0,R)$, given by

$\displaystyle W_{\varepsilon,R}(x):=\left\\{\begin{array}[]{ll}1&0\leq|x|\leq
1+\varepsilon\,,\\\
\frac{|x|^{-\gamma}-R^{-\gamma}}{(1+\varepsilon)^{-\gamma}-R^{-\gamma}}&1+\varepsilon<|x|\leq
R\,,\end{array}\right.$ (7.9)

and $V_{\rho}^{s}$ are as in (1.34).

The following corollary holds:

######  Corollary 7.1.

Suppose that hypotheses ${\rm(H_{1}),(H_{2}),(H_{3})}$ hold. Then, as
$\varepsilon\to 0$,

$\displaystyle
w^{\varepsilon}(x)\to\left\\{\begin{array}[]{lll}0&\mbox{if}&\tau=0\,,\\\
A_{R}W_{0,R}(x)&\mbox{if}&\ \tau\in(0,\infty)\,,\\\
W_{0,R}(x)&\mbox{if}&\tau=\infty\end{array}\right.$ (7.13)

uniformly on compact subsets of $\bar{B}(0,R)\setminus\mathbb{S}^{d-1},$ where
for $\tau\in[0,\infty)$,

$\displaystyle A_{R}=A_{R}(\tau)=\frac{\left(\sigma\tau^{d-p}{\rm
cap}_{p}(K)\right)^{\frac{1}{p-1}}}{\left(\sigma\tau^{d-p}{\rm
cap}_{p}(K)\right)^{\frac{1}{p-1}}+\left({\rm
cap}_{p}(\bar{B}(0,1),B(0,R))\right)^{\frac{1}{p-1}}}\,.$ (7.14)

Furthermore, as $\varepsilon\to 0$,

$\displaystyle{\rm
cap}_{p}(\Gamma_{\varepsilon},B(0,R))\to\left\\{\begin{array}[]{lll}0&\mbox{if}&\tau=0\,,\\\
A_{R}^{p}{\rm cap}_{p}(\bar{B}(0,1),B(0,R))+(1-A_{R})^{p}{\rm
cap}_{p}(K)\sigma\tau^{d-p}&\mbox{if}&\ \tau\in(0,\infty)\,,\\\ {\rm
cap}_{p}(\bar{B}(0,1),B(0,R))&\mbox{if}&\tau=\infty\,.\end{array}\right.$
(7.18)

Moreover,

$\displaystyle\|\nabla w-\nabla
w^{\varepsilon}_{A_{R}}\|_{{L^{p}(\bar{B}(0,R))}}\to 0,$ (7.19)

and

$\displaystyle\|w-w_{A_{R}}\|_{L^{\infty}(\bar{B}(0,R))}\to 0\,.$ (7.20)

The proof of this corollary is a line by line adaptation of the arguments in
the preceding sections.

Acknowledgments. The work of PVG was supported in a part by the Simons
Foundation grant 317882 and US-Israel BSF grant 2020005. The work of FN was
partially supported by the NSF grant DMS-1900008 and US-Israel BSF grant
2020019.

## References