\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{amssymb}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 84, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/84\hfil Grushin fundamental solutions]
{Fundamental solutions to the $p$-Laplace equation in a class
 of Grushin vector fields}

\author[T. Bieske \hfil EJDE-2011/84\hfilneg]
{Thomas Bieske}

\address{Thomas Bieske \newline
Department of Mathematics\\ 
University of South Florida\\
Tampa, FL 33620, USA}
\email{tbieske@math.usf.edu}

\thanks{Submitted December 10, 2010. Published June 29, 2011.}
\subjclass[2000]{35H20, 53C17, 17B70}
\keywords{Grushin-type spaces; $p$-Laplacian}

\begin{abstract}
 We find the fundamental solution to the $p$-Laplace equation
 in a class of Grushin-type spaces.  The singularity occurs
 at the sub-Riemannian points, which naturally corresponds to
 finding the fundamental solution of a generalized Grushin
 operator in Euclidean space. We then use this solution to find
 an infinite harmonic function with specific boundary data and
 to compute the capacity of annuli centered at the singularity.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\newcommand{\ip}[2]{\langle #1,#2\rangle}

\section{Motivation}

The $p$-Laplace equation is the model equation for nonlinear
potential theory.  The Euclidean results of \cite{HKM:NPT} can be
extended into a class of sub-Riemannian spaces possessing an
algebraic group law, called Carnot groups \cite{HH:QR}.
Fundamental solutions to the $p$-Laplace equation in a subclass of
Carnot groups called groups of Heisenberg-type have been found in
\cite{CDG, HH:QR}. The exploration of the $p$-Laplace equation in
sub-Riemannian spaces without an algebraic group law is currently
a topic of interest. In this paper, we will find the fundamental
solution to the $p$-Laplace equation for $1< p < \infty$ in a
class of Grushin-type spaces. The singularity occurs at the
sub-Riemannian points, which naturally corresponds to finding the
fundamental solution of a generalized Grushin operator in
Euclidean space.

\section{Grushin-type spaces}

Before presenting the main theorem, we recall the construction of
such spaces and their main properties.  We begin with $\mathbb{R}^n$,
possessing coordinates $(x_1,x_2,\dots,x_n)$ and vector fields
$$
X_i = \rho_i (x_1,x_2,\dots,x_{i-1}) \frac{\partial}{\partial x_i}
$$
for $i=2,3,\dots, n$ where $\rho_i (x_1,x_2,\dots,x_{i-1})$ is a  (possibly constant)
real-valued function. We decree that $\rho_1 \equiv 1$ so that
$$
X_1=\frac{\partial}{\partial x_1}\,.
$$
A quick calculation shows that when $i < j$ and $\rho_j (x_1,x_2,\dots,x_{j-1})$ is differentiable, the Lie bracket is given by
\begin{equation}\label{bracket}
X_{ij} \equiv [X_i,X_j]= \rho_i (x_1,x_2,\dots,x_{i-1}) \frac{\partial \rho_j (x_1,x_2,\dots,x_{j-1})}{\partial x_i}
\frac{\partial }{\partial x_j}.
\end{equation}
If the $\rho_i$'s are polynomials, at each point there is a finite
number of iterations of the Lie bracket so that $\frac{\partial}{\partial
x_i}$ has a non-zero coefficient. This is easily seen for $X_1$
and $X_2$, and the result is obtained inductively for $X_i$.  (It
is noted that the number of iterations necessary is a function of
the point.)  It follows that H\"{o}rmander's condition is
satisfied by such vector fields.

We may further endow $\mathbb{R}^n$ with an inner product (singular where
the polynomials vanish) so that the collection $\{X_i\}$ forms an
orthonormal basis.  This produces a sub-Riemannian manifold that
we shall call $g_n$, which is also the tangent space to a
generalized Grushin-type space $G_n$.  Points in $G_n$ will also
be denoted by $x=(x_1,x_2,\dots, x_n)$.

Though $G_n$ is not a Lie group, it is a metric space with the
natural metric being the Carnot-Carath\'{e}odory distance, which
is defined for points $x$ and $y$ as follows:
\begin{equation*}
d_C(x,y)= \inf_{\Gamma} \int_{0}^{1} \| \gamma '(t) \|\,dt\,.
\end{equation*}
Here $ \Gamma $ is the set of all curves $ \gamma $ such
that $ \gamma (0) = x, \gamma (1) = y$ and
$$
\gamma '(t) \in \operatorname{span} \{\{X_i(\gamma(t))\}_{i=1}^n\} .
$$
In the case of polynomials, Chow's theorem (see, for example,
\cite{BR:SRG}) states any two points can be joined by such a
curve. In the non-polynomial case, one can explicitly construct
horizontal curves of finite length connecting any two points.
This means $ d_C(x,y) $ is an honest metric.  Using this metric,
we can define a Carnot-Carath\'{e}odory ball of radius $r$
centered at a point $x_0$ by
$$
B_C(x_0,r)=\{p\in G_n : d_C(x,x_0) < r\}
$$
and similarly, we shall denote a bounded domain in $G_n$ by
$\Omega$. The Carnot-Carath\'{e}odory metric behaves differently
at points where the functions $\rho_i$ vanish.  Fixing a point
$x_0$ where the $\rho_i$ are sufficiently differentiable, consider
the $n$-tuple $r_{x_0}=(r^1_{x_0},r^2_{x_0},\dots,r^n_{x_0})$
where $r^i_{x_0}$ is the minimal number of Lie bracket iterations
required to produce
$$
[X_{j_1},[X_{j_2},[\cdots[X_{j_{r^i_{x_0}}},X_i]\cdots](x_0) \neq 0\,.
$$
Note that though the minimal length is unique, the iteration used to
obtain that minimum is not.  Note also that
$$
\rho_i(x_0) \neq 0 \leftrightarrow r^i_{x_0}=0\,.
$$
Using \cite[Theorem 7.34]{BR:SRG} we obtain the local estimate at
$x_0$
\begin{equation} \label{distest}
d_C(x_0,x)  \sim  \sum _{i=1}^n |x_i-x_i^0|^\frac{1}{1+r^i_{x_0}} .
\end{equation}
Given a smooth function $f$ on $G_n$, we define the horizontal gradient
of $f$ as
$$
\nabla_0f(x) = (X_1f(x),X_2f(x),\dots, X_nf(x))
$$
and the symmetrized second order (horizontal)  derivative matrix by
$$
((D^2f(x))^{\star})_{ij} = \frac{1}{2} (X_iX_jf(x)+X_jX_if(x))
$$
for $i,j=1,2,\dots n$.

\begin{definition} \label{def1} \rm
The function $f: G_n \to \mathbb{R}$ is said to be $C_{\rm sub}^1$ if $X_if$
is continuous for all $i=1,2,\dots,n$.  Similarly, the function
$f$ is $C_{\rm sub}^2$ if $X_iX_jf$ is continuous for all
$i,j=1,2,\dots,n$.
\end{definition}

\begin{remark} \label{rmk1} \rm
We note that Euclidean $C^1$ functions are $C_{\rm sub}^1$ functions,
but the class of $C_{\rm sub}^1$ functions is larger than the class
of Euclidean $C^1$ functions. For example, when $n=2$
and $\rho_2(x_1)=x_1$, we have $\sqrt{x_2}$ is $C_{\rm sub}^1$
at the origin, but is clearly not Euclidean $C^1$ at the origin.
The interested reader is directed to \cite{BR:SRG} for a
more complete discussion.
\end{remark}

Using these derivatives, we consider two main operators on $C_{\rm sub}^2$ functions
called the $p$-Laplacian
$$
\Delta_{p}f=\operatorname{div}(\|\nabla_0f\|^{p-2}\nabla_0f)=\sum_{i=1}^nX_i
(\|\nabla_0f\|^{p-2}X_if)
$$
defined for $1 < p < \infty$ and the infinite Laplacian
\[
\Delta_{\infty}f
   =  \sum_{i,j=1}^n X_ifX_jfX_iX_jf
   =   \ip{\nabla_0 f}{(D^2f)^{\star}\nabla_0 f}.
\]

For a more in-depth study of Grushin-type spaces, the reader
is directed to \cite{BR:SRG,B:GG,B:RV} and the
references therein.

\section{The co-area formula and measure theory}

We begin by fixing  $m,n \in \mathbb{N}$ and $k,c \in \mathbb{R}$ so that $m<n$,
$c \neq 0$, and $k\geq0$. We also fix a vector
$a=(a_1,a_2,\dots,a_m)\in \mathbb{R}^m$ and then consider the
following vector fields:
\begin{equation} \label{vfds}
\begin{gathered}
X_i  =  \frac{\partial}{\partial  x_i}\quad  \text{for $i=1$ to $m$}\\
X_{j}  =  c \Big(\sum_{i=1}^m(x_i-a_i)^2\Big)^{k/2}
\frac{\partial}{ \partial  x_{j}}
\quad\text{for $j=m+1$ to $n$.}
\end{gathered}
\end{equation}
Note that this choice corresponds to $\rho_i (x_1,x_2,\dots,x_{i-1})=1$ for $1 \leq i \leq m$ and
$\rho_j (x_1,x_2,\dots,x_{j-1})=c (\sum_{i=1}^m(x_i-a_i)^2)^{k/2}$ for $m+1 \leq j \leq n$.
Additionally,  if $k=0$ and $c=1$, we have the Euclidean space $\mathbb{R}^n$.
Note also that in local coordinates, the $2$-Laplacian operator
is given by
$$
\Delta_2=\sum_{i=1}^m\frac{\partial^2}{\partial x_i^2} +
\sum_{j=m+1}^n c^2\Big(\sum_{i=1}^m(x_i-a_i)^2\Big)^k \frac{\partial^2}{\partial x_{j}^2}.
$$
In place of Fubini's Theorem for iterated integrals, we will make
use of the following Co-Area Formula in the Euclidean
context \cite{C:C}, which
was extended to the Grushin case via \cite[Theorem 4.2]{MSC:co}.

\begin{theorem} \label{thm3.1}
Let $\Omega \subset G_n$ be a bounded domain, and let
$\psi \in C_{\rm sub}^1(\Omega)$ be a smooth, real-valued function which
extends continuously to $\partial \Omega$.  For convenience,
we write $\nabla$ for the Euclidean gradient on $G_n = \mathbb{R}^n$.
Then for any function $g \in L^1(\Omega)$
\begin{equation} \label{coarea}
\iint_\Omega g \|\nabla \psi\| \, d\mathcal{L}_n =
\int_0^\infty \int_{\psi^{-1}\{r\}} g \, d\mathcal{H} dr,
\end{equation}
where $d\mathcal{L}_n$ denotes Lebesgue $n$-measure on $\Omega$, and
 $d\mathcal{H}$ denotes Hausdorff $(n-1)$-measure on $\psi^{-1}(\{r\})$.
\end{theorem}

\begin{corollary} \label{coro3.2}
As above, the theorem also holds for continuous functions $\psi$
which are smooth everywhere except at isolated points.
\end{corollary}

We now consider a point $x_0 \in G_n$ with coordinates
$x_0 = (a_1,\dots,a_m, b_{m+1},\dots, b_n )$ and a non-negative,
continuous radial function $\psi:\mathbb{R}^n\to\mathbb{R}$ that
is  smooth when $x_0\neq x$ and with $\psi(x_0) = 0$.  The
following notation is suggestive for the inverse images of $\psi$.
\begin{gather*}
\mathcal{B}_R(x_0) = \psi^{-1}([0,R))
= \{ x \in \Omega : \psi(x) < R \}, \\
\partial \mathcal{B}_R(x_0) = \psi^{-1}(\{R\})
= \{ x \in \Omega : \psi(x) = R \}.
\end{gather*}
The $x_0$ is omitted when it is clear from the context.
Since $\|\nabla_0\psi\| \lesssim
\|\nabla \psi\|$ and $p>1$, we may apply the Co-Area Formula
to the well-defined function
\[
g = \begin{cases} (\|\nabla_0 \psi\|^p/ \|\nabla \psi\|)\cdot
\|\nabla \psi\| & \|\nabla \psi\| \neq 0 \\
0 & \|\nabla \psi\| = 0
\end{cases}
\]
to obtain the following result.

\begin{proposition} \label{prop3.3}
With the hypotheses as above, let $\mathcal{V}$ be an absolutely continuous
measure to $\mathcal{L}_n$ with Radon-Nikodym
derivative $\|\nabla_0 \psi\|^p = [d\mathcal{V} / d\mathcal{L}_n]$.
Then for sufficiently small $R > 0$,
\begin{equation} \label{coarea2}
\mathcal{V}(\mathcal{B}_R) = \int_{\mathcal{B}_R} d\mathcal{V} =
\int_0^R \int_{\partial \mathcal{B}_r}
\frac{\|\nabla_0 \psi\|^p}{\|\nabla \psi\|} \, d\mathcal{H} dr
\end{equation}
\end{proposition}

In light of the equality in \eqref{coarea2}, we see that the measure
space $(G_n, \mathcal{V})$ is globally Ahlfors $Q$-regular with respect to
balls centered at $x_0$.  In particular, for $R > 0$,
\begin{equation} \label{ahlfors}
\mathcal{V}(\mathcal{B}_R) = \sigma_p R^Q
\end{equation}
where $Q = m+(k+1)(n-m)=k(n-m)+n$ and $\sigma_p = \mathcal{V}(\mathcal{B}_1)$
is a fixed positive constant.

For technical purposes we proceed to study the boundary behavior of
precompact domains $\Omega$.  This now motivates the following
definition.

\begin{definition} \label{def2} \rm
For small values $R>0$, define a measure $\mathcal{S}$ on $\partial \mathcal{B}_R$ as
$$
\mathcal{S}(\partial \mathcal{B}_R) = \int_{\partial \mathcal{B}_R} d\mathcal{S} =
\int_{\partial \mathcal{B}_R} \frac{\|\nabla_0 \psi\|^p}{\|\nabla \psi\|}
\,d\mathcal{H}.
$$
\end{definition}

In particular, $S$ is absolutely continuous with respect to the
Hausdorff $(n-1)$-measure $\mathcal{H}$.  Using previous results,
in particular, the fact that $\psi$ is smooth away from $x_0$,
we now conclude:

\begin{corollary} \label{coro3.4}
\begin{enumerate}
\item $S$ is locally Ahlfors $(Q-1)$-regular and
\begin{equation} \label{ahlfors2}
\mathcal{S}(\partial \mathcal{B}_R) = Q \sigma_p R^{Q-1}.
\end{equation}
\item Let $\varphi$ be a continuous and integrable function on $\mathcal{B}_R$.  Then
as $R \to 0$,
\begin{equation} \label{density}
\frac{R^{1-Q}}{Q\sigma_p} \int_{\partial \mathcal{B}_R} \varphi \,d\mathcal{S} \; \to \;
\varphi(x_0)
\end{equation}
\end{enumerate}
\end{corollary}

\begin{proof}[Sketch of Proof]
Equation \eqref{ahlfors2} follows immediately from differentiating
both Equations \eqref{coarea2} and \eqref{ahlfors}.  Since $\mathcal{S}$ is
absolutely continuous with respect to Hausdorff $(n-1)$-measure
$\mathcal{H}$, it follows that $\mathcal{S}$ is Borel regular.  As a result,
Equation \eqref{density} is the analogue of the Lebesgue Density
Theorem.
\end{proof}

\section{The $p$-Laplace equation}

In this section, we compute the fundamental solution of the
$p$-Laplacian for the previously-defined vector fields
\eqref{vfds} and for $1<p<\infty$. We then use these formulas to
find the explicit formula for a solution to the Dirichlet problem
with specific boundary data.  The following theorem generalizes
\cite{BG} and is the Grushin analog of results in the class of
Carnot groups known as groups of Heisenberg-type
\cite{CDG,HH:QR}.

\begin{theorem}  \label{main}
Let $x_0 = (a_1,a_2,\dots,a_m, b_{m+1},b_{m+2},\dots,b_n )$ be an
arbitrary fixed point. Consider the following quantities, for
$1 <p < \infty$:
\begin{gather*}
w  =  \frac{Q-p}{(2k+2)(1-p)} \quad
\alpha = \frac{Q-p}{1-p} \\
h(x_1, x_2, \dots, x_n)  =   c^2 \Big(\sum_{i=1}^m(x_i-a_i)^{2}\Big)^{k+1}+(k+1)^2\sum_{j=m+1}^n(x_{j}-b_j)^2 \\
f(x_1, x_2, \dots , x_n)  =  [h(x_1, x_2, \dots , x_n)]^w \\
\psi(x_1, x_2, \dots, x_n) = [h(x_1, x_2, \dots, x_n)]^\frac{1}{2k+2} \\
\sigma_p = \int_{\mathcal{B}_1} \|\nabla_0\psi\|^p\,d\mathcal{L}_n \\
C_1 = \alpha^{-1}(Q\sigma_p)^\frac{1}{1-p} \quad
C_2 = (Q\sigma_Q)^\frac{1}{1-Q}.
\end{gather*}
Then, for the constants $C_1$ and $C_2$ as above,
\begin{equation} \label{fund}
\begin{gathered}
\Delta_pC_1f(x_1, x_2, \dots , x_n) =  \delta_{x_0} \quad
\text{when }  p \neq Q \\
\Delta_p(C_2\log{\psi}(x_1, x_2, \dots , x_n)) =  \delta_{x_0} \quad
\text{when }  p = Q
\end{gathered}
\end{equation}
in the sense of distributions.
\end{theorem}

\begin{proof}
We first comment that for the sake of rigor, we should invoke
the regularization of $h$ given by
$$
h_\varepsilon(x_1, x_2, \dots, x_n) = c^2
\Big(\sum_{i=1}^m(x_i-a_i)^{2}+\varepsilon^2\Big)^{k+1}+(k+1)^2
\sum_{j=m+1}^n(x_{j}-b_j)^2
$$
for $\varepsilon >0$ and letting $\varepsilon \to 0$. However, we
shall proceed formally. Suppressing the variables $(x_1, x_2,
\dots , x_n)$, and letting
$$
\Sigma=\sum_{i=1}^m(x_i-a_i)^{2}
$$
 we compute for $p \neq Q$:
\begin{gather*}
X_if  =  \alpha h^{w-1}c^2(x_i-a_i)\Sigma^{k}\ \textmd{for $i=1,2,\dots, m$}\\
X_{j}f  =  c \alpha h^{w-1}\Sigma^{\frac{k}{2}}(k+1)(x_{j}-b_j)\ \textmd{for $j=m+1,\dots, n$}\\
\|\nabla_0f\|^2  =
c^2 \alpha^2h^{2w-1}\Sigma^{k}\\
\|\nabla_0f\|^{p-2}  =
|c\alpha|^{p-2}h^{w(p-2)-\frac{p-2}{2}}\Sigma^{\frac{k(p-2)}{2}}.
\end{gather*}
We then are able to compute, for $i=1,2,\dots, m$,
\[
\|\nabla_0f\|^{p-2}X_if  =  \alpha|\alpha|^{p-2}|c|^p
h^{w(p-1)-\frac{p}{2}}
\Sigma^{\frac{kp}{2}}(x_i-a_i)
\]
and for $j=m+1,m+2,\dots, n$,
\begin{equation*}
\|\nabla_0f\|^{p-2}X_{j}f  =  \alpha|\alpha|^{p-2}c|c|^{p-2}
h^{w(p-1)-\frac{p}{2}}
\Sigma^{\frac{kp}{2}-\frac{k}{2}}(k+1)(x_{j}-b_j).
\end{equation*}
Setting
$$
D_p \equiv \frac{\Delta_pf}{\alpha |\alpha|^{p-2}|c|^p}\quad\text{and}\quad
\Upsilon=w(p-1)-\frac{p}{2}
$$
we can then compute
\begin{align*}
D_p &= \sum_{i=1}^m h^{\Upsilon}\Sigma^{\frac{kp}{2}}+ \sum_{i=1}^m h^{\Upsilon}(kp)\Sigma^{\frac{kp-2}{2}}(x_i-a_i)^2\\
&\quad + \sum_{i=1}^m \Upsilon h^{\Upsilon-1}2c^2(k+1)
 \Sigma^{\frac{kp}{2}+k}(x_i-a_i)^2\\
&\quad+\sum_{j=m+1}^n h^{\Upsilon}(k+1)\Sigma^{\frac{kp}{2}}+ \sum_{j=m+1}^n 2\Upsilon h^{\Upsilon-1}\Sigma^{\frac{kp}{2}}(k+1)^3
  (x_{j}-b_j)^2\\
&=  h^{\Upsilon-1}\Sigma^{\frac{kp}{2}}
  \bigg( mh+(kp)h+(k+1)(n-m)h\\
&\quad+ \big(\alpha(p-1)-p(k+1)\big) \Big(c^2\Sigma^{k+1}+(k+1)^2
  \sum_{j=m+1}^n(x_{j}-b_j)^2\Big)\bigg)\\
&=  h^{\Upsilon}\Sigma^{\frac{kp}{2}}\Big( \alpha(p-1)-p(k+1)+m+(kp)+(k+1)(n-m)\Big)\\
 &=  h^{\Upsilon}\Sigma^{\frac{kp}{2}}((p-Q)-p+Q)=0.
\end{align*}

Note that these computations are valid wherever the function $f$ is
smooth and in particular, these are valid away from the point $x_0$.
We next note that
$$
\|\nabla_0f\|^{p-1} \sim \psi^{1-Q}
$$
and so we conclude that $\|\nabla_0f\|^{p-1}$ is locally integrable on $G_n$ with respect to Lebegue measure.
We then consider $\phi \in C^{\infty}_0$ with compact support in the ball
$$
\mathcal{B}_R = \{x: \psi(x) < R \}.
$$
Let $0 < r < R$ be given so that $\mathcal{B}_r \subset \mathcal{B}_R$.  In the annulus
$\mathcal{A} := \mathcal{B}_R \setminus \overline{\mathcal{B}_r}$ we have, via the Leibniz rule,
\begin{align*}
\operatorname{div}(\phi \|\nabla_0f\|^{p-2}\nabla_0f)
&= \phi \operatorname{div}(\|\nabla_0f\|^{p-2}\nabla_0f) +
\|\nabla_0f\|^{p-2}\ip{\nabla_0f}{\nabla_0\phi} \\
&= 0 + \|\nabla_0f\|^{p-2}\ip{\nabla_0f}{\nabla_0\phi}.
\end{align*}
Let $\mathcal{L}_n$ and $\mathcal{H}$ be the measures from \eqref{coarea} and recall
$$
\Sigma\equiv \sum_{j=1}^m(x_i - a_i)^2.
$$
Applying Stokes' Theorem,
\begin{align*}
& \int_{\mathcal{A}}\|\nabla_0f\|^{p-2}
\ip{\nabla_0f}{\nabla_0\phi}d\mathcal{L}_n \\
&= \int_{\mathcal{A}}\operatorname{div}(\phi \|\nabla_0f\|^{p-2}\nabla_0f)d\mathcal{L}_n \\
&=\int_{\mathcal{A}} \Big( \sum_{i=1}^m X_i[\phi \|\nabla_0f\|^{p-2}X_if]
 + c \Sigma^{\frac{k}{2}} \sum_{j=m+1}^n\frac{\partial}{\partial x_{j}}\Big(\phi \|\nabla_0f\|^{p-2}X_{j}f\Big) \Big) d\mathcal{L}_n \\
&=\int_{\mathcal{A}} \Big( \sum_{i=1}^m X_i[\phi \|\nabla_0f\|^{p-2}X_if]
+ \sum_{j=m+1}^n \frac{\partial}{\partial x_{j}}\Big( c \Sigma^{\frac{k}{2}}\phi \|\nabla_0f\|^{p-2}X_{j}f\Big) \Big) d\mathcal{L}_n \\
&=\int_{\mathcal{A}} \operatorname{div}_{\operatorname{eucl}}
\begin{bmatrix}
\phi \|\nabla_0f\|^{p-2}X_1f \\
\vdots \\
\phi \|\nabla_0f\|^{p-2}X_mf \\
c \Sigma^{\frac{k}{2}}\phi \|\nabla_0f\|^{p-2}X_{m+1}f \\
\vdots \\
c \Sigma^{\frac{k}{2}}\phi \|\nabla_0f\|^{p-2}X_n f
\end{bmatrix} d\mathcal{L}_n \\
&=\int_{\partial\mathcal{A}} \frac{1}{\|\nu\|}
 \Big( \phi \|\nabla_0f\|^{p-2}\sum_{i=1}^mX_if \nu_i
 + c\Sigma^{\frac{k}{2}}\phi \|\nabla_0f\|^{p-2}
 \sum_{j=m+1}^nX_{j}f \nu_{j} \Big) d\mathcal{H} \\
&=-\int_{\partial \mathcal{B}_r} \frac{1}{\|\nu\|}
 \Big( \phi \|\nabla_0f\|^{p-2}\sum_{i=1}^mX_if \nu_i
 + c\Sigma^{\frac{k}{2}}\phi  \|\nabla_0f\|^{p-2}
 \sum_{j=m+1}^nX_{j}f \nu_{j} \Big) d\mathcal{H}
\end{align*}
where $\nu$ is the outward Euclidean normal of $\mathcal{A}$.
Recalling that
$$
\psi(x_1, x_2, \dots, x_n) = [h(x_1, x_2, \dots,
x_n)]^{1/(2k+2)},
$$
and that
$\nu_j = \frac{\partial\psi}{\partial x_j}$,
we may proceed with the computation,
\begin{align*}
& \int_{\mathcal{A}}\|\nabla_0f\|^{p-2}
\ip{\nabla_0f}{\nabla_0\phi}d\mathcal{L}_n \\
&=
-\int_{\partial \mathcal{B}_r} \frac{\alpha \psi^{\alpha-1}}{\|\nu\|} \phi \|\nabla_0\psi^\alpha\|^{p-2}
\Big(\sum_{i=1}^m(\frac{\partial\psi}{\partial x_i})^2 + c^2 \Sigma^k \sum_{j=m+1}^n(\frac{\partial\psi}{\partial x_{j}})^2 \Big) d\mathcal{H} \\
&=
-\int_{\partial \mathcal{B}_r} \frac{\alpha \psi^{\alpha-1}}{\|\nu\|} \phi
\|\nabla_0\psi\|^{p-2}|\alpha|^{p-2} \psi^{(p-2)(\alpha-1)}
\Big( \| \nabla_0\psi\|^2 \Big) d\mathcal{H} \\ &=
-\int_{\partial \mathcal{B}_r} \frac{|\alpha|^{p-2}\alpha \psi^{(p-1)(\alpha-1)}}{\|\nu\|} \phi
\|\nabla_0\psi\|^p d\mathcal{H}.
\end{align*}
Recall that by definition, $\psi \equiv r$ on $\partial \mathcal{B}_r$.
We then have
\[
\int_{\mathcal{A}}\|\nabla_0f\|^{p-2} \ip{\nabla_0f}{\nabla_0\phi} d\mathcal{L}_n
= - |\alpha|^{p-2} \alpha r^{1-Q}
\int_{\partial \mathcal{B}_r} \frac{\phi \|\nabla_0\psi\|^p}{\|\nu\|} \, d\mathcal{H}.
\]
Letting $r \to 0$, we apply \eqref{density} and obtain
\begin{equation} \label{scaling}
\int_{\mathcal{A}}\|\nabla_0f\|^{p-2} \ip{\nabla_0f}{\nabla_0\phi} d\mathcal{L}_n \; \to \;
- |\alpha|^{p-2} \alpha (Q\sigma_p) \phi(x_0).
\end{equation}
We then obtain the case for $p \neq Q$.
The case of $p=Q$ is similar and left to the reader.
\end{proof}

It was shown in \cite{B:GG} and \cite{B:RV} that in Grushin-type spaces, viscosity infinite harmonic functions are
limits of weak $p$-harmonic functions as $p$ tends to infinity.  This
motivates the following corollary.

\begin{corollary} \label{coro4.2}
The function $\psi$, as defined above, is infinite harmonic
in the space $G_n \setminus \{ x_0 \}$.
\end{corollary}

\begin{proof}
We use the formula that for a smooth function $u$,
$$
\Delta_{\infty}u=\frac{1}{2}\nabla_0u \cdot \nabla_0\|\nabla_0u\|^2.
$$
Computing as in the proof of the Theorem, we have
$$
\|\nabla_0\psi\|^2 = c^2\Sigma^k h^\frac{-2k}{2k+2}.
$$
Thus we obtain for $i=1,2,\dots, m$,
\[
X_i\|\nabla_0\psi\|^2
=  2kc^2h^{\frac{-2k}{2k+2}-1}\Sigma^{k-1}(x_i-a_i)
\big(h-c^2 \Sigma^{k+1}\big)
\]
and for $j=m+1,m+2,\dots, n$,
\begin{align*}
X_{j}\|\nabla_0\psi\|^2 = -2kc^3 h^{\frac{-2k}{2k+2}-1}\Sigma^{\frac{3k}{2}}(k+1)(x_{j}-b_j)
\end{align*}
so that using the derivatives as in the proof of the Theorem,
\begin{align*}
\Delta_{\infty}\psi
&=  \sum_{i=1}^m 2kc^4 h^{\frac{-4k-1}{2k+2}-1}\Sigma^{2k-1}(x_i-a_i)^2
 \big(h-c^2 \Sigma^{k+1}\big)\\
& \quad - 2kc^4h^{\frac{-4k-1}{2k+2}-1}(k+1)^2\Sigma^{2k}\sum_{j=m+1}^n(x_{j}-b_j)^2\\
&=  2kc^4h^{\frac{-4k-1}{2k+2}-1}\Sigma^{2k} \Big(h-c^2 \Sigma^{k+1}-(k+1)^2\sum_{j=m+1}^n(x_{j}-b_j)^2\Big)\\
&=   2kc^4h^{\frac{-4k-1}{2k+2}-1}\Sigma^{2k}\times (0).
\end{align*}
\end{proof}

Using the existence-uniqueness of viscosity infinite harmonic functions
\cite{B:GG,B:RV} and the fact that absolute minimizers in Grushin
spaces are viscosity infinite harmonic functions and enjoy comparison
with cones \cite{BDM:BDM}, we conclude the following corollary.

\begin{corollary} \label{coro4.3}
Let $0<s\in \mathbb{R}$.
Define the function $\Psi_s : \partial \mathcal{B}_s(x_0) \cup\{x_0\}
 \to \mathbb{R}$ by
\[
\Psi_s(x_1, x_2, \dots, x_n) =
\begin{cases}
s & \text{on } \partial \mathcal{B}_s(x_0) \\
0 & \text{at }  x_0
\end{cases}
\]
Then $s\cdot \psi$ is the unique absolute minimizer of $\Psi$
into the ball $\mathcal{B}_s(x_0)$.  In addition, $s\cdot\psi$
enjoys comparison with Grushin cones.
\end{corollary}

\section{Spherical capacity}

In this section, we will use previous results to compute the
capacity of spherical rings centered at the point
$x_0=(a_1,a_2,\dots,a_m,b_{m+1},b_{m+2},\dots,b_n )$. We first
recall the definition of $p$-capacity.

\begin{definition} \label{def3} \rm
Let $\Omega \subset G_n$ be a bounded, open set, and
$K \subset \Omega$ a compact subset. For $1 \leq p < \infty$
we define the $p$-capacity as
$$
\operatorname{cap}_p(K, \Omega) =
\inf \Big\{ \int_\Omega \|\nabla_0u\|^p\,d\mathcal{L}_n:
u \in C^\infty_0(\Omega), \; u|K = 1 \Big\}.
$$
\end{definition}

We note that although the definition is valid for $p=1$,
we will consider only $1 < p < \infty$, as in the previous sections.
Because $p$-harmonic functions are minimizers to the energy integral
$$
\int_{G_n} \|\nabla_0f\|^p\;d\mathcal{L}_n
$$
it is natural to consider $p$-harmonic functions when computing
the capacity.  In particular, an easy calculation similar
to the previous section shows
\[
u(x) = \begin{cases}
\frac{\psi(x)^\alpha -  R^\alpha}{ r^\alpha -  R^\alpha} &
\text{when } p \neq Q \\[4pt]
\frac{\log{\psi(x)} - \log{R}}{\log{r} - \log{R}} &
\text{when } p = Q
\end{cases}
\]
is a smooth solution to the Dirichlet problem
\begin{gather*}
\Delta_pu = 0 \quad \text{in }  \mathcal{B}_R(x_0) \setminus \mathcal{B}_r(x_0) \\
u = 1 \quad \text{on }  \partial \mathcal{B}_r(x_0) \\
u = 0 \quad \text{on }  \partial \mathcal{B}_R(x_0)
\end{gather*}
for $1< p< \infty$.

We state the following theorem, which follows from the computations of
the previous section.

\begin{theorem} \label{thm5.1}
Let $0 < r < R$ and $1 < p < \infty$.  Then we have
\[
\operatorname{cap}_p\big(\mathcal{B}_r(x_0), \mathcal{B}_R(x_0)\big)
 = \begin{cases}
\alpha^{p-1} Q \sigma_p \big( r^\alpha - R^\alpha \big)^{1-p} &
\text{when }  1 < p < Q \\[3pt]
Q\sigma_Q [\log{R} - \log{r}]^{1-Q} & \text{when } p = Q \\[3pt]
\alpha^{p-1} Q \sigma_p \big( R^\alpha - r^\alpha \big)^{1-p} &
\text{when }  p > Q.
\end{cases}
\]
\end{theorem}

\subsection*{Acknowledgements}
The author wishes to thank the anonymous referee for his/her
careful reading of this manuscript and for the helpful suggestions
to improve its readability and clarity.

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