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

\AtBeginDocument{{\noindent\small \emph{
Electronic Journal of Differential Equations}, 
Vol. 2010(2010), No. 30, pp. 1--6.\newline 
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu
or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/30\hfil Gradient estimation]
{Gradient estimation of a $p$-harmonic map}

\author[B. Wang, L. Ma \hfil EJDE-2010/30\hfilneg]
{Bei Wang, Li Ma} 

\address{Bei Wang \newline
Department of Mathematics, Jiangsu Institute of Education\\
Nanjing, 210013, China}
\email{jsjywang@126.com}

\address{Li Ma \newline
Institute of Science, PLA University of Science and Technology\\
Nanjing, 211101, China} 
\email{mary96@126.com}

\thanks{Submitted June 18, 2009. Published February 26, 2010.}
\subjclass[2000]{35J70, 49J20, 58G18}
\keywords{Gradient estimate; $p$-harmonic map; spherical coordinates}

\begin{abstract}
 This article presents $L^p$ estimates for the gradient of
 $p$-harmonic maps. Since the system satisfies a natural growth
 condition, it is difficult to use standard elliptic estimates.
 We use spherical coordinates to convert the system into another
 system with angle functions. The new system can be
 estimate by the standard elliptic technique.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}

\section{Results}

Let $G \subset \mathbb{R}^n$ ($n\in \{2,3\}$) be a bounded and simply
connected domain with smooth boundary $\partial G$. Denote
$S^{n-1}=\{x \in \mathbb{R}^n: x_1^2+x_2^2+\dots +x_n^2=1\}$. Let $g$
be a smooth map from $\partial G$ into $S^{n-1}$ satisfying
$\deg(g,\partial G)=d=0$. Denote by $\{e_i\}_{i=1}^{n}$ an
orthogonal basis of $\mathbb{R}^n$. We are concerned with the
estimate of the gradient of $p$-harmonic maps on $G$, where $p>2$.

We call $u \in W^{1,p}(G,S^{n-1})$ a $p$-harmonic map on $G$, if
it is a weak solution of (cf. \cite{h1})
\begin{equation}
-\mathop{\rm div}|\nabla u|^{p-2}\nabla u)=u|\nabla u|^p. \label{e1}
\end{equation}
The $L^p$ estimate of the gradient of the weak solutions of
$p$-Laplace system is essential for the better regularity
(cf. \cite{c1,h1,h2,h3,h4,r1,t1}). Thus, in this paper we  prove the
following theorem.


\begin{theorem} \label{thm1}
If $u$ is a $p$-harmonic map on $G$ and $u=g$ on $\partial G$, then
there exists a constant $C>0$ which only depends on $G,g,p,n$,
such that
$$
\|\nabla u\|_{L^p(G)} \leq C.
$$
\end{theorem}

Different from \cite{t1}, it is not easy to estimate the weak solution
since \eqref{e1}  satisfies the natural growth condition.
In \cite{r1}, a sharp Gagliardo-Nirenberg inequality is used for
obtaining  regularity of the $W^{2,p}$-solution.
For the $W^{1,p}$ weak solution, this estimate can not be used.

To prove the main theorem, we should list some preliminaries.

\begin{proposition} \label{prop2}
The $p$-harmonic map $u$ on $G$ satisfies
\begin{equation}
\int_G|\nabla u|^{p-2}(u \wedge \nabla u)\nabla \zeta dx=0, \quad
\forall \zeta \in W_0^{1,p}(G). \label{e2}
\end{equation}
On the contrary, the function $u\in W^{1,p}(G,S^{n-1})$ satisfying
\eqref{e2} must be a $p$-harmonic map on $G$.
\end{proposition}

\begin{proof}
For simplicity, we only calculate formally. Taking the wedge
product \eqref{e1} with $u$, we have
$$
-u\wedge \mathop{\rm div}|\nabla u|^{p-2}\nabla u)=0.
$$
Noting $\nabla u\wedge \nabla u=0$, we have
$$
-\mathop{\rm div}|\nabla u|^{p-2}u\wedge \nabla u)=0.
$$
It is easy to see that it satisfies \eqref{e2}.
On the contrary, if $u\in W^{1,p}(G,S^{n-1})$ satisfies \eqref{e2},
namely
$$
-\mathop{\rm div}|\nabla u|^{p-2}u\wedge \nabla u)=0,
$$
which is equivalent to
$$
-u\wedge \mathop{\rm div}|\nabla u|^{p-2}\nabla u)=0.
$$
This means that there exists $\lambda \in \mathbb{R}$ such that
$$
-\mathop{\rm div}|\nabla u|^{p-2}\nabla u)=\lambda u.
$$
Taking the inner product with $u$ and  noting $|u|=1$, it is not
difficult to deduce that $\lambda =|\nabla u|^p$ a.e. in $G$.
Thus, $u$ is a $p$-harmonic map on $G$.
\end{proof}

\begin{proposition} \label{prop3}
If $n=2$ and $u$ is a $p$-harmonic map on $G$, and $u=g$ on
$\partial G$, then
$$
\|\nabla u\|_{L^p(G)}^p=\min\{\int_G|\nabla u|^pdx, u \in
W^{1,p}(G,S^{n-1}), u|_{\partial G}=g\}.
$$
\end{proposition}

\begin{proof}
When $n=2$, by virtue of $g \in S^{n-1}$ and $\deg(g,\partial
G)=0$, we can write (cf. \cite[Eq. (7)]{b1})
$$
g=\cos\phi_0 e_1+\sin\phi_0 e_2.
$$
Here $\phi_0 \in C^{\infty}(\partial G,[0,2\pi])$ is a
single-valued function. According to \cite[Proposition 2.4]{l3}, we
know that there exists a unique weak solution $\phi$ of the
boundary value problem
\begin{gather}
-\mathop{\rm div}|\nabla \phi|^{p-2}\nabla \phi)=0,\quad\text{in }
 G, \label{e3} \\
\phi|_{\partial G}=\phi_0. \label{e4}
\end{gather}
Set
\begin{equation}
u=\cos\phi e_1+\sin\phi e_2. \label{e5}
\end{equation}
It is not difficult to verify by Proposition \ref{prop2} that $u$ is a weak
solution of \eqref{e1} with $u|_{\partial G}=g$ if and only if $\phi$
in
\eqref{e5} is a weak solution of \eqref{e3} and \eqref{e4}. Therefore, $u$ in \eqref{e5} is
the unique weak solution.

In view of $d=0$, the class $W_g^{1,p}(G,S^{n-1})=\{v\in
W^{1,p}(G,S^{n-1}), u|_{\partial G}=g\}$ is not empty. In fact,
the smooth harmonic map with the boundary value $g$ belongs to
this class. Consider the minimizing problem
$$
\min\{\int_G|\nabla u|^pdx, u \in W_g^{1,p}(G,S^{n-1})\}.
$$
Clearly, the minimizer exists, and it is also a $p$-harmonic map
on $G$. In view of the uniqueness, this minimizer must be $u$ in
\eqref{e5}. It is easy to see our conclusion.
The proof is complete.
\end{proof}

When $n=3$, we can also convert  \eqref{e1} into the form \eqref{e3}.

\begin{proposition} \label{prop4}
Let $n=3$ and $u$ be a $p$-harmonic map on $G$. Then there exist
single-valued functions $\phi_1(x)\in W^{1,p}(G,[0,\pi])$ and
$\phi_2(x)\in W^{1,p}(G,[0,2\pi])$, such that
\begin{equation}
\begin{aligned}
&\int_G|\nabla u|^{p-2}\big[(\cos\phi_2\nabla\phi_1
-\sin\phi_1\cos\phi_1\sin\phi_2\nabla\phi_2)e_1\\
&+(\sin\phi_2\nabla\phi_1
-\sin\phi_1\cos\phi_1\cos\phi_2\nabla\phi_2)e_2\\
&+\sin^2\phi_1\nabla\phi_2e_3]\nabla \zeta dx=0, \quad \forall
\zeta \in W_0^{1,p}(G).
\end{aligned} \label{e6}
\end{equation}
\end{proposition}

\begin{proof}
Since $G$ is a simply connected domain and $|u|=1$, we have the
formula of $3$-dimension spherical coordinates,
$$
u=\cos\phi_1e_1+\sin\phi_1\cos\phi_2e_2 +\sin\phi_1\sin\phi_2e_3.
$$
Here $\phi_j=d\theta+\psi_j$, $j=1,2$. Both $\psi_1\in
W^{1,p}(G,[0,\pi])$ and $\psi_2\in W^{1,p}(G,[0,2\pi])$ are
single-valued functions (cf. \cite{b2,h3}). In view of $d=0$,
$\phi_j=\psi_j$ must be single-valued. By calculation,
\begin{equation} \label{e7}
\begin{gathered}
\begin{aligned}
\nabla u&=-\sin \phi_1 \nabla\phi_1e_1+(\cos
\phi_1\cos\phi_2\nabla
\phi_1-\sin\phi_1\sin\phi_2\nabla\phi_2)
e_2\\
&\quad +(\cos\phi_1\sin\phi_2\nabla\phi_1
+\sin\phi_1\cos\phi_2\nabla\phi_2)e_3;
\end{aligned} \\
|\nabla u|^2=|\nabla \phi_1|^2+\sin^2\phi_1|\nabla\phi_2|^2;
\\
\begin{aligned}
u\wedge \nabla u
&=\sin^2\phi_1\nabla\phi_2 e_1
-(\sin\phi_2\nabla\phi_1
+sin\phi_1\cos\phi_1\cos\phi_2\nabla\phi_2)e_2\\
&\quad +(\cos\phi_2\nabla\phi_1
-sin\phi_1\cos\phi_1\sin\phi_2\nabla\phi_2)e_3.
\end{aligned}
\end{gathered}
\end{equation}
Inserting this result into \eqref{e2} yields our conclusion.
\end{proof}

Different from the single equation \eqref{e3}, Equation \eqref{e6}
is a system when $n=3$. The uniqueness is not true anymore.
The $L^p$ estimate is more complicate than the case $n=2$.
We shall adopt the idea in \cite{l1} to establish this estimate.

\begin{proposition} \label{prop5}
Let $B(y_0,4R)\subset \subset G$, then for any
$\xi\in C_0^{\infty}(B(y_0,3R))$, there holds
$$
\int_{B(y_0,3R)}|\nabla u|^{p-2}\sin^2\phi_1|\nabla
\phi_2|^2\xi^p dy \leq C\Big(\int_{B(y_0,3R)}|\nabla
u|^p\xi^pdy\Big)^{1-\frac{2}{p}}.
$$
\end{proposition}

\begin{proof}
The equality corresponding with the vector $e_1$ in the integral
system \eqref{e6} is
$$
\int_{B(y_0,3R)}|\nabla u|^{p-2}\sin^2\phi_1\nabla\phi_2 \nabla
\zeta dy=0, \quad \forall \zeta\in W_0^{1,p}(B(y_0,3R)).
$$
Letting $\zeta=\phi_2\xi^p$ where $\xi\in
C_0^{\infty}(B(y_0,3R))$, we have
\begin{align*}
&\int_{B(y_0,3R)}|\nabla u|^{p-2}\sin^2\phi_1|\nabla\phi_2|^2
\xi^p dy\\
&\leq |\int_{B(y_0,3R)}|\nabla u|^{p-2}
\sin^2\phi_1(\xi^{p-1}\phi_2) \nabla \phi_2 \nabla \xi dy|.
\end{align*}
Using H\"older's inequality, we obtain that, for any $\delta\in
(0,1)$,
\begin{align*}
&\int_{B(y_0,3R)}|\nabla u|^{p-2}\sin^2\phi_1|\nabla\phi_2|^2
\xi^pdy\\
& \leq \delta\int_{B(y_0,3R)}|\nabla
u|^{p-2}\sin^2\phi_1|\nabla\phi_2|^2
\xi^pdy\\
&\quad +C(\delta)\int_{B(y_0,3R)}|\nabla
u|^{p-2}\sin^2\phi_1|\nabla\xi|^2\xi^{p-2}\phi_2^2dy.
\end{align*}
Letting $\delta$ be sufficiently small, we obtain
\begin{align*}
\int_{B(y_0,3R)}\xi^p|\nabla
u|^{p-2}\sin^2\phi_1|\nabla\phi_2|^2dy
&\leq C\int_{B(y_0,3R)}|\nabla u|^{p-2}\xi^{p-2}dy\\
&\leq C\Big(\int_{B(y_0,3R)}|\nabla u|^p\xi^pdy\Big)^{1-\frac{2}{p}}.
\end{align*}
The proof is complete.
\end{proof}

\begin{proposition} \label{prop6}
Let $B(y_0,4R)\subset \subset G$, then for any
$\xi\in C_0^{\infty}(B(y_0,3R))$, there holds
$$
\int_{B(y_0,3R)}|\nabla u|^{p-2}|\nabla \phi_1|^2\xi^p dy \leq
C\Big(\int_{B(y_0,3R)}|\nabla u|^p\xi^pdy\Big)^{1-\frac{2}{p}}.
$$
\end{proposition}

\begin{proof}
The equalities corresponding with $e_2$ and $e_3$ in \eqref{e6} are
\begin{gather*}
\int_{B(y_0,3R)}|\nabla u|^{p-2}(\sin\phi_2\nabla\phi_1
-sin\phi_1\cos\phi_1\cos\phi_2\nabla\phi_2) \nabla \zeta dy=0,
\\
\int_{B(y_0,3R)}|\nabla u|^{p-2}(\cos\phi_2\nabla\phi_1
+sin\phi_1\cos\phi_1\sin\phi_2\nabla\phi_2) \nabla \zeta
dy=0.
\end{gather*}
Take $\zeta=\phi_1\xi^p\sin\phi_2$ and
$\zeta=\phi_1\xi^p\cos\phi_2$ in two equalities above,
respectively. Then, adding one to the other, we obtain
\begin{equation} \label{e8}
\begin{aligned}
\int_{B(y_0,3R)}|\nabla u|^{p-2}|\nabla\phi_1|^2 \xi^p dy
&\leq \Big[|\int_{B(y_0,3R)}|\nabla u|^{p-2}\phi_1\nabla
\phi_1\nabla\xi^pdy|\\
&\quad +2|\int_{B(y_0,3R)}|\nabla
u|^{p-2}\sin\phi_1(\nabla\phi_1\nabla \phi_2)\xi^pdy|\\
&\quad +2|\int_{B(y_0,3R)}|\nabla
u|^{p-2}\phi_1\sin\phi_1\nabla\phi_2\nabla\xi^p dy|\Big]\\
&\quad +2|\int_{B(y_0,3R)}|\nabla
u|^{p-2}\phi_1\sin\phi_1\cos\phi_1|\nabla\phi_2|^2\xi^pdy|\\
&:=J_1+J_2.
\end{aligned}
\end{equation}
Similar to the proof of Proposition \ref{prop5}, by applying H\"older's
inequality, we also have
\begin{equation}
J_1 \leq \delta \int_{B(y_0,3R)}|\nabla u|^{p-2}|\nabla
\phi_1|^2\xi^pdy+ C(\int_{B(y_0,3R)}|\nabla
u|^p\xi^pdy)^{1-2/p}. \label{e9}
\end{equation}

To estimate $J_2$, we firstly consider $\phi_1\in[0,\pi/2]$.
Since $\lim_{\phi_1\to 0}\frac{\sin\phi_1}{\phi_1}=1$, we
can find $\delta_0>0$ such that as $0<\phi_1<\delta_0$, there
holds $1-\frac{\sin\phi_1}{\phi_1} \leq 1/2$ which means
$\phi_1 \leq 2\sin\phi_1$. When $\delta_0 \leq
\phi_1\leq\pi/2$, there holds $\sin\phi_1 \geq
\sin\delta_0>0$. Thus, by Proposition \ref{prop5},
\begin{align*}
J_2
&\leq 2\int_{B(y_0,3R)\cap [\phi_1<\delta_0]} |\nabla
u|^{p-2}\sin^2\phi_1|\nabla\phi_2|^2\xi^pdy\\
&\quad +\frac{\pi}{2\sin\delta_0}\int_{B(y_0,3R)\cap [\delta_0\leq
\phi_1\leq\pi/2]} |\nabla
u|^{p-2}\sin^2\phi_1|\nabla\phi_2|^2\xi^pdy\\
&\leq C(\int_{B(y_0,3R)}|\nabla u|^p\xi^pdy)^{1-2/p}.
\end{align*}
When $\phi_1 \in [\pi/2,\pi]$, we can replace $\phi_1$ in the
test functions $\zeta$ by $\pi-\phi_1$. we can also deduce the
same result. Substituting this results and \eqref{e9} into \eqref{e8} and
choosing $\delta$ sufficiently small, we can complete the
proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1}]
\emph{Interior estimate.} Combining Propositions \ref{prop5}
 and \ref{prop6}, and
noting \eqref{e7}, we can derive
$$
\int_{B(y_0,3R)}|\nabla u|^p\xi^p dy \leq
C\Big(\int_{B(y_0,3R)}|\nabla u|^p\xi^pdy\Big)^{1-2/p}.
$$
Using Young's inequality, and letting $\xi=1$ on $B(x,2R)$, we can
deduce that
\begin{equation}
\int_{B(y_0,2R)}|\nabla u|^p dy \leq C. \label{e10}
\end{equation}
The interior estimate is obtained.

In the following, we shall investigate the estimation near the
boundary. Let $y_0\in \partial G$. Since $g$, $G$ are smooth and
$d=0$, we can find single-valued functions $\Phi_1 \in
C^{\infty}(\partial G,[0,\pi])$ and $\Phi_2 \in
C^{\infty}(\partial G,[0,2\pi])$, such that
$$
g=\cos\Phi_1e_1+\sin\Phi_1\cos\Phi_2e_2 +\sin\Phi_1\sin\Phi_2e_3.
$$
Since $\partial G$ is smooth, $\Psi_i$ is extended into $G$ (a
neighborhood of $\partial G$). Replacing $\phi_i$ by
$\phi_i-\Phi_i$ in the test function $\zeta$ as we deal with the
interior estimation just now, and arguing as above, we can also
deduce that $\int_{G\cap B(y_0,R)}|\nabla u|^p dy \leq C$, where
$C>0$ only depends on $n,G,R,p$ and $g$. Combining this with (10),
we complete the proof.
\end{proof}

\subsection*{Remark}
Similar to the argument of $n=3$, we can generalize Theorem \ref{thm1} to
the case $n \geq 4$. In fact, we can write a $S^{n-1}$-valued map
$w$ under the spherical coordinates as
\begin{align*}
w&=\cos \theta_1 e_1+\sin
\theta_1\cos\theta_2e_2+\sin\theta_1\sin\theta_2\cos\theta_3e_3
+\dots \\
&\quad +\sin\theta_1\dots \sin\theta_{n-2}\cos\theta_{n-1}e_{n-1}
+\sin\theta_1\dots \sin\theta_{n-2}\sin\theta_{n-1}e_n,
\end{align*}
where $(\theta_1,\dots ,\theta_{n-1})\in
[0,\pi]\times\dots \times[0,\pi]\times[0,2\pi]$, and each
$\theta_i\in W^{1,p}(G)$. Hence, we have a result as \eqref{e7},
\begin{equation}
\begin{aligned}
|\nabla w|^2
&=|\nabla\theta_1|^2+\sin^2\theta_1|\nabla\theta_2|^2+
\sin^2\theta_1\sin^2\theta_2|\nabla\theta_3|^2+\dots \\
&\quad + \sin^2\theta_1\dots \sin^2\theta_{n-2}|\nabla\theta_{n-1}|^2.
\end{aligned} \label{e11}
\end{equation}
Thus, \eqref{e2} becomes a system on $\theta_i (i=1,2,\dots ,n-1)$, which
contains $\frac{n(n-1)}{2}$ single equations. Using the idea in
 \cite[\S2]{l2}, we also estimate $L^{p/2}$-norm of each term of the
right hand side of \eqref{e11} by choosing some equations from the system
properly.

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\end{document}