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\def\rightheadline{EJDE--2000/45\hfil
Some constancy results for harmonic maps
\hfil\folio}
\def\leftheadline{\folio\hfil Kewei Zhang
\hfil EJDE--2000/45}
\def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt %
Electronic Journal of Differential Equations,
Vol.~{\eightbf 2000}(2000), No.~45, pp.~1--13.\hfil\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfill\break
ftp ejde.math.swt.edu (login: ftp)\bigskip} }
\topmatter
\title
Some constancy results for harmonic maps \\
from non-contractable domains into spheres
\endtitle
\thanks
{\it 1991 Mathematics Subject Classifications:} 58E20 35J65 35J60 \hfil\break\indent
{\it Key words:} harmonic maps, uniqueness, Pohozaev identity, closed geodesics,
\hfil\break\indent tubular neighbourhoods.\hfil\break\indent
\copyright 2000 Southwest Texas State University and
University of North Texas.\hfil\break\indent
Submitted March 31, 2000. Published June 14, 2000.
\endthanks
\author Kewei Zhang \endauthor
\address
Kewei Zhang \hfill\break\indent
Department of Mathematics, Macquarie University, Sydney Australia
\endaddress
\email kewei\@ics.mq.edu.au
\endemail
\abstract
We use the Pohozaev identity on sub-domains of a
Euclidean $r$-neighbourhood for a closed or broken curve to show
that harmonic maps from such domains into spheres with constant
boundary value remain constant.
\endabstract
\endtopmatter
\document
\head \S 1. Introduction\endhead
In this paper we generalize
a constancy result for harmonic maps from a non-star shaped domain
in $\Bbb R^3$ to the sphere $S^2$ obtained by Chou and Zhu
\cite{CZ}. In \cite{CZ} a special class of non-star shaped domains
was constructed by rotating a curve which is carefully designed by
using inversions in Euclidean spaces. The first result of the
present paper is to generalize this result to domains including
all smooth rotational ones (Theorem 1). For domains in $\Bbb R^m$
with $m\geq 3$,
we can show that
the same result holds on a tubular neighbourhood (see e.g.
\cite{S, I. Cha.9})
of a closed planar curve under a nondegeneracy condition
for closed geodesic in planar domains (Theorem 3). One such
example is the tubular neighbourhood
of a closed convex curve such as the solid torus. When $m\geq 4$,
we can show that the same claim is true for a thin tubular
neighbourhood of any smooth embedded curve with an orthogonal
moving frame. We state the results only for $u:\Omega\subset \Bbb
R^3\to S^2$ although they can be easily proved for higher
dimensional cases. The only exception is Theorem 4 where we can only prove
the result for domains at least in $\Bbb R^4$.
It is well known that if either $\Omega\subset \Bbb R^2$ is
contractable \cite{L} or
$\Omega\subset \Bbb R^m$ is star-shaped with $m\geq 3$ \cite{W},
the constancy result holds. If
one perturb a star-shaped domain in a $C^2$ manner, one expect to
have the so-called `nearly star-shaped' domains and the constancy
result is still true \cite{DZ}. It is also known that if the
boundary of the domain $\partial\Omega$ is disconnected, the
constancy result fails \cite{BBC}.
The method we use is the Pohozaev identity (see \cite{P,PS,CZ}).
We carefully divide the original domain into sub-domains which are
thin slices of the original domain
such that each sub-domain is star-shaped
with respect to some specific point on a curve. We apply Pohozaev
identity on each of these sub-domains and use the the constacy
condition $u=u_0$ only on part of its boundary. We then obtain
an inequality on each sub-domains. We sum up the resulting terms
and use the definition of Riemann integral. In the limit, we
obtain an inequality connecting two volume integrals. We reach
our conclusions by comparing quatities on both sides of the
inequality. Some results on shortest path in an Euclidean domain
in $\Bbb R^2$ (or geodesics in such a domain) are used.
A smooth mapping $u: \Omega\subset\Bbb R^m\to S^n$ is harmonic if
$$-\Delta u=u|Du|^2\quad\text{in}\quad \Omega\tag 1$$
and $u$ is a critical point of the total energy
$E(u)=\int_\Omega |Du|^2dx$.
Let $\Omega\subset \Bbb R^m$ be piecewise smooth and $u\in
C^2(\Omega, S^n)\cap C^1(\bar\Omega,S^n)$ be a smooth solution of
(1). Let $\nu(x)=(\nu_1(x),\cdots,\nu_m(x))$ be the outward
normal vector at $x\in\partial\Omega$, and let $h=(h_1,h_2,\cdots,
h_m)$ be a smooth vector field on $\bar\Omega$. Then (see
\cite{CZ})
$$\frac{\partial}{\partial x_\alpha} \left(
h_\alpha|Du|^2-2h_\beta\frac{\partial u_k}{\partial x_\alpha}
\frac{\partial u_k}{\partial x_\beta}\right)=\frac{\partial
h_\alpha}{\partial x_\alpha}|Du|^2 -2\frac{\partial
h_\beta}{\partial x_\alpha}\frac{\partial u_k}{\partial x_\alpha}
\frac{\partial u_k}{\partial x_\beta},\tag 2
$$
where the summation
convention is assumed with $1\leq \alpha,\,\beta\leq m$ and $1\leq
k\leq n+1$. Recall that a domain $\Omega$ is star-shaped if there
is a point $x_0\in \Omega$ such that the line segment
$\overline{x_0x}$ is contained in $\Omega$. For convenience, we
call $x_0$ the {\it central point} of $\Omega$ if $\Omega$ is
star-shaped with respect to $x_0$.
\head \S 2. Main Results\endhead
Theorem 3 below covers Theorems 1 and 2. However, since the
proofs of both theorems are needed for establishing Theorem 3, we
prove them separately.
\proclaim{Theorem 1} Suppose $\Omega\subset R^3$ is a smooth
domain and the orthogonal projection of the domain to the first
component is an interval $[a,b]$. We assume that there is a
$\delta>0$, such that for all $a\leq t_10$ is a smooth function defined in $[a,b]$, then the
rotation of the the two dimensional region bounded by $f$ and
$x_1$ axis around $\Bbb R^{m-2}$ defines the domain. In
particular, the domain we defined is much more general that the
one given by \cite{CZ}.
\endremark
Let $\gamma:[0,l]\to \Bbb R^m$ be an simple, smooth and convex
curve with bounded curvatures. Then it is easy to see that the
$r$-neighbourhood $$\Omega_r=\{ x\in \Bbb R^m,\;
\operatorname{dist}(x,\gamma)0$. Let $\Omega_r$ be
the $r$-neighbourhood of $\gamma$ in $\Bbb R^3$ with $00$ and the shortest path connecting $\gamma(0)$ and $\gamma(l)$
is unique \cite{BR}. Furthermore, in both cases, the geodesics are
of class $C^{1,1}$ (see, for example \cite{C1,C2}).
The geometric descriptions for geodesics in domains of $\Bbb R^n$
can be found, for example, in \cite{AB}: A geodesic contacting the
boundary in a segment is a geodesic of the boundary (in $\Bbb
R^2$, it is part of the boundary); a geodesic segment not touching
the boundary is a
straight line segment. A segment on the boundary joins a segment in the ambient space in a
differentiable join. An endpoint on the boundary of a segment not
touching the boundary is called a {\it switching point}. The
accumulation points of switching points are called {\it
intermittent points}.
We need some technical conditions on the tubular domains which
exclude the intermittent points. The reason for such assumptions
is purely for avoiding technical complications.
\proclaim{ Hypothesis (H1)} If $\gamma\subset \Bbb R^2\times
\{0\}$ is closed and $\gamma_0$ is a closed geodesic
in $\bar\Omega_r\cap \Bbb R^2\times \{0\}$
which is homotopic to $\gamma$, where $\Omega_r\subset \Bbb R^3$
is a tubular neighbourhood of $\gamma$. Then
\roster
\item"(i)" $\gamma_0$ has finite number of switching points,
hence it does not have intermittent points;
\item"(ii)" there is a $\delta>0$, such that for every
straight line segment $\mu\subset \gamma_0$ lying inside
$\Omega_r\cap\Bbb R^2\times\{0\}$ and any $p,\, q\in\mu$ with
$|p-q|\leq \delta$, the sub-domain of $\Omega_r$ bounded by
normal planes of $\mu$ passing through $p$ and $q$ respectively is a star-shaped domain
with any point on $\mu$ between $p$ and $q$ a central point.
\endroster
\endproclaim
\proclaim{Hypothesis (H2)} If $\gamma:[0,l]\to \Bbb R^2$ is broken
with $\gamma(0)=p\neq q=\gamma(l)$. Let $p^\prime = p-\dot
\gamma(0) r$, $q^\prime=q+\dot \gamma(l)r$. Then for $r>0$
sufficiently small, $p^\prime,\,
q^\prime\in\partial(\Omega_r\cap\Bbb R^2\times\{0\})$. Let
$\gamma_0$ be the geodesic in $\overline{\Omega_r\cap\Bbb
R^2\times\{0\}}$ connecting $p^\prime$ and $q^\prime$. Then
\roster
\item"(i)" $\gamma_0$ has finite number of switching points, hence it does not have
intermittent points;
\item"(ii)" there is a $\delta>0$, such that for every
straight line segment $\mu\subset \gamma_0$ lying inside
$\Omega_r\cap\Bbb R^2\times\{0\}$ and any $a,\, b\in\mu$ with
$|a-b|\leq \delta$, the sub-domain of $\Omega_r$ bounded by
normal planes of $\mu$ passing through $p$ and $q$ respectively is a star-shaped domain
with any point on $\mu$ between $p$ and $q$ a central point.
\endroster
\endproclaim
\proclaim{Theorem 3} $\gamma\subset R^2$ be a smooth closed or
broken curve with maximal curvature $k_0>0$. Let $\Omega_r$ be
the $r$-neighbourhood of $\gamma$ in $\Bbb R^3$ with $00$, the only smooth harmonic map $u$ from
$\bar\Omega_r$ to $S^n$ with constant boundary value $u_0$ is
$u\equiv u_0$.
\endproclaim
\head \S 3. Proofs of the main results \endhead
\demo{Proof of Theorem 1} We divide $[a,b]$ evenly as
$a=t_00$
such that $|\ddot\gamma(s)|\leq C_0$ for all $s\in [0,l]$.
Therefore we also have $$\aligned &\left|\langle \frac{1}{2}\ddot
\gamma_r(x_{i+1})(s^\prime_{i+1}-s_{i+1})^2 -\frac{1}{2}\ddot
\gamma_r(\eta_{i+1})(s_{i+1}-s^\prime_{i})^2,\rangle\right|\\
&\leq \frac{1}{2}C_0\left[
(s^\prime_{i+1}-s_{i+1})^2+(s_{i+1}-s^\prime_{i})^2\right]\\ &\leq
C_0(s^\prime_{i+1}-s^\prime_{i})^2.
\endaligned
\tag 15$$ Similarly, we have $$\aligned &\langle
\gamma_r(s^\prime_{i+1})-\gamma_r(s^\prime_{i}),Du_k\rangle\\
&=\langle \dot \gamma (s_{i+1}),Du_k\rangle
(1-rk(s_{i+1})(s^\prime_{i+1}-s^\prime_{i})\\ &+\langle
\frac{1}{2}\ddot
\gamma_r(x^\prime_{i+1})(s^\prime_{i+1}-s_{i+1})^2
-\frac{1}{2}\ddot
\gamma_r(\eta^\prime_{i+1})(s_{i+1}-s^\prime_{i})^2,Du_k\rangle,
\endaligned
\tag 16$$ with $$\aligned &\left| \langle \frac{1}{2}\ddot
\gamma_r(x_{i+1})(s^\prime_{i+1}-s_{i+1})^2 -\frac{1}{2}\ddot
\gamma_r(\eta_{i+1})(s_{i+1}-s^\prime_{i})^2,Du_k\rangle\right|\\
&\leq C_0|Du_k|(s^\prime_{i+1}-s^\prime_{i})^2.
\endaligned\tag 17
$$ Now we can estimate the second sum in (12) as follows
$$\aligned &A_2=\\ &\sum^{N-1}_{i=0}
\int_{\Gamma_{i+1}}\left(|Du|^2 \langle
\gamma_r(s^\prime_{i+1})-\gamma_r(s^\prime_{i}),\,\nu\rangle
-2\langle Du_k,
\gamma_r(s^\prime_{i+1})-\gamma_r(s^\prime_{i})\rangle \langle
Du_k,\nu\rangle \right) dS\\ &\leq \int_{\Gamma_{i+1}}\left(|Du|^2
- 2\langle Du_k,\dot\gamma(s_{i+1})\rangle^2\right)dS
[1-rk(s_{i+1})](s^\prime_{i+1}-s^\prime_{i})\\
&+C_0(s^\prime_{i+1}-s^\prime_{i})\frac{l}{N}\int_{\Gamma_{i+1}}3|Du|^2dS,
\endaligned
\tag 18$$ where we have used the fact that $|\langle
Du_k,\nu\rangle|\leq |Du_k|$. Now we can estimate the two sums
$A_1$ and $A_2$ in (12).
$$\aligned \sum^{N-1}_{i=0}I_i\leq &
A_1+A_2\\ \leq& \frac
{3rk_0l^2}{16N^2}\int_{\partial\Omega}\left|\frac{\partial
u}{\partial \nu}\right|^2 dS \\
&+\int_{\Gamma_{i+1}}\left(|Du|^2 - 2\sum^3_{k=1}\langle
Du_k,\dot\gamma(s_{i+1})\rangle^2\right)dS
[1-rk(s_{i+1})](s^\prime_{i+1}-s^\prime_{i})\\
&+C_0(s^\prime_{i+1}-s^\prime_{i})\frac{l}{N}\int_{\Gamma_{i+1}}3|Du|^2dS.
\endaligned
\tag 19
$$
Passing to the limit $N\to\infty$ in (19) and noticing
that $$\lim_{N\to\infty}A_1\to 0,\quad
\lim_{N\to\infty}C_0\sum^{N-1}_{i=0}(s^\prime_{i+1}-s^\prime_{i})\frac{l}{N}\int_{\Gamma_{i+1}}3|Du|^2dS=0$$
because
$$\sum^{N-1}_{i=0}(s^\prime_{i+1}-s^\prime_{i})\int_{\Gamma_{i+1}}3|Du|^2dS$$
converges to an integral while $l/N\to 0$, we have
$$\aligned
&\limsup_{N\to\infty}\sum^{N-1}_{i=0}I_i\\ &\leq
\int_0^l\int_{\Gamma_s}|Du|^2(1-rk(s))dS\,ds-
2\int_0^l\int_{\Gamma_s}\sum^{3}_{k=1}\langle
Du_k,\dot\gamma\rangle^2(1-rk(s))dS\,ds,
\endaligned
\tag 20
$$
where $$\Gamma_s=\{
\gamma(s)+t\beta(s)+ze_3,\;t^2+z^2\leq r^2\}.$$ Now we sum up the
right hand side of (10): $$ \sum^{N-1}_{i=0}J_i=
\int_{\omega_i}|Du|^2 dx =\int_{\Omega_r} |Du|^2 dx.\tag 21$$ We
now change variables $$x=\gamma(s)+t\beta(s)+ze_3,$$ to obtain $$
\int_{\Omega_r}|Du|^2 dx
=\int^l_0\int_{\Gamma_s}|Du|^2(1-tk(s))dS\,ds. \tag 22$$ Finally we
obtain, from (20) and (22), $$\aligned
&\int^l_0\int_{\Gamma_s}|Du|^2(1-tk(s))dS\,ds\\
&\leq\int_0^l\int_{\Gamma_s}|Du|^2(1-rk(s))dS\,ds-
2\int_0^l\int_{\Gamma_s}\sum^{3}_{k=1}\langle
Du_k,\dot\gamma\rangle^2(1-rk(s))dS\,ds,
\endaligned
$$ so that $$\int^l_0\int_{\Gamma_s}|Du|^2(r-t)k(s)dS\,ds\leq
-2\int_0^l\int_{\Gamma_s}\sum^{3}_{k=1}\langle Du_k,\dot\gamma\rangle^2(1-rk(s))dS\,ds.
\tag 23$$ The first consequence of (23) is
$$\int_0^l\int_{\Gamma_s}\sum^{3}_{k=1}\langle
Du_k,\dot\gamma\rangle^2(1-rk(s))dS\,ds=0,$$ so that $\langle
Du_k,\dot\gamma\rangle=0$ hence for each fixed $(t,z)$,
$u(\gamma(s)+t\beta(s)+ze_3)$ is independent of $s$. Now, at
least in an interval $[a,b]\subset [0,l]$ with $a**0$
hence the left hand side of (23) gives $|Du|^2=0$ for $s\in
[a,b]$. Therefore in $$\{\gamma(s)+t\beta(s)+ze_3,\, s\in [a,b],
t^2+z^2\leq r\},$$ we have $u=u_0$. Since $u$ is independent of
$s$, we see that $u\equiv u_0$. \hfill $\boxed{\,}$
\enddemo
\demo{Proof of Theorem 3} If the curve $\gamma$ is closed and
$\Omega_r\subset \Bbb R^3$ is its open $r$-neighbourhood with
$rk_0<1$, we let $\gamma_0$ be a closed geodesic homotopic to
$\gamma$. If $\gamma_0$ does not have switching point, $\gamma_0$
must be the inner curve $\gamma_r$ of $\Omega_r$ defined in the
proof of Theorem 2 and it must be convex. Otherwise it is not the
locally shortest geodesic. Therefore, from Theorem 2, $u\equiv
u_0$. It is also obvious that if $\gamma_0$ has switching points,
it must have at least two such points. Since we assumed that
$\gamma_0$ has finitely many switching points, we denote them by
$p_1,\, p_2,\cdots p_m, p_{m+1}$ with $p_1=p_{m+1}$ such that
$p_k$ and $p_{k+1}$ are two consequent switching points along
$\gamma_0$. We parameterize $\gamma_0$ by its arc-length
$\gamma_0:[0,l]\to \bar\Omega_r$ with $\gamma_0(0)=p_1$. let
$0=s_1\leq s_2<\cdots 0$ is sufficiently small. We
now extend $\gamma$ to $\partial\Omega$ smoothly ($C^1$) by
defining
$$
\gathered \gamma_-(s)=p-s\dot\gamma(0),\quad 0\leq s<\leq r,\\
\gamma_+(s)=q+s\dot\gamma(l),\quad 0\leq s\leq r.
\endgathered
$$
Then $\gamma_-\subset \bar B_r^-$, and $\gamma_+\subset \bar
B_r^+$. We see that $\gamma_-(r),\,
\gamma_+(r)\in\partial\Omega_r$. If we divide $ B_r^-$ along
$\gamma_-$ by using normal planes of $\gamma_-$, each sub-domain
between two planes is star-shaped with respect to points on
$\gamma_-$ in the sub-domain. Similarly, we can do the same for
$B_r^+$. Let $\gamma_0=\gamma_-\cup\gamma\cup\gamma_+$. Then if we
divide $\Omega_r$ along $\gamma_0$ by using normal planes of
$\gamma_0$ and use the Pohozaev identity on each sub-domain and
follow the argument for the case of closed curves, we may
conclude the proof. \hfill $\boxed{\,}$
\enddemo
\demo{Proof of Theorem 4} We use a similar idea as that in the
proof of theorem 2 and theorem 3. If $\gamma$ is closed, we divide
$\Omega_r$ along $\gamma$ itself instead of $\gamma_r$. if
$\Omega\subset \Bbb R^m$ with $m\geq 4$, formula (4') should be
changed into $$\int_{\partial\Omega_i}\left(|Du|^2\langle
x-x^i,\nu\rangle -2\langle Du_k,x-x^i\rangle\langle
Du_k,\nu\rangle\right)=(m-2)\int_{\Omega_i}|Du|^2dx.$$ Recall that
the Jacobian of the mapping $$(s,x_2,x_3,\cdots, x_m)\to
\gamma(s)+x_2e_2(s)+\cdots, x_me_m(s)$$ is $1-x_2k_1(s)$, we may
follow the arguments similar to the proof of Theorem 2 by using
$\gamma$ as the central curve of $\Omega_r$ to obtain
$$\aligned
(m-2)\int_{\Omega_r}|Du|^2dx=&(m-2)\int^l_0\int_{\Gamma_s}|Du|^2(1-x_2k_1(s))dS\,ds\\
\leq&\int_0^l\int_{\Gamma_s}|Du|^2dS\,ds-
2\int_0^l\int_{\Gamma_s}\sum^{n+1}_{k=1}\langle
Du_k,\dot\gamma\rangle^2dS\,ds,
\endaligned
\tag 25$$
where
$$\Gamma_s=\{
\gamma_s+x_2e_2(s)+\cdots,x_me_m(s),\; x^2_2+\cdots+x_m^2\leq
r\}.$$
Since $1-rk_0\leq 1-x_2k_1(s)\leq 1$, we estimate the right
hand side of (25) as follows: $$\aligned
&\int_0^l\int_{\Gamma_s}|Du|^2dS\,ds-
2\int_0^l\int_{\Gamma_s}\sum^{n+1}_{k=1}\langle
Du_k,\dot\gamma\rangle^2dS\,ds\\ &\leq
\frac{1}{1-rk_0}\int_0^l\int_{\Gamma_s}|Du|^2(1-x_2k_1(s))dS\,ds
-2\int_0^l\int_{\Gamma_s}\sum^{n+1}_{k=1}\langle
Du_k,\dot\gamma\rangle^2dS\,ds\\ &=
\frac{1}{1-rk_0}\int_{\Omega_r}|Du|^2dx-2\int_0^l
\int_{\Gamma_s}\sum^{n+1}_{k=1}\langle
Du_k,\dot\gamma\rangle^2dS\,ds,
\endaligned
\tag 26$$
Combining (25) and (26) we obtain $$\left(
m-2-\frac{1}{1-rk_0}\right)\int_{\Omega_r}|Du|^2dx \leq
-2\int_0^l\int_{\Gamma_s}\sum^{n+1}_{k=1}\langle
Du_k,\dot\gamma\rangle^2dS\,ds.\tag 27$$
Now, since $m\geq 4$,
$m-2\geq 2$, we may have
$$ m-2-\frac{1}{1-rk_0}>0,\quad\text{if }\;
00$ is
sufficiently small, so that $|Du|^2=0$ in $\Omega_r$ hence $u=u_0$
in $\bar\Omega_r$. \hfill $\boxed{\,}$
\enddemo
\remark{Remark 3} The methods for proving Theorem 4 can be used to
establish similar uniqueness results for the Dirichlet problem
$-\Delta u+|u|^{p-1}u=0$ with $u=0$ on $\partial \Omega$ \cite{Z}
at least for $p>(n+1)/(n-3)$ in a tubular neighbourhood of a
closed or broken curve in $\Bbb R^n$ with $n\geq 4$. One can only
divide the domain along the
central
curve because the corresponding energy density is
$F(u,Du)=|Du|^2/2-|u|^{p+1}/(p+1)$ which is not necessarily
positive. Therefore the approach in Theorem 2 and Theorem 3 of
using the shortest path does not improve the result.
\endremark
\Refs \widestnumber\key{ABB} \ref\key{AA} \by R. Alexander and S.
Alexander \paper Geodesics in Riemann manifolds-with-boundary\jour
Indiana Univ. Math. J.\vol 30\yr 1981\pages 481-488\endref
\ref\key {AB} \by F. Albercht and I. D. Berg\paper Geodesics in
Euclidean spaces with analytic obstacle\jour Proc. AMS\vol 113 \yr
1991\pages 201-207\endref \ref\key{ABB} \by S. Alexander, I. D.
Berg and R. L. Bishop\paper The Riemann obstacle problem \jour
Illinois J. Math. \vol 31 \yr 1987 \pages 167-184\endref \ref\key
{BBC} \by F. Bethuel, H. Brezis and J. M. Coron \paper Relaxed
energies for harmonic maps \paperinfo in Variational Methods, H.
Berestycki, J. M. Coron and I. Ekeland eds. \yr 1990
\pages \publ Birkh\"auser\endref
\ref\key {BH} \by R. D. Bourgin and S. E. Howe\paper Shortest
curves in planar regions with
curved boundary\jour Theor. Comp. Sci. \vol 112 \yr 1993 \pages 215-253\endref
\ref\key{BR} \by R. D. Bourgin and P. L. Renz\paper Shortest paths
in simply connected regions in $R^2$ \jour Adv. Math. \vol 76\yr
1989\pages 260-295\endref \ref\key{C1} \by A. Canino\paper
Existence of a closed geodesic on $p$-convex sets \jour Ann. Inst.
H. Poincar\'e Anal. Non Lin. \vol 5 \yr 1988\pages 501-518\endref
\ref\key{C2} \by A. Canino\paper Local properties of geodesics on
$p$-convex sets\jour Ann. Mat. pura appl.\vol CLIX\yr 1991\pages
17-44\endref \ref\key{CZ} \by K.S.Chou and X.P.Zhu\paper Some
constancy results for nematic
liquid crystals and harmonic maps\jour
Anal. Nonlin. H. Poncar\'e Inst.\ 12 \yr 1995 \pages 99-115\endref
\ref\key {DZ} \by E. N. Dancer and K. Zhang\paper Uniqueness of
solutions
for some elliptic equations and systems in nearly star-shaped domains \paperinfo
To appear in \jour Nonlin. Anal. TMA\endref \ref\key{L} \by L.
Lemaire \paper Applications harmoniques de surfaces
riemanniennes\jour J. Diff. Geom. \vol 13\yr 1978 \pages
51-78\endref
\ref \key {M} \by E. Mitidieri\paper A Rillich Type identity and applications
\jour Comm. PDEs \vol 18\yr 1993\pages 125-151\endref \ref\key {P}
\by S. I. Pohozaev \paper Eigenfunctions of the equation $\Delta
u+\lambda f(u)=0$ \jour Soviet Math. Dokl.\vol 6\yr 1965\pages
1408-1411\endref \ref\key {PS} \by P. Pucci and J. Serrin \paper A
general variational identity\jour
Indiana Univ. Math. J. \vol 35\yr 1986\pages 681-703\endref
\ref\key {S} \by M. Spivak\book Differential Geometry \vol I-II
\publ Publish or Perish\yr 1979\endref \ref\key {V} \by R. C. A.
M. van der Vorst \paper Variational identities
and applications to differential systems
\jour Arch. Rational Mech. Anal.\vol 116 \yr 1992 \pages
375-398\endref \ref\key {W} \by J. C. Wood\paper Non-existence of
solutions to certian
Dirichlet problems for harmonic maps \paperinfo preprint Leeds Univ.\yr 1981\endref
\ref\key{Z} \by K. Zhang \paper Uniqueness of a semilinear
elliptic equation in non-contractable domains under supercritical
growth conditions\jour EJDE\vol 1999\issue 33 \yr 1999 \pages
1-10\endref
\endRefs
\enddocument
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