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\markboth{\hfil Existence and regularity results \hfil EJDE--1998/36}
{EJDE--1998/36\hfil Masashi Misawa\hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 1998}(1998), No.~36, pp. 1--17. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu \quad ejde.math.unt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
Existence and regularity results for the gradient flow
for p-harmonic maps
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35K45, 35K65.
\hfil\break\indent
{\em Key words and phrases:} p-harmonic map, gradient flow, degenerate 
parabolic system.
\hfil\break\indent
\copyright 1998 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Submitted August 29, 1998. Published December 21, 1998.} }
\date{}
\author{ Masashi Misawa}
\maketitle

\begin{abstract} 
We establish existence and regularity for a solution of
the evolution problem associated to p-harmonic maps
if the target manifold has a nonpositive sectional curvature.
\end{abstract}

\makeatletter
\renewcommand{\theequation}{\thesection.\arabic{equation}}\@addtoreset{equation}{section}
\makeatother
%
\newtheorem{lm}{Lemma}[section]
\newtheorem{thm}{Theorem}[section]


\section{Introduction}

Let $M$ and $N$ be compact, smooth Riemannian manifolds without boundary, of
dimensions $m$ and $k$, with metrics $g$ and $\gamma$, respectively.
Since $N$ is compact, by Nash's embedding theorem
we can regard $N$ as being isometrically embedded in
a Euclidean space ${\mathbb R}^n$ for some $n$.
For a $C^1-$map $u:M \rightarrow N \subset R^n$,
we define the $p$-energy $E(u)$ by
\begin{equation}
E(u) = \int_M
\mbox{\small $\frac{1}{p}$}|Du|^p dM, \quad p \ge 2,
\label{original-func}
\end{equation}
where,
in local coordinates on $M$,
$$
dM = \sqrt{|g|} dx,
\quad|Du|^2 =\sum_{\alpha, \beta = 1}^m
\sum_{i = 1}^n
g^{\alpha \beta}
D_\alpha u^i
D_\beta u^i,
$$
with
$\left(g^{\alpha \beta}\right) = (g_{\alpha \beta})^{-1}$,
$|g| = |\det (g_{\alpha \beta})|$
and
$D_\alpha = \partial/\partial x^\alpha$,
$\alpha = 1, \cdots,  m$.


The Euler-Lagrange equation of the $p$-energy is
%
\begin{equation}
- \triangle_p u + A_p (u) (Du, Du) = 0\,,
\label{euler-eqn}
\end{equation}
%
where $\triangle_p$ denotes the $p$-Laplace operator
$$
\triangle_p u =\mbox{\small
$\frac{1}{\sqrt{|g|}}$
}
D_\alpha
\biggl(
\sqrt{|g|} g^{\alpha \beta}
|D u|^{p - 2}
D_\beta u
\biggr)
$$
on $M$, which is a degenerate elliptic operator,
and where
$A_p(u)(D u, D u)$ is given by
$$
A_p(u) (Du, Du)=|D u|^{p - 2}g^{\alpha \beta}
A(u)(D_\alpha u, D_\beta u)
$$
in terms of the second fundamental form $A(u)(D u, D u)$ of 
$N$ in ${\mathbb R}^n$ at $u$.

Here and in what follows, the summation notation
over repeated indices is adopted.

We call (weak) solutions of (\ref{euler-eqn})
(weakly) $p$-harmonic maps.

One method to look for $p$-harmonic maps is
to exploit the gradient flow
related to the $p$-energy,
which is called $p$-harmonic flow.
The gradient flows are described by a system of
second order nonlinear degenerate parabolic partial differential equations
%
\begin{eqnarray}
&\partial_t u  - \triangle_p u + A_p(u)(Du, Du) = 0
\quad\mbox{in}\quad(0, \infty) \times M,
\label{eqn}\\
& u(0, x) = u_0(x)
\quad\mbox{for $x \in M$.}
\label{init}
\end{eqnarray}

For $p = 2$, Eells and Sampson showed in \cite{eells-sam}
that there exists a global smooth solution
provided that the target manifold $N$ has nonpositive
sectional curvature and that the solution converges to
a harmonic map suitably as $t_k \rightarrow \infty$.
This result concerns the homotopy problem,
that is, to find a harmonic map homotopic to a given map.
When the target manifold $N$ is of non-positive sectional curvature
and $p > 2$, the homotopy problem was solved by
Duzzar and Fuchs \cite{duzzar-fuchs}
by applying the direct method in the calculus of variations
for the regularized $p$-energy functional
(see (\ref{reg-ene}) below)
and using $C^1_\alpha-$estimates for solutions of
the Euler-Lagrange equation (\ref{euler-eqn}).
In this paper we establish the global existence and
$C^{0, 1}_\alpha -$regularity of a weak solution to
the $p$-harmonic flow provided that the target manifold $N$ has
non-positive sectional curvature.
The regularity of weak solutions of
degenerate parabolic systems with only principal terms
was discussed and the $C^{0,1}_\alpha -$regularity of solutions
was established in\cite{chen-dibene,dibene-fried1,dibene-fried2,dibene-fried3}.
(Also see \cite{choe1,choe2,tolks,uhlen}
for corresponding elliptic systems.)
The global existence of a weak solution to
the p-harmonic flow was shown when
the target manifold is a sphere in \cite{chen-hong},
and, more generally, a homogeneous space in \cite{hunger2,hunger3}.  
For $p = m$, the global existence of a partial $C^{0, 1}_\alpha -$
weak solution was established in \cite{hunger4}.
For the regularity of harmonic maps and flows,
we refer to \cite{schoen-uhlen,giaq-giusti,struwe,chen-struwe}.

To state our results, we need some preliminaries.
Let us define the metric $\delta_q$, $q \ge 1$,
by
$$
\delta_q(z_1, z_2)
=\max \{ |t_1 - t_2|^{1/q}, |x_1 -x_2| \}
$$
for any
$z_i = (t_i, x_i)$
$\in (0, \infty) \times R^m$,
$i = 1, 2$.
If $q = 2$, the metric $\delta_2$ is the usual parabolic metric.
For a bounded domain $\Omega \subset R^m$,
we use the usual function spaces
$C^k_\alpha(\Omega, R^n)$,
$L^q(\Omega, R^n)$
and
$W^1_q(\Omega, R^n)$.
For any $T > 0$, denote by 
$C^{\alpha/q, \alpha}([0, T] \times \Omega, R^n)$
the space of functions
defined on $[0, T] \times \Omega$
with values in ${\mathbb R}^n$, H\"older continuous 
with respect to the metric $\delta_q$
with an exponent $\alpha$,
$0 < \alpha < 1$.
In particular,
$C^{1/q, 1}([0, T] \times \Omega, R^n)$
is the space of functions
with values in ${\mathbb R}^n$ that are Lipschitz continuous 
with respect to the metric $\delta_q$.
We also use the notation
%
\begin{eqnarray*}
&
C^{1, 2}_\alpha([0, T] \times \Omega, R^n)
=C^0_{\alpha/2}([0, T]; C^2_\alpha(\Omega, R^n))
\cap
C^1_{\alpha/2}([0, T]; C^0_\alpha(\Omega, R^n)),
\\
&C^{0, 1}_\alpha([0, T] \times \Omega, R^n)
=C^0_{\alpha/2}([0, T]; C^1_\alpha(\Omega, R^n)).&
\end{eqnarray*}
%
If the domain is a compact, smooth Riemannian manifold $M$,
then, for
$z_i = \left(t_i, x_i\right)
\in \left(0, \infty\right) \times M$,
$i = 1, 2$,
we replace the metric $\delta_q$,
$q \ge 1$,
by
$$
\max \left\{\left|t_1 - t_2\right|^{1/q},
{\mbox{dist}}_M\left(x_1, x_2\right)
\right\}\,,$$
where ${\mbox{dist}}_M\left(x_1, x_2\right)$
means the geodesic distance of $x_1, x_2 \in M$
with respect to the metric $g$
on the manifold $M$,
and we
define $C^k_\alpha(M, R^n)$,
$C^{1/q, 1}_\alpha([0, T] \times M, R^n)$,
$C^{\alpha/q, \alpha}_\alpha([0, T] \times M, R^n)$,
$C^{1, 2}_\alpha([0, T] \times M, R^n)$
and
$C^{0, 1}_\alpha([0, T] \times M, R^n)$
to be the spaces of functions
belonging to the corresponding spaces above
with $\Omega = U$ 
for any local coordinate neighborhood $U$ on $M$.
%
We now define a set of Sobolev mappings from $M$ to $N$,
which is called the energy space:
%
\[
W^{1, p}(M, N)
=\{
u \in W^{1, p}(M, R^n):
u(x) \in N
\quad\mbox{for almost all $x \in M$}
\},
\]
equipped with the topology inherited from
the one of the linear Sobolev spaces
$W^{1, p}(M, R^n)$.

We are interested in a global weak solution
$u \in$
$L^\infty((0, \infty); W^{1, p}$
$(M, N))$
$\cap W^{1, 2}((0, \infty);L^2(M, R^n))$
of (\ref{eqn}) and (\ref{init}), satisfying,
for all \newline
$\phi \in
L^{p'}((0, \infty); W^{1, p'}(M, R^n))
\cap
L^\infty (
$
$
(0, \infty)
$
$
\times M, R^n)
$
with $p'$ the dual exponent of $p$,
the support of which is compactly contained in
$(0, \infty) \times U$
for a coordinate chart $U$ on $M$,
%====================%
\begin{equation}
\int_{ (0, \infty) \times M}
\left\{
\phi \cdot
\partial_t u
+ |D u|^{p - 2}
g^{\alpha \beta}
D_\beta u
\cdot
D_\alpha \phi
+ \phi
\cdot
A_p(u) (D u, D u)
\right\}
\,dM\,dt
=0,
\label{w-eqn}
\end{equation}
%
and satisfying the initial condition
%
\begin{equation}
|u(t) - u_0|_{L^2(M)}
\rightarrow 0,
\quad t \rightarrow 0.
\label{init-cond2}
\end{equation}

Our main theorem is the following:

\begin{thm}
\label{main-th}
Assume that
the sectional curvature of the target manifold $N$
is nonpositive.
Let $u_0 \in C^2_\beta(M, N)$
with $0 < \beta < 1$,
the image of which is contained in
a geodesic ball ${\cal B}\left(a_0\right)$
in $N$ around a point $a_0 \in N$.
Then there exists a global weak solution
$u \in $
$L^\infty((0$,
$\infty); W^{1, p}(M, N))$
$\cap$
$W^{1, 2}((0, \infty);L^2(M, R^n))$
with the energy inequality
%
\begin{equation}
\int_{(0, T) \times M}
|\partial_t u|^2
dM dt
+ \sup_{0 \le t \le T}
E(u(t))
\le
E(u_0)
\quad\mbox{for all $T > 0$}.
\label{energy}
\end{equation}
%
Moreover,
for a positive number
$\alpha, 0 < \alpha < 1$,
$
u \in
C^{\alpha/p, \alpha}_{\mbox{\rm loc}} ((0, \infty) \times M, R^n)
$
and
$
D u \in
C^{\alpha/2, \alpha}_{\mbox{\rm loc}} ((0, \infty) \times M, R^n).
$
%
\end{thm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The regularized $p$-energy}
%
%
First we will make a special isometric embedding of
$(N^k, \gamma)$ in
$(R^n, h)$.
(Refer to \cite{hunger4}.)
Let us define a metric $h$ as follows.
%
%====new metric======%
Since $N$ is compact, we can use
the standard Nash embedding of $N$ in ${\mathbb R}^n$
and choose a tubular neighborhood
${\cal O}_{2 \delta}(N) \subset R^n$
of $N$
such that
${\cal O}_{2 \delta}(N) = \{
x \in R^n: \mbox{dist}(x, N) < 2 \delta
\}$,
where $\delta$ is a sufficiently small positive constant,
and $\mbox{dist}$ is the usual Euclidean distance.
Then let us put
%
$({\widetilde \gamma}_{ij})
=(\gamma_{ij})
\otimes
(\delta_{ij})
$
locally
on
$N \times B^{n - k}_{2 \delta}$,
where
$B^{n - k}_{2 \delta}$
is a ball in ${\mathbb R}^{n - k}$
with a radius $2 \delta$.
We can extend
${\widetilde \gamma}_{ij}$
smoothly
to ${\mathbb R}^n$
by defining
$h_{ij}
=\phi {\widetilde \gamma}_{ij}
+ (1 - \phi) \delta_{ij}
$
for
$\phi \in C^\infty_0(R^n, R)$
with support in
${\cal O}_{2 \delta}(N)$
and
$\phi \equiv 1$
on
${\cal O}_\delta(N)$.
%
By such an embedding of $N$ into ${\mathbb R}^n$,
we have an involutive isometry $\pi$
from a tubular neighborhood ${\cal O}_\delta$
to itself,
which has exactly the target manifold $N$
for its fixed points.

For $u \in R^n$, let
%
\begin{equation}
\Gamma_{ik}^l(u)
=\mbox{\small $\frac{1}{2}$}
h^{ij}
\biggl(
\mbox{\small $\frac{d h_{jk}}{d u^i}(u)$}
-\mbox{\small $\frac{d h_{ik}}{d u^j}(u)$}
+ \mbox{\small $\frac{d h_{ij}}{d u^k}(u)$}
\biggr),
\quad\left(h^{i j}\right)
=\left(h_{i j}\right)^{- 1},
\end{equation}
be the Christoffel symbol for the metric
$(h_{ij})$.
%
%=======owari new metric=========
For $\epsilon > 0$,
the regularized $p$-energy
(refer to \cite{duzzar-fuchs}, \cite{hunger4}) of
a map
$u:(M, g) \rightarrow (R^n, h)$
is defined by
%
\begin{equation}
E_\epsilon(u)
=\int_M
e_\epsilon(u)
dM,
\quad e_\epsilon(u)
=\mbox{\small $\frac{1}{p}$}
\left(
\epsilon + |D u|^2
\right)^{
\frac{p}{2}
},
\label{reg-ene}
\end{equation}
%
where, in local coordinates
$( x^\alpha )$
of $M$
and
$( u^i )$
of ${\mathbb R}^n$,
%
\begin{equation}
|D u|^2
=g^{\alpha \beta}(x)
h_{i j}(u)
D_\alpha u^i
D_\beta u^j.
\label{dens-dirich}
\end{equation}
%
We consider the gradient flow for $E_\epsilon$,
described by the parabolic system
%
\begin{equation}
\partial_t u
-\Delta_p^\epsilon u
-\Gamma_p^\epsilon(u)(Du, Du)
= 0,
\label{reg-eqn}
\end{equation}
%
where, in local coordinates of $M$ and ${\mathbb R}^n$,
%
\begin{eqnarray}
&&\Delta_p^\epsilon u
=\mbox{\small $\frac{1}{\sqrt{|g|}}$}
D_\alpha
\left(
(\epsilon + |D u|^2)^{
\frac{p}{2} - 1
}
\sqrt{|g|}
g^{\alpha \beta}
D_\beta u
\right),
\nonumber
\\
&&\Gamma_p^\epsilon(u)(Du, Du)
=(\epsilon + |D u|^2)^{
\frac{p}{2} - 1
}
g^{\alpha \beta}
\Gamma_{i j}^l(u)
D_\alpha u^i
D_\beta u^j.
\end{eqnarray}
%
\par
\parindent0pt
%
Recall that
$u_0$ is a member of
$C^2_\beta(M, N)$,
$0 < \beta < 1$, and has image
in the geodesic ball ${\cal B}\left(a_0\right) \subset N$
around the point $a_0 \in N$.
Let us consider
the initial value problem for the equation
(\ref{reg-eqn})
with
(\ref{init}).
We apply the Leray-Schauder fixed point theorem
to show the existence of
a solution $u_\epsilon$
to the problem
for any $\epsilon$, $0 < \epsilon < 1$.
%
%
\par
\parindent0pt
%
For this purpose we introduce the linearized parabolic system:
Let us take $T > 0$ arbitrarily.
For any $\tau$,
$0 \le \tau \le 1$,
and
$w \in$
$C^{0, 1}_\alpha([0, T] \times M, R^n)$,
we find a classical solution
$u \in $
$C^{1, 2}_\alpha([0, T] \times M, R^n)$
of the linear parabolic system
%
\begin{eqnarray}
&
\partial_t u^i
=A_{i j}^{\alpha \beta}(t, x)
D_\alpha D_\beta
u^j
+ B_{i j}^\beta(t, x)
D_\beta u^j
\quad\mbox{in $(0, T) \times M$},
\quad i = 1, \cdots, n,&
\nonumber \\
& u =  \exp_{a_0} \left(
\tau \exp_{a_0}^{- 1}\left(u_0\right)
\right)
\quad\mbox{on $\{ t = 0 \} \times M$,}&
\label{lin-eq}
\end{eqnarray}
%
where $\exp_{a_0} \left(\cdot\right)$
is the exponential map defined on a Euclidean ball
$B(0) \subset R^k$ around the origin
with values in ${\cal B}\left(a_0\right) \subset N$,
and the coefficients are, in local coordinates
of $M$ and ${\mathbb R}^n$,
%
\begin{eqnarray}
%==========coefficient of D^2 u==========
A_{i j}^{\alpha \beta}(t, x)
&=&(p e_\epsilon(w))^{1 -\frac{2}{p}
}
\left(
g^{\alpha \beta}
\delta_{i j}
+ (p - 2)
\mbox{\small
$\frac{
g^{\beta \nu}
D_\nu w^k
h_{j k}(w)
g^{\alpha \mu}
D_{\mu} w^i
}
{
\left(p e_\epsilon(w)\right)^{
\frac{2}{p}}
}
$
}
\right),
\nonumber
\\
%======coefficient of Du==========
B_{i j}^\beta(t, x)
&=&\delta_{i j}
(p e_\epsilon(w))^{1 -\frac{2}{p}
}
\left\{
\mbox{\small
$\frac{1}{\sqrt{|g|}}$
}
D_\alpha
\left(
\sqrt{|g|}
g^{\alpha \beta}
\right)
\right.
\nonumber
\\
%==========%2=============
&&+\left. \left(
\mbox{\small $\frac{p}{2}$}
- 1
\right)
\mbox{\small
$\frac{g^{\alpha \beta}
D_\mu w^k
D_\nu w^l}
{
\left(p e_\epsilon(w)\right)^{\frac{2}{p}}
}
$}
\left(
\mbox{\small
$\frac{d g^{\mu \nu}
}{
d x^\alpha
}(x)
$
}
h^{k l}(w)
+ g^{\mu \nu}
D_\alpha w
\cdot
\mbox{\small
$\frac{d h^{k l}
}
{d u}(w)
$
}
\right)
\right\}
\nonumber
\\
%========% 3===============
&&+\left(p e_\epsilon(w)\right)^{1 -\frac{2}{p}
}
g^{\alpha \beta}
\Gamma_{j k}^i(w)
D_\alpha w^k.
\label{lin-coef}
\end{eqnarray}
%
The equation (\ref{lin-eq}) is written as
%
\begin{equation}
h_{i l}(w)
\partial_t u^i
=h_{i l}(w)
A_{i j}^{\alpha \beta}(t, x)
D_\alpha D_\beta
u^j
+ h_{i l}(w)
B_{i j}^\beta(t, x)
D_\beta u^j,
\end{equation}
%
in which
%
\begin{eqnarray*}
\lefteqn{ h_{i l}(w)
A_{i j}^{\alpha \beta}(t, x) }\\
&=& (p e_\epsilon(w))^{1 -\frac{2}{p}}
\left(g^{\alpha \beta}h_{j l}(w)
+(p - 2) \mbox{\small $\frac{ g^{\beta \nu} D_\nu w^k h_{j k}(w)
g^{\alpha \mu} D_{\mu} w^i h_{i l}(w)}
{
\left(p e_\epsilon(w)\right)^{
\frac{2}{p}}}$}\right),
\end{eqnarray*}
%
which is a positive definite matrix.
%
Here we note the relation for the principal term of
(\ref{reg-eqn}) with $0 \le \epsilon < 1$:
%
\begin{eqnarray*}
\lefteqn{
\left(
\Delta_p u^j + \left(\Gamma_p^\epsilon(u)(D u, Du)\right)^j
\right)
h_{i j}(u) }  \\
&=& \mbox{\small
$\frac{1}{
\sqrt{\left|g\right|}
}$
}
D_\alpha \left(
\left(p e_\epsilon\left(u\right)\right)^{1 -\frac{2}{p}
}
\sqrt{\left|g\right|}
g^{\alpha \beta}
h_{i j}(u)
D_\beta u^j
\right)  \\
&&-\mbox{\small$\frac{1}{2}$}
\left(p e_\epsilon\left(u\right)\right)^{1 -\frac{2}{p}}
g^{\alpha \beta}
\mbox{\small
$\frac{d h_{j k}}{d u^i} (u)$
}
D_\alpha u^jD_\beta u^k.
\end{eqnarray*}
%
We fix an ``approximating number'' $\epsilon$,
$0 < \epsilon < 1$.
We define an operator
$P$:
$[0, 1] \times C^{0, 1}_\alpha([0, T] \times M, R^n)$
$\ni (\tau, w)$
$\mapsto$
$u = P(\tau, w)$
$\in C^{0, 1}_\alpha([0, T] \times M, R^n)$
such that
$u = P(\tau, w)$
is a classical solution to
(\ref{lin-eq}).
%
The exponent $\alpha$, $0 < \alpha < 1$,
will be stipulated later.

To exploit the Leray-Schauder fixed point theory,
we have to verify the following conditions:
\begin{enumerate}
\item  There exists a unique classical solution to (\ref{lin-eq}),
which implies that the operator $P$ is well-defined.

\item The operator $P$ is continuous and compact on
$[0, 1] \times C^{0, 1}_\alpha([0, T] \times M, R^n)$.

\item If $\tau = 0$, there exists a unique solution
determined uniformly on all
$w \in C^{0, 1}_\alpha([0, T] \times M, R^n)$.

\item Fixed points $u_\tau$ of the operator $P(\tau, \cdot)$,
which are solutions to the equation with $w = u_\tau$
in (\ref{lin-eq}), are uniformly bounded in
$C^{0, 1}_ \alpha([0, T] \times M, R^n)$
with respect to $\tau$, $0 \le \tau \le 1$ (and $\epsilon$, $0 < \epsilon < 1$).
\end{enumerate}

In the following sections, we will show the validity of the above statements.

%
%===============Linearized parabolic system=============%
%
\section{Linearized parabolic system}
%
In this section, we prove the existence of
a classical solution to the linearized parabolic system (\ref{lin-eq}),
and show that the corresponding operator $P$ is continuous and compact.

Let the exponent $\alpha$ be $0 < \alpha \le \beta$,
where $\beta$ is a H\"older exponent of
the initial value $u_0$.
%
\begin{lm}
\label{ex-lin}
There exists a unique classical solution to the linearized parabolic system 
(\ref{lin-eq}).
\end{lm}
%
%
Noting (\ref{lin-coef}), we immediately see that
the coefficients $A_{i j}^{\alpha \beta}$
and $B_{i j}^\alpha$, $\alpha, \beta = 1, \cdots, m;$
$i, j = 1, \cdots, n$,
are H\"older continuous in $[0, T] \times M$
with the exponent $\alpha$
and the H\"older constant depending only on
$\left(g^{\alpha \beta}\right)$,
$\left(h_{i j}\right)$,
$\epsilon, p$
and $\left|w\right|_{C^{0, 1}_\alpha}$,
and that
%
\begin{equation}
\epsilon^{
\frac{p}{2} - 1}
\left|\xi\right|^2
\le
A_{i j}^{\alpha \beta}
\xi^j_\beta
\xi^k_\alpha
h_{k i}(w)
\le
\mbox{\small
$\left(
\epsilon + \sup_{[0, T] \times M} \left|D w\right|^2
\right)^{\frac{p}{2} - 1}
$}
\left|\xi\right|^2
\end{equation}
%
holds for any
$\left(t, x\right) \in
[0, T] \times M$
and
$\xi = \left(\xi^i_\alpha\right)
\in R^{m n}$,
where
$$\left|\xi\right|^2
=\sum_{\alpha = 1}^m
\sum_{i = 1}^n
(\xi_\alpha^i)^2\,.$$
The parabolic system of the same type as (\ref{lin-eq})
is investigated in \cite{misawa}
and the maximum principal for a classical solution
is obtained.
By combination of it
with the Schauder estimates in \cite{schlag}(see \cite{misawa}),
we have the uniform boundedness in
$C^{1, 2}_\alpha([0, T] \times M, R^n)$
for classical solutions $u$:
%
\begin{equation}
|u|_{C^{1, 2}_\alpha}
\le
\gamma
\left(
|f|_{C^{\alpha/2, \alpha}}
+ |u_0|_{C^2_\alpha}
\right),
\label{max-schau}
\end{equation}
%
where a positive constant $\gamma$ depends only on
the H\"older constant of
$\left(A^{\alpha \gamma}_{j l}\right)$
and
$\left(B^\beta\right)$
and hence $\gamma$
depends on $p, \epsilon$
and $\left|w\right|_{C^{0, 1}_\alpha}$.
%
Thus we conclude the following result.
%
\begin{lm}
\label{Max-Schau}
Let $u \in C^{1, 2}_\alpha([0, T] \times M, R^n)$
be a solution to the parabolic system (\ref{lin-eq}).
Then there exists a positive constant
$\gamma$ depending only on
$|w|_{C^{\alpha/2, \alpha}}$,
$|u_0|_{C^2_\alpha}$,
$\epsilon, p$,
$(g_{\alpha \beta})$
and
$(h_{i j})$
such that
%
\begin{equation}
|u|_{C^{1, 2}_\alpha}
\le
\gamma.
\end{equation}
%
\end{lm}
%
As in \cite{misawa},
we can prove the existence of a classical solution
of (\ref{lin-eq}).
 
Now we prove the continuity and compactness of the operator
$P$.
%
\begin{lm}
The operator $P$
is continuous and compact in
$[0, 1] \times C^{0, 1}_\alpha([0, T]\times M, R^n)$.
\end{lm}

\paragraph{Proof.} ({\bf Compactness})
For all
$w \in X := C^{0, 1}_\alpha([0, T] \times M, R^n)$
such that
$|w|_X \le U$
with a uniform positive constant $U$,
and all $\tau$, $0 \le \tau \le 1$,
let $u = P(\tau, w)$.
Then, by Lemma~\ref{Max-Schau},
we have
%
\begin{equation}
|u, D u, D^2 u, \partial_t u|_{
C^{\alpha/2, \alpha}
}
\le \gamma,
\end{equation}
with a positive constant $\gamma$
depending only on
$U$,
$|u_0|_{C^2_\alpha}$,
$\epsilon$ and $p$.
%
Here we note that
the coefficients in (\ref{lin-coef})
are Lipschitz continuous in $w$ and $D w$
with a Lipschitz constant depending on $\epsilon$.
By the uniform boundedness of
$D^2 u$ and $\partial_t u$,
we can apply Lemma $3.1$ in \cite[pp.78-9]{lady} 
with $\alpha = \beta = 1$ to find that
$|D u|_{C^{1/2, 1}([0, T] \times M)}$
is uniformly bounded.
The family $\{ u \}$ of such functions
is actually a compact set in $X$,
since $\alpha < 1$.
Consequently, the operator
$P(\tau, \cdot)$, $0 \le \tau \le 1$,
maps a bounded set in $X$
into a compact set in $X$.

({\bf Continuity})
Take $w_1, w_2 \in X$ satisfying, for $\delta > 0$,
%
\begin{equation}
|w_1 - w_2|_X \le \delta
\end{equation}
%
and let
$u_1 = P(\tau, w_1)$ and $u_2 = P(\tau, w_2)$
for any $\tau$,
$0 \le \tau \le 1$.
Subtract the equation for $u_1$ from the one for $u_2$
to obtain, for $u = u_2 - u_1$,
%
\begin{equation}
\partial_t u
=A(x, w_2, D w_2) \cdot D^2 u + B(x, w_2, D w_2) \cdot D u
+ F(t, x),
\end{equation}
%
where
$A(x, v, D v)$
and
$B(x, v, D v)$
are
$\left(A^{\alpha \gamma}_{j l}\right)$
and
$\left(B^\beta\right)$
in (\ref{lin-coef})
with
$w = v$,
respectively, and
%
\begin{eqnarray*}
F(t, x) &=&\left(
A(x, w_2, D w_2) - A(x, w_1, D w_1)
\right) \cdot D^2 u_1 \\
&&+\left(
B(x, w_2, D w_2) - B(x, w_1, D w_1)\right) \cdot D u_1\,.
\end{eqnarray*}
%
%===dai
Noting the Lipschitz continuity
in the variables $w, D w$ of the coefficients \newline
$A(x, w, D w)$
and
$B(x, w, D w)$,
we obtain, from (\ref{max-schau}),
%
\begin{equation}
|u|_{C^{1, 2}_\alpha}
\le
\gamma
|F|_{
C^{\alpha/2, \alpha}
},
\label{Schau0}
\end{equation}
%
where
we note that $u = 0$
on $\{t = 0 \} \times M$,
and that the positive constant $\gamma$
is determined by
$|A|_{C^{\alpha/2, \alpha}}$
and
$|B|_{C^{\alpha/2, \alpha}}$,
and hence
$\gamma$ depends only on
$
|w_2|_{C^{0, 1}_\alpha},
\epsilon,
$
$(g_{\alpha \beta})$
and
$(h_{i j})$.
%
$F$ is estimated from above by
%
\begin{equation}
|F|_{
C^{\alpha/2, \alpha}
}
\le
\gamma
|w_1 - w_2|_X,
\end{equation}
%
where the positive constant
$\gamma$
depends only on
$|D u_1|_{C^{\alpha/2, \alpha}}$,
$|D^2 u_1|_{C^{\alpha/2, \alpha}}$,
$\epsilon$,
$(g_{\alpha \beta})$
and
$(h_{i j})$.
Thus, we choose a positive constant
$\gamma$
depending only on
$|w_1|_X$,
$|u_0|_{C^2_\alpha}$,
$\epsilon$,
$(g_{\alpha \beta})$
and
$(h_{i j})$
such that
%
\begin{equation}
|u_1 - u_2|_X
\le |u|_{
C^{1, 2}_\alpha
}
\le
\gamma
\delta.
\end{equation}
%
As above, we can verify that
$P(\tau, w)$
is continuous on $\tau$
for each $w \in X$:
%
For $\tau_1, \tau_2$,
$0 \le \tau_1, \tau_2 \le 1$,
we put
$u_1 = P\left(\tau_1, w\right)$
and
$u_2 = P\left(\tau_2, w\right)$
for fixed $w \in X$.
Then $u = u_2 - u_1$ satisfies the equation
%
\begin{eqnarray}
& \partial_t u
=A(x, w, D w)
\cdot D^2 u
+ B(x, w, D w)
\cdot
D u
\quad\mbox{in $[0, T] \times M$}, &\nonumber\\
&u(0) =\exp_{a_0}\left(
\tau_2
\exp_{a_0}^{- 1} \left(u_0\right)
\right)
-\exp_{a_0}\left(
\tau_1
\exp_{a_0}^{- 1} \left(u_0\right)
\right). &
\label{toch20-lin}
\end{eqnarray}
%
Noting the definition of the exponential map
$\exp_{a_0}\left(\cdot\right)$,
we have, with a positive constant
$\gamma$ depending only on
$\left(h_{i j}\right)$,
%
\begin{equation}
\left|u(0)\right|_{C^2_\alpha}
\le
\gamma
\left|\tau_2 - \tau_1\right|
\left|u_0\right|_{C^2_\alpha}.
\label{toch21-lin}
\end{equation}
%
Applying Schauder estimates
(\ref{max-schau}) and (\ref{toch21-lin})
for (\ref{toch20-lin}),
we obtain
%
\begin{equation}
\left|u\right|_{C^{1, 2}_\alpha}
\le
\gamma
\left|\tau_2 - \tau_1\right|
\left|u_0\right|_{C^2_\alpha},
\end{equation}
%
where the positive constant $\gamma$
depends only on
$p, \epsilon$,
$\left|w\right|_{C^{0, 1}_\alpha}$
and $\left(h_{i j}\right)$.
%
Consequently, we find
that the operator $P$
is continuous in
$[0, 1] \times X$.
 
We now consider the case $\tau = 0$.
If $\tau = 0$, then,
for any $w \in X$,
$u = P(0, w)$ is a solution of
(\ref{lin-eq})
with the initial condition
%
\begin{equation}
u = a_0
\quad\mbox{on $\{t = 0\} \times M$}.
\nonumber
\end{equation}
%
By the uniqueness of the solution
of (\ref{lin-eq}) with this initial condition,
$P\left(0, w\right) = a_0$
for all $w \in X$.
Thus, $P(0, \cdot)$ maps all $w \in X$
into the constant map $a_0$.

%===============A Priori estimates==========Boundedness of D u=========%

\section{Uniform boundedness of $D u$}

Now we consider a priori estimates for fixed points
of the operator $P(\tau, \cdot)$,
$0 \le \tau \le 1$,
which are solutions to the parabolic system
%
\begin{eqnarray}
\partial_t u
&=&
\mbox{\small 
$\frac{1}{\sqrt{|g|}}$
}
D_\alpha
\left(
(p e_\epsilon(u))^{1 -\frac{2}{p}
}
\sqrt{|g|}
g^{\alpha \beta}
D_\beta u
\right)
\nonumber
\\
& &
\quad
+
(p e_\epsilon(u))^{1 -\frac{2}{p}
}
g^{\alpha \beta}
\Gamma_{i j}(u)
D_\alpha u^i
D_\beta u^j
\quad
\mbox{in $
(0, T] \times M,
$
}
\label{quasi-eq}
\\
u
&=&
\exp_{a_0}\left(\tau \exp_{a_0}^{- 1}(u_0)\right)
\quad\mbox{on $\{ t = 0 \} \times M$}.
\end{eqnarray}
%
First we establish an energy inequality
for solutions of (\ref{quasi-eq}).
%
\begin{lm}
\label{ene-inequ}
Let $u \in C^{1, 2}_0([0, T] \times M, R^n)$
be a solution to (\ref{quasi-eq}).
Then the energy inequality
%
\begin{equation}
\int_{(t_0, t_1) \times M}
|\partial_t u|^2
dM dt
+ E_\epsilon (u(t_1))
\le
E_\epsilon (u(t_0))
\label{ene-ineq}
\end{equation}
%
holds for all $t_0, t_1$,
$0 \le t_0 < t_1 \le T$.
\end{lm}

\paragraph{Proof.}
We multiply (\ref{quasi-eq}) by $h_{i j}(u) \partial_t u^i$.
For the right hand side of the resulting equality, we use
(refer to \cite[pp.558-9, pp.564-5]{struwe1})
%
\begin{eqnarray}
\lefteqn{
\mbox{\small
$\frac{1}{\sqrt{|g|}}$
}
D_\alpha
\left(
(p e_\epsilon(u))^{1 -\frac{2}{p}
}
\sqrt{|g|}
g^{\alpha \beta}
D_\beta u^j
\partial_t u^i
h_{i j}(u)
\right) } \nonumber \\
&=&\mbox{\small
$\frac{1}{\sqrt{|g|}}$
}
D_\alpha
\left(
(p e_\epsilon(u))^{1 -\frac{2}{p}
}
\sqrt{|g|}
g^{\alpha \beta}
D_\beta u^j
\right)
\partial_t u^i
h_{i j}(u)  \label{ene-toch1} \\
&&+ (p e_\epsilon(u))^{1 -\frac{2}{p}
}
g^{\alpha \beta}
D_\beta u^j
D_\alpha
\left(
\partial_t u^i
h_{i j}(u)
\right) \nonumber\\
&=&\partial_t e_\epsilon(u)
+ \left(
\Delta_p^\epsilon u^j
+ (p e_\epsilon(u))^{1 -\frac{2}{p}
}
\Gamma^j(u)
(D u, D u)
\right)
\partial_t u^i
h_{i j}(u). \nonumber
\end{eqnarray}
%
Integrate (\ref{ene-toch1}) on $[t_0, t_1] \times M$
to obtain
$$ \int_{(t_0, t_1) \times M}
h_{i j}(u)
\partial_t u^i
\partial_t u^j
dM dt
+ \int_M
\left\{
e_\epsilon(u(t_1))
-e_\epsilon(u(t_0))
\right\}
dM
= 0
$$ 
and hence the desired estimate.
In particular, noting that
$D u(0) = \tau D u_0$ in $M$,
we have obtained (\ref{ene-ineq}) with
$E_\epsilon\left(u(t_0)\right)$
replaced by
$E_\epsilon\left(\tau u_0\right)$
for all $t_1$, $0 \le t_1 \le T$.

\begin{lm} \label{bound}
Let $u \in C^{1, 2}_0([0, T] \times M, R^n)$
be a solution to \mbox{\rm (\ref{quasi-eq})}.
Suppose that the image of $u$ is contained in the target manifold $N$.
Then we have, with a positive constant $\gamma$
depending only on
$M, N, T$ and $\sup_M |D u_0|$,
%
\begin{equation}
\sup_{(0, T) \times M}
|D u|
\le
\gamma
=\gamma
\left(
M, N, T, \sup_M |D u_0|
\right).
\label{uni-bound}
\end{equation}
\end{lm}

For solutions to (\ref{quasi-eq}),
we have the Bochner formula (refer to \cite[pp.134-135]{duzzar-fuchs1}
and \cite[pp.128-131]{hamilton}):
Put
$v = (\epsilon + |D u|^2)/2$.
%
Then we have,
in $(0, T) \times M$,
%
\begin{eqnarray}
\lefteqn{
\partial_t v
-\mbox{\small $\frac{1}{\sqrt{|g|}}$}
D_\alpha
\left(
(2 v)^{\frac{p}{2}-1}
a^{\alpha \beta}
D_\beta v
\right)
+ (p - 2)
(2 v)^{
\frac{p}{2} - 2
}
g^{\alpha \beta}
D_\alpha v
D_\beta v }\nonumber \\
\lefteqn{ + (2 v)^{
\frac{p}{2} - 1
}
g^{\gamma {\bar \gamma}}
g^{\beta {\bar \beta}}
D_\gamma D_\beta u^i
D_{\bar \gamma} D_{\bar \beta} u^j
h_{i j}(u)
+ (2 v)^{
\frac{p}{2} - 1
}
R_M^{\alpha \beta}
D_\alpha u^i
D_\beta u^j
h_{i j}(u) } \nonumber \\
&=&(2 v)^{\frac{p}{2}-1}
g^{\alpha {\bar \alpha}}
g^{\beta {\bar \beta}}
R_{i j k l}^N
D_\alpha u^i
D_\beta u^j
D_{\bar \alpha} u^k
D_{\bar \beta} u^l, \hspace{3cm}
\label{boch0}
\end{eqnarray}
%
where we put
$$
a^{\alpha \beta}\left(t, x\right)
=\sqrt{\left|g\right|}
\left(
g^{\alpha \beta} + \left(p - 2\right)
\mbox{\small
$\frac{
g^{\alpha \mu}
g^{\beta \nu}
D_\mu u^i
D_\nu u^j
h_{i j}(u)
}
{2 v}$
}
\right).
$$ 
Since we assume that the sectional curvature of $N$
is nonpositive, we have
%
\begin{equation}
g^{\alpha {\bar \alpha}}
g^{\beta {\bar \beta}}
R_{i j k l}^N
D_\alpha u^i
D_\beta u^j
D_{\bar \alpha} u^k
D_{\bar \beta} u^l
\le 0.
\label{sect-cond}
\end{equation}
%
Thus we obtain,
from
(\ref{sect-cond})
and
(\ref{boch0}),
with a positive constant $\gamma$
depending only on
$\left(g_{\alpha \beta}\right)$
and the derivative,
%
\begin{equation}
\partial_t v
-\mbox{\small
$\frac{1}{\sqrt{|g|}}$
}
D_\alpha
\left(
(2 v)^{\frac{p}{2}-1}
a^{\alpha \beta}
D_\beta v
\right)
\le
\gamma
(2 v)^{
\frac{p}{2}
}
\quad\mbox{in $(0, T) \times M$}.
\label{boch2}
\end{equation}
%
For brevity, we assume that
$(g_{\alpha \beta}) = Id$.
(We can argue similarly in the general case.)
Then the formula (\ref{boch2}) becomes
%
\begin{equation}
\partial_t v
-D_\alpha
\left(
(2 v)^{\frac{p}{2}-1}
a^{\alpha \beta}
D_\beta v
\right)
\le
\gamma
v^{p/2}.
\label{boch-sim}
\end{equation}
%
Let $k$ be
$k \ge
\hat {k}
=\max
\{1,
\sup_M
|D u_0|^2
\}
$
and put
$M_t = (0, t) \times  M$
for $0 < t < T$.
Then we substitute a test function
$\phi = (v - k)^+$
$= \max \{v - k, 0 \}$
into the formula (\ref{boch-sim})
to obtain
%
\begin{equation}
\int_{M_t}
\left\{
\partial_t v
(v - k)^+ + \left(2 v\right)^{
\frac{p}{2}
- 1}
a^{\alpha \beta}
D_\beta v
D_\alpha
(v - k)^+ \right\}
dz
\le
\gamma
\int_{M_t}
v^{p/2}
(v - k)^+ dz.
\label{har0}
\end{equation}

Now we estimate
$
\int_{M_t}
v^{p/2}
(v - k)^+ dz.
$
First we deform
$
v^{p/2}
(v - k)^+ $
as
$$
((v - k)^+ )^{\frac{p}{2}+1}+ k^{\frac{p}{2} + 1}.
$$
We estimate the quantity
$
\int_{M_t}
(
(v - k)^+ )^{p/2 + 1}
dz
$
by using the H\"older and Sobolev inequalities.
Set
$V = (v - k)^+$.
%
Then
%
\begin{eqnarray*}
\lefteqn{ \int_{M_t}
V^{\frac{p}{2}+1}dz }\\
&\le& \sup_{0 \le \tau \le t}
\biggl(
\int_{\{\tau \} \times M}
V^2
dx
{\biggr)}^{1/a}
\sup_{0 \le \tau \le t}
\biggl(
\int_{\{\tau \} \times M}
V^{
\frac{p}{2}
}
dx
{\biggr)}^{1/b} \times  \\
&& \int_0^t
\biggl(
\int_{\{\tau \} \times M}
V^{\frac{m}{m - 2}(
\frac{p}{2} + 1)}
dx
{\biggr)}^{\frac{1}{c}}
d \tau \\
%========3============
&\le&
\sup_{
0 \le \tau \le t
}
\biggl(
\int_{\{\tau \} \times M}
V^2
dx
{\biggr)}^{1/a}
\sup_{
0 \le \tau \le t
}
\biggl(
\int_{\{\tau \} \times M}
V^{p/2}
dx
{\biggr)}^{1/b} \times \\
%=========4=============
&&
\gamma
\left( m, |M|^{-\frac{1}{m}} \right)
t^{
\frac{c(m - 2)}{(c - 1)m - 2 c}
}
\biggl(
\int_{M_t}
\left(
V^{\frac{p}{2} + 1}
+ |D V^{
\frac{1}{2}
\left( \frac{p}{2} + 1 \right)
}|^2
\right)
dz
{\biggr)}^{
\frac{m}{c(m - 2)}
},
\end{eqnarray*}
%
where the exponents $a, b$ and $c$ satisfy
%
\begin{equation}
\frac{1}{a}
=\frac{2p}{m(p - 2) + 2 p},
\quad\frac{1}{b}
=\frac{2(p - 2)}{m(p - 2) + 2 p},
\quad\frac{1}{c}
=\frac{(m - 2)(p - 2)}{m(p - 2) + 2p}.
\label{exp0}
\end{equation}
%
Noting that $1/a + m/c(m - 2) = 1$,
we have
%
\begin{eqnarray*}
\int_{M_t} V^{\frac{p}{2} + 1} dz
&\le&
\gamma
\left( m, p, |M|^{- \frac{1}{m}} \right)
t^{\frac{c (m - 2)}
{(c - 1)m - 2c}
}
\sup_{
0 \le \tau \le t
}
\biggl(
\int_{M_t}
V^{p/2}
 dx
{\biggr)}^{1/b} \times \\
&&\biggl\{
\sup_{
0 \le \tau \le t
}
\int_{ \{ \tau \} \times M}
V^2
dx
+ \int_{M_t}
\left(
V^{\frac{p}{2} + 1}
+ \left|
D V^{
\frac{p + 2}{4}
}
\right|^2
\right)
dz
\biggr\}.
\end{eqnarray*}
%
Using the energy inequality (\ref{ene-ineq})
and choosing $t > 0$ to be small,
we estimate
%
\begin{eqnarray*}
\lefteqn{
\gamma
\left( m, p, |M|^{-\frac{1}{m}} \right)
t^{
\frac{c (m - 2)}
{(c - 1)m - 2c}
}
\sup_{
0 \le \tau \le t
}
\biggl(
\int_{M_t}
V^{p/2}
dx
{\biggr)}^{
\frac{1}{b} } } \\
&\le&
\gamma
\left(m, p, |M|^{-\frac{1}{m}} \right)
t^{
\frac{c (m - 2)}
{m(c - 1) - 2c}
}
\biggl(
\int_{M_t}
|D u_0|^p
dx
{\biggr)}^{1/b}
\le
\frac{1}{2},
\end{eqnarray*}
%
where we note that
$c(m - 2)/(m(c- 1) - 2c) > 0$
and that the positive number $t$
depends only on $E(u_0)$
and
$\gamma(m, p, |M|^{-1/m})$.
%
Thus we have
%
\begin{eqnarray}
\int_{M_t}
V^{\frac{p}{2} + 1}dz
&\le&
{\widetilde \gamma}
\left( m, p, |M|^{-\frac{1}{m}} \right)
t^{
\frac{c (m - 2)}
{(c - 1)m - 2c}
}
\sup_{
0 \le \tau \le t
}
\biggl(
\int_{M_t}
V^{p/2}
 dx
{\biggr)}^{1/b} \times \nonumber \\
&&\biggl\{
\sup_{
0 \le \tau \le t
}
\int_M
V^2
dx
+ \int_{M_t}
\left|
D V^{\frac{p + 2}{4}}
\right|^2
dz
\biggr\}.
\label{har1}
\end{eqnarray}

 
Next we treat
$k^{p/2 + 1} |M_t \times \{ v > k \}|$.
%
By H\"older's inequality, we have
%
\begin{eqnarray}
k^{\frac{p + 2}{2}}
|M_t \times \{v > k \}|
\le
k^{2 \delta}
\sup_{0 \le \tau \le t}
\biggr(
\int_M
v^{p/2}
dx
\biggr)^{1/b}
\int_0^t
|
\{v > k\}
|^{\frac{1}{a} + \frac{1}{c}}
d \tau,
\label{har2}
\end{eqnarray}
%
where the exponent
$\delta$
is determined by
%
\begin{equation}
2 \delta =\frac{
(p - 2)(p + 2) + 8 p
}
{
2(m(p - 2) + 2p)
}.
\label{exp1}
\end{equation}
%
Now we note that, if we take the exponents
$\kappa, q$ and $r$ to satisfy
%
\begin{equation}
\frac{2(1 + \kappa)}{r} = 1,
\quad\frac{r}{q} = \frac{1}{a} + \frac{1}{c},
\quad\frac{1}{r} + \frac{m}{2q} = \frac{m}{4},
\label{exp2}
\end{equation}
%
then
$$ 
\kappa > 0,
\quad0 < \delta < 1 + \kappa.
$$ 
Combining (\ref{har1}) with (\ref{har2})
and substituting the resulting inequalities
into (\ref{har0}), we have
%
\begin{eqnarray}
\lefteqn{
\sup_{0 \le \tau \le t}
\int_{M_\tau}
((v - k)^+)^2
dx
+ \int_{M_t}
v^{\frac{p}{2} - 1}
|D (v - k)^+ |^2
dz } \nonumber \\
&\le & \gamma
\left( m, p, |M|^{-\frac{1}{m}} \right)
t^{
\frac{
c (m - 2)
}
{
m(c - 1) - 2c
}
}
\sup_{0 \le \tau \le t}
\int_{\{ \tau \} \times M}
((v - k)^+)^{
\frac{p}{2}
}
dx
\biggr)^{1/b} \times  \nonumber\\
&&\biggl(
\sup_{0 \le \tau \le t}
\biggl(
\int_{\{\tau\} \times M}
((v - k)^+)^2
dx
+ \int_{M_t}
\left| D
((v - k)^+)^{
\frac{p + 2}{4}
}
\right|^2
dz \biggr) \label{har4} \\
&&+ \gamma(m, p)
\sup_{0 \le \tau \le t}
\biggl(
\int_{\{\tau\} \times M}
v^{p/2}
dx
\biggr)^{1/b}
k^{2 \delta}
\int_0^t
|\{v > k\}|^{\frac{1}{a} + \frac{1}{c}}
dt, \nonumber
\end{eqnarray}
%
where we used the facts that
the matrix $\left(a^{\alpha \beta}\right)$
is positive definite
and that
$v \le \max\{
1, \sup_M \left|D u_0\right|^2 \}$
on
$\left\{t = 0\right\}
\times M$.
\par
\parindent0pt
%
Using (\ref{ene-ineq})
and
noting that
$c(m - 2)/(m(c - 1) - 2 c) > 0$,
we choose
$t_1 = t > 0$ 
to satisfy
%
\begin{equation}
t^{
\mbox{\small
$\frac{c (m - 2)}
{m(c - 1) - 2c}$
}
}
\biggl(
\frac{p + 2}{4}
\biggr)^2
\gamma
\left( m, p, |M|^{\frac{1}{m}} \right)
E_1(u_0)^{
\frac{1}{b}
}
\le
\mbox{\small $\frac{1}{2}$}.
\label{small-cond}
\end{equation}
%
Then we obtain, from (\ref{har4}), with a positive constant
$\gamma$ depending only on $m$ and $p$,
%
\begin{eqnarray}
\lefteqn{
\sup_{0 \le \tau \le t_1}
\int_{\{\tau \} \times M}
((v - k)^+)^2
dx
+ \int_{M_{t_1}}
|D (v - k)^+ |^2
dz   } \label{iterate} \\
&\le&
\gamma(m, p)
\sup_{0 \le \tau \le t_1}
\biggl(
\int_{\{\tau\} \times M}
v^{p/2}
dx
\biggr)^{1/b}
k^{2 \delta}
\int_0^{t_1}
|\{v > k\}|^{\frac{1}{a} + \frac{1}{c}}
dt, \nonumber
\end{eqnarray}
%
where we used that $k \ge 1$ and
$$ 
\int_{M_{t_1}}
\left|
D ( (v - k)^+)^{
\frac{p + 2}{4}
}
\right|^2
dz
\le
\left(
\mbox{\small $\frac{p + 2}{4}$}
\right)^2
\int_{M_{t_1}}
v^{\frac{p}{2} - 1}
|D (v - k)^+ |^2
dz.
$$ 
Now apply Theorem 6.1
in \cite[pp.102-103]{lady}
for (\ref{iterate})
to obtain
$$
\sup_{M_{t_1}} v \le
\gamma(m, p) \max \left\{1,\sup_M |D u_0|^2 \right\}.
$$
Noting that, by (\ref{small-cond}),
the positive number $t_1$ depends on
$E_1(u_0), |M|, m$ and $p$,
and arguing as in
\cite[p.186]{lady},
we have
$$
\sup_{(0, T) \times M}
v
\le
\gamma(m, p)
\max \left\{
1,
\sup_M |D u_0|^2
\right\}.
$$
%=======the uniqueness of solutions of==================
%========= the regularized p-harmonic flow=============%

Once we have the uniform boundedness (\ref{uni-bound}),
we can argue as in
\cite[p.245, Theorem 1.1; p.291, 14, pp.217--218]{dibene-text}
(also see \cite{choe2})
to arrive at the following:

\begin{lm} \label{conti}
Let $u \in C^{1, 2}_0([0, T] \times M, R^n)$
be a solution of
(\ref{quasi-eq}).
We can choose positive constants
$\gamma$,
depending only on
$M, N, p, \sup_{(0, T) \times M} |D u|$,
and ${\widetilde \alpha}, 0 < {\widetilde \alpha} < 1$,
depending only on $m$ and $ p$,
such that
%
\begin{equation}
|u|_{C^{{\tilde \alpha}/p, {\tilde \alpha}}}
+ |D u|_{C^{{\tilde \alpha}/2, {\tilde \alpha}}}
\le
\gamma.
\label{uni-conti}
\end{equation}
\end{lm}

 
We now specify the value of the exponent $\alpha$,
$0 < \alpha \le \beta$,
which has not yet been determined.
We set
$\alpha = \min \{{\widetilde \alpha}, \beta \}$,
where
${\widetilde \alpha}$ is selected in Lemma~\ref{conti}.
 
Now we prove the uniqueness of a solution of
(\ref{quasi-eq}).

\begin{lm}\label{Unique}
Let $u_1, u_2 \in C^{1, 2}_0([0, T] \times, R^n)$
be two solutions to
(\ref{quasi-eq})
with the same initial value
$\exp_{a_0}\left(\tau \exp_{a_0}^{- 1}(u_0)\right)$.
Then $u_1 \equiv u_2$
in $[0, T] \times M$.
\end{lm}

\paragraph{Proof.}
We consider only the case $\tau = 1$, since
$u(0) = \exp_{a_0}\left(\tau \exp_{a_0}^{- 1}(u_0)\right)
\in N$ on $M$
and the case $0 \le \tau < 1$ is investigated
similarly.
Let $u \in C^{1, 2}_\alpha([0, T] \times M, R^n)$
be a solution to (\ref{quasi-eq}) with $\tau = 1$.
Then $u(0) = u_0$ in $M$.

Since the image of $u_0$
is contained in the target manifold $N$,
we can choose a positive number
${\widetilde T} = {\widetilde T}(u)$
such that
%
$u \in {\cal O}_\delta(N)$
in $[0, {\tilde T}] \times M$.
%
Then, by the definition of the metric $(h_{i j})$ of ${\mathbb R}^n$,
we find that
%
\begin{equation}
g^{\alpha {\bar \alpha}}
g^{\beta {\bar \beta}}
R^N_{i j k l}(u)
D_\alpha u^i
D_\beta u^j
D_{\bar \alpha} u^k
D_{\bar \beta} u^l
\le 0
\quad\mbox{in $[0, {\widetilde T}] \times M$,}
\end{equation}
%
since the sectional curvature of $N$ is nonpositive.
%
Thus, by Lemma~\ref{bound},
we have
(\ref{uni-bound}) with replacing $T$ by $\widetilde T$.
%
Let $u_1, u_2 \in C^{1, 2}_\alpha([0, T] \times M, R^n)$
be two solutions to (\ref{quasi-eq}) with $\tau = 1$.
Set
${\widetilde T} = \min \{ {\widetilde T}(u_1), {\widetilde T}(u_2) \}$.
Subtract the equation for $u_1$
from the one for $u_2$
and
take a test function $u_2 - u_1$
in the resulting equation
for $t$,
$0 \le t \le {\widetilde T}$
to obtain,
with $v = u_2 - u_1$,
%
\begin{eqnarray*}
\lefteqn{
\int_{M_t}v \cdot\partial_t v\,dM\, dt  } \\
\lefteqn{ + \int_{M_t}
\left\{ (p e_\epsilon(u_2))^{1 - \frac{2}{p}}
h_{i j}(u_2)
D_\beta u_2^j
-(p e_\epsilon(u_1))^{1 - \frac{2}{p}}
h_{i j}(u_1)
D_\beta u_1^j
\right\}
g^{\alpha \beta}
D_\alpha v^i
dM dt
}\\ 
&=& \int_{M_t}
g^{\alpha \beta}
\bigg\{
(p e_\epsilon(u_2))^{1 - \frac{2}{p}}
\Gamma_{i j}(u_2)
(D_\alpha u_2^i, D_\beta u_2^j) \\
&& \hspace{1.5cm} -(p e_\epsilon(u_1))^{1 - \frac{2}{p}}
\Gamma_{i j}(u_1)
(D_\alpha u_1^i, D_\beta u_1^j)
\bigg\}
\cdot
v\,dM \,dt\,.\hspace{2.4cm}
\end{eqnarray*}
%
We estimate each term of this equality.
Put
$w(s) = (1 - s) u_1 + s u_2$
for
$s$, $0 \le s \le 1$.
Then
%
\begin{eqnarray*}
\lefteqn{
\left(
(p e_\epsilon(u_2))^{1 - \frac{2}{p}}
h_{i j}(u_2)
D u_2^j
-(p e_\epsilon(u_1))^{1 - \frac{2}{p}}
h_{i j}(u_1)
D u_1^j
\right)
g^{\alpha \beta}
D_\alpha v^i  }\\
&=&\int_0^1
\biggl\{
(p e_\epsilon(w(s)))^{1 - \frac{2}{p}}
|D v|^2
+ (p - 2)
(p e_\epsilon(w(s)))^{1 - \frac{4}{p}}
\langle
D v, D w(s)
\rangle^2  \\
&&+ (p e_\epsilon(w(s)))^{1 - \frac{2}{p}}
g^{\alpha \beta}
D_\beta v^j
D_\alpha w^i(s)
\mbox{\small $\frac{d h^{i j}}{d u}(w(s))$}
\cdot v\\
&& +\mbox{\small ${\frac{p - 2}{2}}$}
(p e_\epsilon(w(s)))^{1 - \frac{4}{p}}
g^{\alpha \beta}
D_\beta w^j(s)
D_\alpha w^i(s)
v
\cdot
\frac{d h^{i j}}{d u}(w(s))
\langle
D w(s), D v
\rangle
\biggr\}
ds.
\end{eqnarray*}
%
The third and fourth terms on the right hand side
are bounded from above by
%
\begin{eqnarray*}
\lefteqn{
\gamma
\biggl(
p, N,
\sup_{M_{\widetilde T}} |D u_1|,
\sup_{M_{\widetilde T}} |D u_2|
\biggr)
\int_0^1
|v|^2
ds } \\
&&
+ \mbox{\small $\frac{1}{2}$}
\int_0^1
(p e_\epsilon(w(s)))^{1 - \frac{2}{p}}
\biggl(
|D v|^2
+ (p - 2)
\mbox{\small
$\frac{
\langle
D w(s), D v
\rangle^2
}
{
(p e_\epsilon(w(s)))^{\frac{2}{p}}
}
$
}
\biggr)
ds\,.
\end{eqnarray*}
%
As above, we have
%
$$
g^{\alpha \beta}
\left(
(p e_\epsilon(u_2))^{1 - \frac{2}{p}}
\Gamma_{i j}(u_2)
(D_\alpha u_2^i, D_\beta u_2^j)
-(p e_\epsilon(u_1))^{1 - \frac{2}{p}}
\Gamma_{i j}(u_1)
(D_\alpha u_1^i, D_\beta u_1^j)
\right)
\cdot
v
$$
%
\begin{eqnarray*}
&\le& \gamma
\biggl(
p, M, N,
\sup_{M_{\widetilde T}}|D u_1|,
\sup_{M_{\widetilde T}}|D u_2|
\biggr)
\int_0^1
|v|^2
ds
\nonumber
\\
&&
+ \mbox{\small $\frac{1}{2}$}
\int_0^1
(p e_\epsilon(w(s)))^{1 - \frac{2}{p}}
\left(
|D v|^2
+ (p - 2)
\mbox{\small
$\frac{
\langle
D w(s), D v
\rangle^2
}
{
(p e_\epsilon(w(s)))^{\frac{2}{p}}
}
$
}
\right)
ds.
\end{eqnarray*}
%
As a result we have
%
\begin{eqnarray}
\lefteqn{
\int_{M_t}
\left\{
v \cdot \partial_t v
+ \mbox{\small $\frac{1}{2}$}
\int_0^1
(p e_\epsilon(w(s)))^{1 - \frac{2}{p}}
\left(
|D v|^2
+ (p - 2)
\mbox{\small
$\frac{
\langle
D w(s), D v
\rangle^2
}
{
(p e_\epsilon(w(s)))^{\frac{2}{p}}
}
$
}
\right)
ds
\right\}
dM
dt } \nonumber \\
&\le&\gamma
\biggl(
p, M, N,
\sup_{M_{\widetilde T}} |D u_1|,
\sup_{M_{\widetilde T}} |D u_2|
\biggr)
\int_{M_t}
|v|^2\,dM\, dt\,. \hspace{3.5cm}
\label{uniq1}
\end{eqnarray}
%
Putting
%
$F(t) = \int_{M_t} |v|^2 dM dt$ for any $t$,
$0 \le t \le {\widetilde T}$,
and noting $v (0) = 0$,
we find from (\ref{uniq1}) that
%
$$
\mbox{\small $\frac{d}{d t}$}
F(t)
\le
\gamma
\biggl(
p, M, N,
\sup_{M_{\widetilde T}} |D u_1|,
\sup_{M_{\widetilde T}} |D u_2|
\biggr)
F(t)
$$
for all $0 \le t \le {\tilde T}$, 
from which it follows that
%
$ \exp (- \gamma t) F(t) \le 0$
for all $t \in [0, {\tilde T}]$.
Therefore we have
%
$F({\tilde T}) = 0$,
which implies that
%
$v = 0$
in
$[0, {\widetilde T}] \times M$.
%
%
%========u \in N=========== 
Now we observe that
the images of $u_1$ and $u_2$
are in the target manifold $N$.
We consider $u = u_1$.
%
Take a positive number ${\tilde T} = {\tilde T}(u)$
such that
$u \in {\cal O}_\delta(N)$
in $[0, {\tilde T}] \times M$. 
%
We use the involutive isometry
$\pi$ from ${\cal O}_\delta(N)$ to itself
such that the fixed point set of $\pi$ is exactly
the target manifold $N$.
%
Compare $\pi(u)$ with $u$:
%
Since the image of $u_0$ is imposed on $N$,
$\pi(u)(0) = u(0)$ in $M$.
%
Noting that the operator
$\pi:{\cal O}_\delta(N) \rightarrow {\cal O}_\delta(N)$
is isometry,
we know that $\pi(u)$ satisfies (\ref{quasi-eq})
with $\tau = 1$,
of which $u$ is also a solution.
%
By the arguments above,
we find that
$\pi(u) \equiv u$
in $[0, {\widetilde T}] \times M$
and that the image of $u$ in
$[0, {\widetilde T}] \times M$
is on the fixed point set $N$ of $\pi$.
Therefore we have verified that $u_1 = u_2 \in N$
in $[0, {\tilde T}] \times M$.

Replacing an initial value $u_0$
with $u_1({\widetilde T})(= u_2({\widetilde T}))$
and repeating the above argument,
we conclude our uniqueness assertion:
$u_1 \equiv u_2$ in $[0, T] \times M$.
In addition, we have proven the following:


\begin{lm} \label{Image}
Let $u \in C^{1, 2}_0([0, T] \times M, R^n)$
be a solution to (\ref{quasi-eq}).
Then $u \in N$ in $[0, T] \times M$.
\end{lm}

By combination of Lemmata~\ref{bound}, \ref{conti}
with Lemma~\ref{Image},
we conclude that (\ref{uni-bound}) and
(\ref{uni-conti}) hold uniformly for
all solutions
$u \in C^{1, 2}_0([0, T] \times M, R^n)$
of (\ref{quasi-eq}).

%=======the limit as \epsilon -> 0==================%

\section{The limit $\epsilon \rightarrow 0$}

First we claim the existence and uniqueness of the regularized
$p$-harmonic flow, which is a solution of (\ref{quasi-eq})
with $\tau = 1$.
By the arguments in Sect.$3$ and Sect.$4$,
we can apply the Leray-Schauder fixed point theorem
and obtain a unique fixed point $u_\epsilon$ in
$C^{1, 2}_\alpha([0, T] \times M, N)$
of the operator $P_1$.

\begin{lm} \label{exist-quasi}
For any $\epsilon$, $0 < \epsilon < 1$,
there exists a unique solution
$u_\epsilon$ in \newline $C^{1, 2}_\alpha([0, T] \times M, N)$
of \mbox{\rm (\ref{reg-eqn})} with the initial value \mbox{\rm(\ref{init})}.
\end{lm}
%

We now explain how to pass to the limit $\epsilon \rightarrow 0$
and show the validity of Theorem~\ref{main-th}.
 
By Lemma~\ref{ene-inequ}, we choose a subsequence
$\{ u_k \}$ with $u_k = u_{\epsilon_k}$,
$0 < \epsilon_k < 1$,
and a function $u$ defined on
$(0, T) \times M$
with value in ${\mathbb R}^n$ such that, as $\epsilon_k \rightarrow 0$,
%
\begin{eqnarray}
&
D u_k \rightarrow D u
\quad\mbox{weakly* in $L^\infty((0, T);L^p(M))$,} &\nonumber\\
&
\partial_t u_k \rightarrow \partial_t u
\quad\mbox{weakly in $L^2 ((0, T) \times M)$,}&
\label{weak-conv}
\end{eqnarray}
%
Noting Lemmata~\ref{bound} and \ref{conti},
we apply the Ascoli-Arzela theorem to obtain
%
\begin{equation}
u_k \rightarrow u
\quad\mbox{strongly in $C^{0, 1}_0([0, T] \times M, R^n)$.}
\label{strong-conv}
\end{equation}
%
By Lemma~\ref{Image} and (\ref{strong-conv}),
we know that
%
\begin{equation}
u \in N
\quad\mbox{in $[0, T] \times M$.}
\label{image-cond}
\end{equation}
%
By (\ref{weak-conv}) and (\ref{strong-conv}),
we can take the limit $\epsilon_k \rightarrow 0$
in the weak form of the equation (\ref{reg-eqn})
with a test function
$\phi \in C^\infty([0, T] \times M, R^n)$:
%
\begin{eqnarray*}
&
\int_{(0, T) \times M}
\bigl\{
\phi \cdot \partial_t u_k
+ (p e_{\epsilon_k}(u_k))^{1 - \frac{2}{p}}
g^{\alpha \beta}
D_\beta u_k
\cdot
D_\alpha \phi
\\
& \hspace{2cm}
-(p e_{\epsilon_k}(u_k))^{1 - \frac{2}{p}}
g^{\alpha \beta}
\Gamma_{i j}(u_k)
D_\alpha u_k^i
\cdot
D_\beta u_k^j
\bigr\}\,dM \,dt
= 0&
\end{eqnarray*}
%
and find that the limit function $u$ satisfies (\ref{w-eqn}),
where we note (\ref{image-cond}).
Using (\ref{weak-conv}) in the energy inequality (\ref{ene-ineq})
with $\epsilon = \epsilon_k$ and $u = u_k$,
we have (\ref{energy}).
Lemma~\ref{conti} with (\ref{strong-conv})
implies the H\"older continuity of $u$ and $D u$
in the statement of Theorem~\ref{main-th}
with the H\"older exponent
$\alpha = \min \{{\widetilde \alpha}, \beta \}$.
 
Finally, we use the energy inequality (\ref{ene-ineq})
to make the estimate
%
\begin{equation}
\int_M
\left|
u_k(t) - u_0
\right|^2
dM
\le
t
\int_{(0, t) \times M}
\left|
\partial_t u_k
\right|^2
dM dt
\le
t E_1(u_0).
\label{init-check}
\end{equation}
%
By (\ref{strong-conv}),
we take the limit $k \rightarrow \infty$
in (\ref{init-check})
to show the validity of (\ref{init-cond2}).
%
\paragraph{Acknowlegements.}
The author would like to thank
Professor Robert M. Hardt, Rice University,
for his interest in this work
and his valuable comments.
The author is also grateful to
Professor Norio Kikuchi, Keio University,
for his constant encouragement.
%

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\end{thebibliography} \bigskip

{\sc  Masashi Misawa} \\
Department of Computer Science and Information Mathematics \\
Faculty of Electro-Communications \\
The University of Electro-Communications, Japan \\
Email address: misawa@im.uec.ac.jp         
         
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