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\markboth{\hfil Quantitative uniqueness and vortex degree \hfil EJDE--2000/61}
{EJDE--2000/61\hfil Igor Kukavica \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.~{\bf 2000}(2000), No.~61, pp.~1--15. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
 Quantitative, uniqueness, and vortex degree estimates
 for solutions of the  \\ Ginzburg-Landau equation
%  
\thanks{ {\em Mathematics Subject Classifications:} 35B05, 35J25, 35J60, 35J65, 35Q35.
\hfil\break\indent
{\em Key words:} Unique continuation, vortices, Ginzburg-Landau equation.
\hfil\break\indent
\copyright 2000 Southwest Texas State University. \hfil\break\indent
Submitted June 23, 2000. Published Ocotber 2, 2000.} }
\date{}
%
\author{ Igor Kukavica }
\maketitle

\begin{abstract} 
 In this paper, we provide a sharp upper bound for the 
 maximal order of vanishing for non-minimizing solutions 
 of the Ginzburg-Landau equation
 $$
   \Delta u=-{1\over\epsilon^2}(1-|u|^2)u
 $$
 which improves our previous result \cite{Ku2}. An
 application of this result is a sharp upper bound for the
 degree of any vortex.  
 We treat Dirichlet  (homogeneous and non-homogeneous) 
 as well as Neumann boundary conditions.
\end{abstract}

\def\theequation{\thesection.\arabic{equation}}
\newtheorem{Theorem}{Theorem}[section]
\newtheorem{Lemma}[Theorem]{Lemma}
\newtheorem{Remark}[Theorem]{Remark}

\def\diam{\mathop{\rm diam}\nolimits}    %diameter
\def\dist{\mathop{\rm dist}\nolimits}    %distance


\section{Introduction}
\setcounter{equation}{0} \label{sec1}

In this paper, we provide vortex
degree estimates for solutions of the Ginzburg-Landau equation
  \[
   \Delta u= -{1\over\epsilon^2}
   \bigl( 1-|u|^2\bigr)u\,.
  \]
The vortices of solutions of this equation were studied by Bethuel,
Brezis, and H\'elein in \cite{BBH}. (We recall that $x_0$ is a vortex
if it is an isolated zero of $u$, and if the degree of $u$ at $x_0$ is
nonzero.) They prescribed nonhomogeneous boundary conditions
$u|_{\partial\Omega}=g$ with $g\colon \partial \Omega\to S^{1}$ such that
$\deg g=d>0$. If $\Omega$ is convex, and if
$\epsilon$ is sufficiently small, they proved that a minimizing
solution $u$ has
precisely $d$ distinct vortices of degree $1$. 
This result has been extended to include all bounded smooth domains by
Struwe \cite{S}.
It was further shown in \cite{BBH} that there exist non-minimizing
solutions of the Ginzburg-Landau equation whose vortex at the origin
is an arbitrarily prescribed nonzero integer.

In this paper, we find a sharp upper bound in terms of $1/\epsilon$
for the degree of vortices
for solutions which are not necessarily
minimizing. 
Chanillo and Kiessling proved in
\cite[Lemma~6]{CK} that if $x_0$ is a vortex of degree
$d\in{\mathbb N}$, then the vanishing order of $u$ is at least
$d$. Therefore, 
we may use
unique continuation methods to address this problem.
Using the result of Chanillo and Kiessling, the paper
\cite{Ku2} implies
that in the homogeneous Dirichlet and periodic cases
the degree of $u$
at any vortex $x_0$ is less than
$C\epsilon^{-2}$ where $C$ depends only on $\Omega$. In Theorem~\ref{t3.1}
below
we improve this bound to $C\epsilon^{-1}$ and then show that this
bound is best
possible. The main tool in
the proof is a new logarithmically convex quantity for the Laplacian
operator. More precisely, for any $\alpha>-1$ and any harmonic
function $u$, the quantity
  \[
   H(r)=\int_{B_r(0)}u(x)^2 \bigl(
                             r^2-|x|^2
                            \bigr)^{\alpha}\,dx
  \]
is logarithmically convex, i.e., $\log H(r)$ is a convex function of
$\log r$. Due to cancellations of terms involving $\alpha$ in
(\ref{f24})--(\ref{f14}) below, and due to  a gradient structure of
the Ginzburg-Landau equation,
we can choose an appropriate optimal $\alpha$ which gives our
result. Inspired
by an example in \cite{BBH}, we construct in Remark~\ref{r3.3}
a solution which shows that our bound $C\epsilon^{-1}$ can not be
improved upon. Theorem~\ref{t3.2} contains an
estimate concerning the boundary condition $u|_{\partial \Omega}=g$ where
$|g|=1$, while Theorem~\ref{t3.1neu} covers the Neumann case.

For properties of stationary Ginzburg-Landau equation, 
cf.~\cite{BBH,DF,L,LR,S} and to \cite{AEK,A,Al,B,GL,K,Ku2,Ku3} for 
various results on logarithmic convexity and unique continuation. 

\section{Quantitative uniqueness for systems}
\setcounter{equation}{0}
\label{sec2}
In this section, we
consider nontrivial solutions $u$  of the system
  \begin{eqnarray}
     & \Delta u
     = F'\bigl(|u|^2\bigr)u &\nonumber\\
     & u|_{\partial \Omega}=0 &\label{f16}
  \end{eqnarray}
where 
$u\in C^2(\Omega,{\mathbb R}^{D})\cap C(\overline\Omega, {\mathbb R}^{D})$ 
with $D\in{\mathbb N}$. We assume that
$\Omega\subseteq{\mathbb R}^{d}$, where $d\ge2$, and one of the
following:
\begin{itemize}
\item[(a)] $\Omega$ is a convex bounded domain;
\item[(b)] $\Omega$ is a Dini domain; Dini domains are bounded
domains with the following property: Around any point there is a
neighborhood $N$,
such that after a rotation of coordinates $\Omega\cap N$ lies below
a graph
of a function whose normal is Dini continuous (see \cite{KN} for
details);
\item[(c)] $\Omega$ is a periodic cube $[0,L]^{d}$; in this case,
$\partial\Omega=\emptyset$.
\end{itemize}

As in \cite{Ku2}, we are mainly interested in periodic boundary
conditions; the papers \cite{AEK} and \cite{KN} enable us to consider
homogeneous Dirichlet conditions without much change. As far as the
Ginzburg-Landau equation is concerned, the nonhomogeneous boundary
conditions $u|_{\partial\Omega}=g$ (with $|g|=1$) and homogeneous
Neumann conditions $(du/ d\nu)|_{\partial\Omega}=0$ are more
physically relevant and more widely studied.
Theorem~\ref{t3.1} addresses the homogeneous Dirichlet boundary
conditions; the non-homogeneous boundary conditions are considered in
Theorem~\ref{t3.2}, while
Theorem~\ref{t3.1neu} covers the Neumann case.
Let $M=\max_{\overline{\Omega}}|u|^2$. On
$F\colon[0,M]\to{\mathbb R}$, we make 
the following assumptions:
\begin{itemize}
\item[(i)] $F\in C^{1}\bigl([0,M]\bigr)$ and
  \begin{equation}
   \bigl|
    F'(x)
   \bigr|\le \lambda
    \,,\quad x\in[0,M]
    \label{f01}
  \end{equation}
for some
$\lambda>0$;
\item[(ii)] $F(0)=0$;
\item[(iii)] $F$ is convex on $[0,M]$.
\end{itemize}

Conditions (ii) and (iii) imply
  \begin{equation}
   F(x)\le xF'(x)
    \,,\quad x\in[0,M]
    \label{f18}
  \end{equation}
and
  \begin{equation}
   \bigl|
    F(x)
   \bigr|\le \lambda x
   \,,\quad x\in [0,M]\,.   
    \label{f02}
  \end{equation}

The following is the main result of this section.

We recall that the order of vanishing 
at $x_0\in\overline{\Omega}$ is defined as the largest
integer $n\in{\mathbb N}_0=\{0,1,\ldots\}$ such that
  \[
   {
    1
    \over
    |B_r(x_0) \cap \Omega|
   }\int_{B_r(x_0) \cap \Omega} |u|^2={\mathcal O}(r^{2n})
    \,,\quad {\rm as } r\to0\,.
  \]
(Here and in each subsequent occurrence, one needs to replace
$B_r(x_0) \cap \Omega$ with $B_r(x_0)$ in the case of periodic
boundary conditions (c).) In particular, if $u$ does not vanish 
at $x_0$, then the order of vanishing is $0$. We also add that $u$ may
not have any zeros in $\Omega$.

\begin{Theorem}
\label{t2.1}
Let $x_0\in\overline{\Omega}$. The order of
vanishing of $u$ at $x_0$ is less than $C(\sqrt{\lambda}+1)$ where $C$
is a constant depending only on $\Omega$.
\end{Theorem}

If $\lambda$ is sufficiently small, and if
$\Omega$ satisfies (a) or (b), then there are no nontrivial solutions
of (\ref{f16}). In these cases,
the bound $C(\sqrt{\lambda}+1)$ may be
replaced by $C\sqrt{\lambda}$.

Let $x_0\in\overline{\Omega}$ and $R>0$ be such that $B_R(x_0)\cap
\Omega$ is starshaped with respect to $x_0$. For an arbitrary
$\alpha>-1$ and $r>0$, denote
  \[
   H_{x_0}(r)=  \int_{B_{r}(x_0)\cap \Omega} \bigl|u(x)\bigr|^2
                    \bigl(
                     r^2-|x-x_{0}|^2
                    \bigr)^{\alpha}\,dx
  \]
where $|u|^2=u_ju_j$. We will omit the dependency on $x_0$ when it is
clear from the context.

\begin{Lemma}
\label{l2.2}
Let $q\ge1$, and let $0<r_1<r_2$ be such that
$qr_2\le R$. Then
  \[
   \log{
        H_{x_0}(qr_1)
        \over
        H_{x_0}(r_1)
       }       \le
   \log{
        H_{x_0}(qr_2)
        \over
        H_{x_0}(r_2)
       }       
       +
       {
        (q^2-1)r_2^2d\lambda
        \over
        \alpha+1
       }\,.
  \]
\end{Lemma}

\paragraph{Proof of Lemma~\ref{l2.2}}
Without loss of generality,
$x_0=0$. Let $r\in(0,R)$ be arbitrary. Denoting $B_r=B_r(0)$, we get
  \begin{eqnarray}
     H'(r)&=&2\alpha r
     \int_{B_r\cap \Omega} |u|^2 \bigl(r^2-|x|^2\bigr)^{\alpha-1}
     \,dx \nonumber\\
     &=&
     {2\alpha\over r}H(r)
     - {1\over r}
     \int_{B_r\cap \Omega}
        x_j |u|^2 \,\partial_j\bigl(
                             \bigl(r^2-|x|^2\bigr)^{\alpha}
                            \bigr)\,dx \label{f19}
  \end{eqnarray}
whence, by the divergence theorem,
  \begin{equation}
   H'(r)
   = {
      2\alpha+d
      \over
      r
     }H(r)
     +
     {
      1
      \over
      (\alpha+1)r
     }I(r)
    \label{f04}
  \end{equation}
where
  \begin{equation}
   I(r)
   =2(\alpha+1)
   \int_{B_r\cap \Omega}
   x_j u_k \,\partial_{j}u_k \bigl(r^2-|x|^2\bigr)^{\alpha}\,dx
    \label{f20}\,.   
  \end{equation}
By the divergence theorem,
  \begin{eqnarray*}
     I(r)&=&
        -\int_{B_r\cap \Omega}
            u_k \,\partial_{j}u_k
            \,\partial_{j}\bigl(
                         \bigl(r^2-|x|^2\bigr)^{\alpha+1}
                        \bigr)
                        \,dx \nonumber\\
     &=&
     \int_{B_r\cap \Omega}\,\partial_{j}u_k\,\partial_{j}u_k
        \bigl(r^2-|x|^2\bigr)^{\alpha+1}\,dx \\
     && + \int_{B_r\cap \Omega}|u|^2
        F'(|u|^2)\bigl(r^2-|x|^2\bigr)^{\alpha+1}
        \,dx \,. 
  \end{eqnarray*}
As in (\ref{f19}) above, we get
  \begin{eqnarray*}
     I'(r)&=&
       {
        2(\alpha+1)
        \over
        r
       }
       \int_{B_r\cap \Omega}\,\partial_{j}u_k\,\partial_{j}u_k
          \bigl(r^2-|x|^2\bigr)^{\alpha+1}\,dx \nonumber\\
     &&- {1\over r}
     \int_{B_r\cap \Omega}
       x_m\,\partial_{j}u_k\,\partial_{j}u_k
       \,\partial_{m}\bigl(
                      \bigl(r^2-|x|^2\bigr)^{\alpha+1}
                     \bigr)
                     \,dx \nonumber\\
     &&+ 2(\alpha+1)r
     \int_{B_r\cap \Omega}|u|^2 F'(|u|^2)
     \bigl(r^2-|x|^2\bigr)^{\alpha}\,dx\,. 
  \end{eqnarray*}
Using the divergence theorem on the second integral leads to
  \begin{eqnarray*}
     I'(r)&=&
     {
      2(\alpha+1)+d
      \over
      r
     }
     \int_{B_r\cap \Omega}
     \,\partial_{j}u_k
     \,\partial_{j}u_k \bigl(r^2-|x|^2\bigr)^{\alpha+1}\,dx \nonumber\\
     &&+ {
        2
        \over
        r
       }
       \int_{B_r\cap \Omega}x_m
       \,
       \partial_{j}u_k
       \,
       \partial_{jm}u_k
       \bigl(r^2-|x|^2\bigr)^{\alpha+1}\,dx \nonumber\\
     &&-{1\over r}
     \int_{B_r\cap \partial\Omega} x_m
        \,\partial_{j}u_k\,\partial_{j}u_k
        \bigl(r^2-|x|^2\bigr)^{\alpha+1}
        \nu_m\,d\sigma(x) \nonumber\\
     &&+2(\alpha+1)r
     \int_{B_r\cap \Omega}|u|^2F'(|u|^2)
     \bigl(r^2-|x|^2\bigr)^{\alpha}\,dx 
  \end{eqnarray*}
where $\nu=(\nu_1,\ldots,\nu_d)$ 
denotes the outward unit normal and where $d\sigma(x)$
stands for the surface measure on $\partial\Omega$. Applying the
divergence theorem in the second integral, this time with respect to
the variable $x_j$, we get
  \begin{eqnarray}
     I'(r)&=&
     {
      2\alpha+d
      \over
      r
     }
     \int_{B_r\cap \Omega}
     \,\partial_{j}u_k
     \,\partial_{j}u_k
     \bigl(r^2-|x|^2\bigr)^{\alpha+1}\,dx \nonumber\\
     &&-{2\over r}\int_{B_r\cap \Omega}
        x_m\partial_{m}u_k
        \,F'\bigl(|u|^2\bigr)u_k\bigl(r^2-|x|^2\bigr)^{\alpha+1}\,dx  \nonumber\\
     &&+ {4(\alpha+1)\over r}\int_{B_r\cap \Omega}x_m\partial_{m}u_k
       \,x_j\partial_{j}u_k
       \bigl(r^2-|x|^2\bigr)^{\alpha}\,dx \nonumber\\
     && + {2\over r}
     \int_{B_r\cap \partial\Omega}
       x_m\partial_{m}u_k\, \partial_{j}u_k
       \bigl(r^2-|x|^2\bigr)^{\alpha+1} \nu_j \,d\sigma(x) \nonumber\\
     &&-{1\over r}
     \int_{B_r\cap \partial\Omega}
        x_m \partial_{j}u_k\,\partial_{j}u_k
        \bigl(r^2-|x|^2\bigr)^{\alpha+1}\nu_m\,d\sigma(x) \nonumber\\
     &&+ 2(\alpha+1)r
     \int_{B_r\cap \Omega}|u|^2F'\bigl(|u|^2\bigr)
     \bigl(r^2-|x|^2\bigr)^{\alpha}\,dx\,.     \label{f03}
  \end{eqnarray}
The second term on the right hand side of (\ref{f03}) equals
  \begin{eqnarray*}
    \lefteqn{-{1\over r}
     \int_{B_r\cap \Omega}x_m
     \,\partial_{m}\bigl(
                    F\bigl(|u|^2\bigr)
                   \bigr)
                   \bigl(r^2-|x|^2\bigr)^{\alpha+1}\,dx }\\
     &=&{d\over r}
      \int_{B_r\cap
      \Omega}F\bigl(|u|^2\bigr)\bigl(r^2-|x|^2\bigr)^{\alpha+1}\,dx  \\     
      &&-{2(\alpha+1) \over  r }
      \int_{B_r\cap \Omega}F\bigl(|u|^2\bigr)|x|^2
        \bigl(r^2-|x|^2\bigr)^{\alpha}\,dx\,;
  \end{eqnarray*}
since the boundary integral vanishes by $F(0)=0$.
On the other hand, the sum of the fourth and the fifth term is
  \[
   {1\over r}\int_{B_r\cap \partial\Omega}
     x_k\nu_k
     {
      \partial u_m
      \over
      \partial\nu
     }
     {
      \partial u_m
      \over
      \partial\nu
     }
     \bigl(r^2-|x|^2\bigr)^{\alpha+1}
     \,d\sigma(x)
  \]
which is due to the fact $\,\partial_{k}u=(\partial
u/\partial\nu)\nu_k$ resulting from $u|_{\partial\Omega}=0$. This
integral is nonnegative
since
$B_r\cap \Omega$ is starshaped with respect to $0$.
Then
  \begin{eqnarray}
     I'(r)&\ge&
     {
      2\alpha+d
      \over
      r
     }I(r)
     -
     {
      2\alpha+d
      \over
      r
     }
     \int_{B_r\cap
     \Omega}|u|^2F'\bigl(|u|^2\bigr)
          \bigl(r^2-|x|^2\bigr)^{\alpha+1} \,dx \nonumber\\
     &&+
     {d\over r}
     \int_{B_r\cap \Omega} F\bigl(|u|^2\bigr)
     \bigl(r^2-|x|^2\bigr)^{\alpha+1}\,dx \nonumber\\
     &&- {
        2(\alpha+1)
        \over
        r
       }
       \int_{B_r\cap \Omega} F\bigl(|u|^2\bigr)
         |x|^2\bigl(r^2-|x|^2\bigr)^{\alpha}\,dx \nonumber\\
     &&+ {4(\alpha+1)
\over
r}\int_{B_r\cap \Omega}
         x_m\,\partial_{m}u_k
         x_j\,\partial_{j}u_k
         \bigl(r^2-|x|^2\bigr)^{\alpha}
         \,dx \nonumber\\
     &&+
     2(\alpha+1)r
       \int_{B_r\cap \Omega}|u|^2F'\bigl(|u|^2\bigr)
         \bigl(r^2-|x|^2\bigr)^{\alpha}\,dx\,.     \label{f21}
  \end{eqnarray}
The sum of the second, the third, the fourth, and the sixth term is 
  \[
   {1\over r}
   \int_{B_r\cap \Omega}E(x)\bigl(r^2-|x|^2\bigr)^{\alpha}\,dx
  \]
where
  \begin{eqnarray}
     E(x)&=&
      - (2\alpha+d)
      |u|^2F'\bigl(|u|^2\bigr)\bigl(r^2-|x|^2\bigr)
      + d F\bigl(|u|^2\bigr)\bigl(r^2-|x|^2\bigr) \nonumber\\     &&-
     (2\alpha+2)F\bigl(|u|^2\bigr)|x|^2
     + (2\alpha+2)F'\bigl(|u|^2\bigr)|u|^2r^2\,.  \label{f24}
  \end{eqnarray}
We get
  \begin{eqnarray}
     E(x)
     &=&
     r^2\Bigl(
       -2\alpha|u|^2F'\bigl(|u|^2\bigr)
       -d|u|^2F'\bigl(|u|^2\bigr)
       + d F\bigl(|u|^2\bigr) \nonumber\\ &&  +
       2\alpha F'\bigl(|u|^2\bigr)|u|^2
       + 2F'\bigl(|u|^2\bigr)|u|^2
     \Bigr) \nonumber\\     &&+
     |x|^2\Bigl(
      2\alpha F'\bigl(|u|^2\bigr)|u|^2
      + dF'\bigl(|u|^2\bigr)|u|^2  \nonumber\\ && - dF\bigl(|u|^2\bigr)
      - 2\alpha F\bigl(|u|^2\bigr)-2F\bigl(|u|^2\bigr)
     \Bigr) \label{f13}
  \end{eqnarray}
from where, using (iii), 
  \begin{eqnarray}
     E(x)&
     \ge&
     r^2\Bigl(
      (2-d)|u|^2F'\bigl(|u|^2\bigr)+dF\bigl(|u|^2\bigr)
     \Bigr)
     -2|x|^2F\bigl(|u|^2\bigr) \,. \label{f14}
  \end{eqnarray}
By (\ref{f01}) and (\ref{f02}) and using $|x|^2\le r^2$, we get
  \[
   E(x)\ge -4d\lambda r^2|u|^2
  \]
which leads to
$$
   I'(r) \ge { 2\alpha+d \over  r } I(r)
   + { 4(\alpha+1) \over r }
   \int_{B_r\cap \Omega} x_m\partial_{m}u_k\, x_j\partial_{j}u_k
      \bigl(r^2-|x|^2\bigr)^{\alpha}  \,dx
    - 4d\lambda r H(r)\,.
$$ 
Now, let $N(r)=I(r)/H(r)$. Then
  \begin{eqnarray*}
     N'(r)&\ge& -4d\lambda r + { 4(\alpha+1) \over rH(r)^2 }
     \Biggl( \int_{B_r\cap \Omega}
            x_m\partial_{m}u_k
            \,x_j\partial_{j}u_k
            \bigl(r^2-|x|^2\bigr)^{\alpha}\,dx  \\
     &&\times
        \int_{B_r\cap \Omega}|u|^2\bigl(r^2-|x|^2\bigr)^{\alpha}\,dx
      - \big(
        \int_{B_r\cap \Omega}
        x_j u_k \,\partial_{j}u_k
        \bigl(r^2-|x|^2\bigr)^{\alpha} \,dx
       \big)^2
         \Biggr) 
     \end{eqnarray*}
whence, by the Cauchy-Schwarz inequality,
$   N'(r)\ge-4d\lambda r$, i.e.,
  \begin{equation}
   N(r_2)-N(r_1)
   \ge
   -2d\lambda(r_2-r_1)(r_2+r_1)
    \,,\quad 0<r_1\le r_2\le R\,.
    \label{f05}    
  \end{equation}
Let $q\ge1$, and let $0<r_1\le r_2\le R$ be such that $qr_2\le
R$. Dividing (\ref{f04}) by $H(r)$ and integrating the resulting
equality between $r_1$ and $qr_1$ leads to
  \begin{eqnarray*}
     \log{
          H(qr_1)
          \over
          H(r_1)
         }
     &=&
     (2\alpha+d)\log q
     +
     {1\over \alpha+1}
     \int_{r_1}^{qr_1}
     {
      N(\rho)
      \over
      \rho
     }
     \,d\rho \nonumber\\
     &=&
     (2\alpha+d)\log q
     +
     {
      1
      \over
      \alpha+1
     }
     \int_{r_2}^{qr_2}
     {
      N(r_1\rho/r_2)
      \over
      \rho
     }\,d\rho\,.
  \end{eqnarray*}
Using (\ref{f05}), we get
  \begin{eqnarray*}
     \log{ H(qr_1)\over H(r_1)}&\le&
     (2\alpha+d)\log q
     + (r_2^2-r_1^2){
                     (q^2-1)d\lambda
                     \over
                     \alpha+1
                    }
       + {1\over \alpha+1}
         \int_{r_2}^{qr_2}
         {
          N(\rho)
          \over
          \rho
         }\,d\rho \nonumber\\
     &=&
     \log{
          H(qr_2)
          \over
          H(r_2)
         }
       +
       {
        (r_2^2-r_1^2)(q^2-1)d\lambda
        \over
        \alpha+1
       } 
       \end{eqnarray*}
and the lemma follows.
\hfill$\Box$

Again, let $x_0\in\overline\Omega$ be fixed, and denote
  \[
   h_{x_0}(r)
   =
   \int_{B_r(x_0)\cap \Omega}|u(x)|^2\,dx\,.   
  \]
Let $R>0$ be such that $B_{R}(x_0)\cap\Omega$ is starshaped with
respect to $x_0\in\overline\Omega$.

\begin{Lemma}
\label{l2.3}
Let $\alpha\ge0$, $0<r_1<4r_2/3$ and $4r_2\le
R$. Then, with the above assumptions,
  \[
   \log{
        h_{x_0}(2r_1)
        \over
        h_{x_0}(r_1)
       }
     \le
   \log{
        h_{x_0}(4r_2)
        \over
        h_{x_0}(r_2)
       }
     +
     C \left(
        \alpha+{
                dr_2^2\lambda
                \over
                \alpha+1
               }
       \right)
  \]
where $C$ is a universal constant.
\end{Lemma}

\paragraph{Proof of Lemma~\ref{l2.3}}
Denote $h(r)=h_{x_0}(r)$ and
$H(r)=H_{x_0}(r)$.
If $0<r<\rho$, then
  \begin{equation}
   H(r)\le r^{2\alpha}h(r)
    \label{f06}   
  \end{equation}
and
  \begin{equation}
   h(r)
   \le
   {
    H(\rho)
    \over
    (\rho^2-r^2)^{\alpha}
   }
    \label{f07}
  \end{equation}
Therefore,
  \[
   \log
   {
    h(2r_1)
    \over
    h(r_1)
   }
   \le 
   \log{
        H(3r_1)
        \over
        H(r_1)
       }
    -
    \alpha\log 5
    \le
    \log{
         H(3r_1)
         \over
         H(r_1)
        }
  \]
since $\alpha\ge0$. By Lemma~\ref{l2.2},
  \[
   \log{
        h(2r_1)
        \over
        h(r_1)
       }
    \le
    \log{
         H(4r_2)
         \over
         H(4r_2/3)
        }
     +
     {
      Cr_2^2d\lambda
      \over
      \alpha+1
     }\,.
  \]
Using (\ref{f06}) and (\ref{f07}) again, we get our assertion.
\hfill$\Box$

In Theorem~\ref{t3.2} below, we will need an interior version of the
above lemma, which we state for sake of completeness.

\begin{Lemma}
\label{l2.3int}
Let $u$ be a solution of 
$\Delta u = F'\bigl(|u|^2\bigr)u$, where $F$ is as above, in
$B_R(x_0)$. Let
$\alpha\ge0$, $0<r_1<4r_2/3$ and $4r_2\le R$
Then
  \[
   \log{
        h_{x_0}(2r_1)
        \over
        h_{x_0}(r_1)
       }
     \le
   \log{
        h_{x_0}(4r_2)
        \over
        h_{x_0}(r_2)
       }
     +
     C \left(
        \alpha+{
                dr_2^2\lambda
                \over
                \alpha+1
               }
       \right)
  \]
where  $
   h_{x_0}(r)
   =
   \int_{B_r(x_0)}|u(x)|^2\,dx
  $
and $C$ is a universal constant.
\end{Lemma}

\paragraph{Proof of Lemma~\ref{l2.3int}}
The proof is the same as that of Lemma~\ref{l2.3}.
\hfill$\Box$

Next lemma will be used in the overlapping chain of balls argument.

\begin{Lemma}
\label{l2.4}Let $\alpha\ge0$. Assume that
$x_1,x_2\in\overline\Omega$ 
and $r>0$
are such that $B_{20r}(x_1)\cap\Omega$ is starshaped with respect
to $x_2\in\overline\Omega$. If $B_r(x_1)$ and $B_r(x_2)$ intersect, and
if
  \[
   \int_{\Omega} |u(x)|^2\,dx\le K H_{x_1}(r)
  \]
for some $K\ge0$, then
  \[
   \int_{\Omega} |u(x)|^2\,dx\le K^{3}
      \exp\left(
           C\left(
             \alpha+{
                     \lambda
                     \over
                     \alpha+1
                    }
            \right)
          \right)H_{x_2}(r)
  \]
where $C$ is a constant which depends only on $d$ and $
\diam(\Omega)$.
\end{Lemma}


\paragraph{Proof of Lemma~\ref{l2.4}}
It is easy to check that
  $
   H_{x_1}(r)\le
   H_{x_2}(4r)\,.
  $
Therefore,
  \begin{equation}
   \int_{\Omega}|u|^2
   \le
   K H_{x_1}(r)
   \le
   K H_{x_2}(4r)
    \label{f09}
  \end{equation}
which, by (\ref{f06}), implies
  \[
   H_{x_2}(8r)
   \le
   (8r)^{2\alpha}h(8r)
   \le
   (8r)^{2\alpha}   \int_{\Omega}|u|^2
   \le
   C^{\alpha} K H_{x_2}(4r)
  \]
where $C$ denotes a generic constant which depends only on $d$ and
$\diam \Omega$. Lemma~\ref{l2.2} then implies
  \begin{equation}
   \log{
        H_{x_2}(4r)
        \over
        H_{x_2}(2r)
       }
     \le
     \log K
     + C\left(
         \alpha+{
                 \lambda
                 \over
                 \alpha+1
                }
        \right)
    \label{f10}
  \end{equation}
and similarly
  \begin{equation}
   \log{
        H_{x_2}(2r)
        \over
        H_{x_2}(r)
       }
     \le
     2\log K
     + C\left(
         \alpha+{
                 \lambda
                 \over
                 \alpha+1
                }
        \right)\,.
    \label{f11}
  \end{equation}
The inequalities (\ref{f09}), (\ref{f10}), and (\ref{f11}) then give
  \[
   \log
   {
    \int_{\Omega}|u|^2
    \over
    H_{x_2}(4r)
   }
   \le
   3\log K
   +
   C\left(
         \alpha+{
                 \lambda
                 \over
                 \alpha+1
                }
        \right)
  \]
which gives our assertion.
\hfill$\Box$


\paragraph{Proof of Theorem~\ref{t2.1}}
In the cases (a) and (c), we can take
$R$ to be arbitrarily large. Note that, in the case (a),
  \begin{equation}
   \log{
        h_{x_0}(4r)
        \over
        h_{x_0}(r)
       }=0
    \,,\quad x_0\in \overline\Omega
    \label{f08}
  \end{equation}
provided $r\ge\diam\Omega$. Therefore, by Lemma~\ref{l2.3}, there is a
numerical constant $C$ such that
  \[
   \log{
        h_{x_0}(2r_1)
        \over
        h_{x_0}(r_1)
       }
       \le
       C
       \left(
        \alpha
        +
        {
         d \diam(\Omega)^2 \lambda
         \over
         \alpha+1
        }
       \right)
    \,,\quad x_0\in \overline\Omega
  \]
for every $\alpha\ge0$ and  $r_1\in(0,\diam \Omega)$. Choosing
$\alpha=\sqrt{d\lambda}\diam \Omega$, we get
  \[
   \log{
        h_{x_0}(2r)
        \over
        h_{x_0}(r)
       }
       \le
       C
       \sqrt{\lambda d}\,\diam \Omega
    \,,\quad x_0\in \overline\Omega
  \]
for $r\in(0,\diam \Omega)$, and Theorem~\ref{t2.1} follows. In the case (c),
the argument is the same. The
only difference is that (\ref{f08}) is replaced by
  \[
   \log{
        h_{x_0}(4r)
        \over
        h_{x_0}(r)
       }\le C
    \,,\quad x_0\in \overline\Omega
  \]
provided $r\ge\diam \Omega$, where $C$ is a constant depending only on
$d$. In this case we therefore obtain
  \[
   \log{
        h_{x_0}(2r)
        \over
        h_{x_0}(r)
       }
       \le
       C
       (1+\sqrt{\lambda}\,\diam \Omega)
    \,,\quad x_0\in \overline\Omega
  \]
for $r\in(0,\diam \Omega)$, where $C$ is a constant which depends only
on dimension $d$.

The proof in the case (b) involves a standard argument employing
overlapping chain 
of balls (cf.\ \cite{Ku1, Ku3}).
Below, the symbol $C$ denotes a generic constant depending
only on $\Omega$. First, we choose $r>0$ and
$x_1,\ldots,x_m\in\overline\Omega$ such that
\begin{itemize}
\item[(1)] $B(x_1,r/2),\ldots,B(x_m,r/2)$ cover $\overline\Omega$;
\item[(2)] for every $j\in\{1,\ldots,m\}$, the region $\Omega\cap
B(x_j,10r)$ is starshaped with respect to $x_j$;
\item[(3)] if $B(x_j,10r)$ intersects $\partial \Omega$, it is assumed
that the variation of the normal $\nu$ is sufficiently small
(cf.~\cite[p.~444]{KN}).
\end{itemize}
We fix $\alpha=\sqrt \lambda + 1$. There exists
$j_0\in\{1,\ldots,m_0\}$ such that
  \[
   \int_{B_{r/2}(x_{j_{0}})} |u|^2
   \ge
   {1\over m}\int_{\Omega}|u|^2
  \]
whence
  \[
   \int_{\Omega}|u|^2 \le C^{\alpha}H_{x_{j_{0}}}(r)\,.
  \]
For every $j\in\{1,\ldots,m\}$, there exists an overlapping chain of
(distinct) balls from (1) connecting $B_{r}(x_j)$ and
$B_r(x_{j_{0}})$. Repeated use of Lemma 2.4 then gives
  \[
   \int_{\Omega}|u|^2
   \le
   C^{\sqrt\lambda+1} H_{x_{j}}(r)
    \,,\quad j=1,\ldots,m\,.    
  \]
Therefore,
  \[
  H_{x_j}(2r)
  \le
  C^{\sqrt \lambda+1}
  H_{x_j}(r)
  \,,\quad j=1,\ldots,m\,.
  \]
An argument parallel to \cite[p.~445]{KN} then
leads to
  \[
  H_{x}(2\rho)
  \le
  C^{\sqrt \lambda+1}
  H_{x}(\rho)
  \]
for every $x\in \overline \Omega$ and arbitrary $\rho\in(0,r/2)$. Using
(\ref{f06}) and (\ref{f07}), we get the theorem.
\hfill$\Box$

\section{The degree of Ginzburg-Landau vortices}
\setcounter{equation}{0}\label{sec3}

Now, we apply
Theorem~\ref{t2.1} to the Ginzburg-Landau equation
  \begin{eqnarray}
     &\Delta u
     = -{ 1 \over \epsilon^2 }
      \bigl(
       1-|u|^2
      \bigr)u \nonumber\\
     &u|_{\partial \Omega}=0 \,,&
    \label{f17}
  \end{eqnarray}
where $u\colon \overline \Omega\to{\mathcal C}$ is assumed to be nontrivial. The
domain $\Omega\subseteq {\mathbb R}^2$ is as in the beginning of Section~\ref{sec2} and
$\epsilon>0$.

\begin{Theorem}
\label{t3.1}
The order of vanishing of $u$ at
$x_0\in\overline\Omega$  is less than
  \begin{equation}
   C\left(
     {
      1
      \over
      \epsilon
     }
     +1
    \right)
    \label{f12}
  \end{equation}
where $C$ is a constant which depends only on $\Omega$.
\end{Theorem}

As it was pointed out in the remark following Theorem~\ref{t2.1}, the above
bound (\ref{f12}) can be replaced by $C/\epsilon$ if $\Omega$
satisfies (a) or (b).

By \cite{CK}, (\ref{f12}) then provides an estimate for the the degree
of $u$ at any 
vortex $x_0\in\Omega$. (Recall that $x_0$ is a vortex if $u(x_0)=0$ and the
degree of $u$ at $x_0$ is nonzero.) Namely, by \cite[Lemma~6]{CK} and
our Theorem~\ref{t3.1}, the degree at every vortex is less than (\ref{f12}).



\paragraph{Proof}
By the maximum principle, we conclude
$\bigl|u(x)\bigr|\le1$ for $x\in\overline\Omega$, i.e., $M=1$. Taking
  \[
   F(x)=-{1\over\epsilon^2}x+{1\over2\epsilon^2}x^2
  \]
we easily verify that (i)--(iii) are satisfied with
$\lambda=\epsilon^{-2}$. Theorem~\ref{t3.1} then follows from
Theorem~\ref{t2.1}.
\hfill$\Box$

Next, we present
a result concerning the nonhomogeneous
boundary conditions $u|_{\partial \Omega}=g$ where
$g\colon\partial\Omega\to S^{1}$ is sufficiently regular, e.g.\
continuous. We assume
that $\Omega$ is starshaped. In this case, Bethuel, ~Br\'ezis,
and H\'elein proved in \cite[Lemma~X.1]{BBH} that
  \begin{equation}
   \int_{\Omega}
     \bigl(
      1-|u|^2
     \bigr)^2
     \le C_0 \epsilon^2
    \label{f15}
  \end{equation}
where $C_0$ depends only on $g$ and $\Omega$.

\begin{Theorem}
\label{t3.2}
The order of vanishing of $u$ at $x_0\in\Omega$
is less than $C/\epsilon$ where $C$ depends on $\Omega$, the boundary
function $g$, and the distance from $x_0$ to $\partial \Omega$.
\end{Theorem}

\paragraph{Proof}
It is easy to check that if $\epsilon$ is sufficiently
large, then $u$ does not vanish. (For instance, we may use the
inequality 
$\bigl|\nabla u(x)\bigr|\le C/\epsilon$ from \cite{BBH} where $C$ depends
on $g$ and $\Omega$.)
Let $x_0\in\Omega$, denote $R=\dist(x_0, \partial \Omega)$ and
$r_0=R/4$. We distinguish two cases.

Case 1: $\epsilon\ge R^2/(C\cdot C_0)$ where $C$ is a large enough
numerical constant and $C_0$ is as in (\ref{f15}). In this case, we can
use analyticity arguments to show that the order of vanishing is
bounded by a constant depending only on $\Omega$, $g$, and $R$
(cf.~\cite{Ku2}).

Case 2: $\epsilon\le R^2/(C\cdot C_0)$ where $C$ is large enough. Then
(\ref{f15}) implies
  \[
   \int_{B_{R/4}(x_0)}|u|^2 \ge {
                                 R^2
                                 \over
                                 C
                                }
  \]
as can be readily checked. Since also $\max_{\overline\Omega}|u|=1$, we get
  \[
    \int_{B_{R}(x_0)}|u|^2
    \le
    C
    \int_{B_{R/4}(x_0)}|u|^2
  \]
where $C$ depends on $\Omega$, $g$, and $R$.
Since $R=\dist(x_0,\partial \Omega)$, we have 
$B_R(x_0)\cap\partial\Omega=\emptyset$. Therefore,
by Lemma~\ref{l2.3int}, we get
  \[
   \log{
        \int_{B_{2r}(x_0)}|u|^2
        \over
        \int_{B_{r}(x_0)}|u|^2
       }
     \le
      C+C\left(
          \alpha+{2R^2\epsilon^{-2}\over \alpha+1}
         \right)
  \]
for all $\alpha\ge0$ provided $r<R/3$. Choosing
$\alpha=R/\epsilon$, we get
  \[
   \log{
        \int_{B_{2r}(x_0)}|u|^2
        \over
        \int_{B_{r}(x_0)}|u|^2
       }
     \le
        C\left(
          {R\over\epsilon}+1
         \right)\,.
  \]
Note that $1\le CR/\epsilon$ due to the fact
$\epsilon\le R^2/CC_0$. Therefore,
  \[
   \log{
        \int_{B_{2r}(x_0)}|u|^2
        \over
        \int_{B_{r}(x_0)}|u|^2
       }
     \le
        C
        {\dist(x_0,\partial \Omega)\over\epsilon}
  \]
and the statement follows.
\hfill$\Box$


\begin{Remark}
\label{r3.3}
{\rm Here we show by means of an example that
Theorem~\ref{t3.1} is sharp. Let
$\Omega=B_1(0)$. We shall show that there exists $\epsilon_0>0$ with
the following property: For every $\epsilon\in(0,\epsilon_0)$, there
exists a solution $u$ of (\ref{f17}) such that the degree of $u$
at $0$ is at least $1/C\epsilon$.

We seek this solution in the form $u(x)=f(r)e^{id\theta}$, where
$x=re^{i\theta}$, with a suitable fixed integer $d$. We find $f$ as
a global minimizer of the functional
  \[
   \Phi(f)= \int_{0}^{1}
      \left(
       rf'^2
       + {d^2\over r}f^2
       + {r\over2\epsilon^2}(f^2-1)^2
      \right)
      \,dr
  \]
in the space
  \[
   V=
   \left\{
    f\in H_{\rm loc}^{1}(0,1):
     \sqrt{r}f', {
                  f
                  \over
                  \sqrt{r}
                 }
                 \in L^2(0,1),
      f(1)=0
   \right\}\,.
  \]
What remains to be shown is that if $d$ is suitably chosen, then the
minimizer $f$ is not identically zero. Choose an arbitrary $g\in V$
such that $0<g(r)<1$ for $r\in(0,1)$, say $g(r)=r(1-r)$. Then if 
$\epsilon\in(0,\epsilon_0)$, where $\epsilon_0$ is sufficiently small,
and if $d=[1/C\epsilon]$ where $C$ is large enough, then
  \[
   \Phi(g)<{
            1
            \over
            2\epsilon^2
           }
           \int_{0}^{1}r\,dr
           =\Phi(0)
  \]
and 0 can not be the global minimizer.

Now, $u(x)=f(r)e^{id\theta}$, where $f$ and $d$ are as above, is not
identically 0, it satisfies 
$u\in C(B_1)\cap C^{\infty}\bigl(B_1\backslash\{0\}\bigr)$ and solves
the Ginzburg-Landau equation
for $x\not=0$. But then it can be readily checked that $0$ is a
removable singularity,
and consequently (\ref{f17}) holds. It also
follows immediately that $0$ is an isolated
zero. Indeed, in the opposite case, there would be a sequence
$r_1,r_2,\ldots$ which converges to $0$ such that $u(x)=0$ if
$|x|=r_j$ for $j\in{\mathbb N}$. Therefore, $0$ would be a zero of infinite
order, which is not possible since $u\not\equiv0$.}
\end{Remark}

The rest of the paper is concerned with the Ginzburg-Landau equation
  \begin{equation}
   \Delta u =
   -{1\over\epsilon^2}
    \bigl(1-|u|^2\bigr)u
    \label{f22}
  \end{equation}
with homogeneous Neumann boundary conditions
  \begin{equation}
   {du\over d\nu}\biggm|_{\partial \Omega}=0
    \label{f23}
  \end{equation}
where $\Omega$ is a connected bounded $C^3$ domain. The treatment is
similar to 
(but not completely the same as)
the Dirichlet case. We need to consider
a conformal straightening of the boundary, which, in turn, leads us to
consider the following analog of (\ref{f16}). Let
  \begin{eqnarray*}
     &\Delta u
     = vF'\bigl(|u|^2\bigr)u &\\
     &{du\over d\nu}\biggm|_{\partial' \Omega}=0 \,,&
  \end{eqnarray*}
where
$\Omega=B_{R_0}^{+}(0)=
\bigl\{
 (x_1,\ldots,x_d)\in B_{R_0}=B_{R_0}(0): x_d>0
\bigr\}
$
and
$\partial'\Omega=
\bigl\{
 (x_1,\ldots,x_d)\in B_{R_0}: x_d=0
\bigr\}
$. As
in Section~\ref{sec2}, we denote
$M=\max_{\overline \Omega}|u|^2$ and we make same assumptions on
$F\colon[0,M]\to{\mathbb R}$ as before. We assume that $v$ is
a nonnegative function such that
$\max_{x\in\Omega}  v(x)\le M_0$
and
$\max_{x\in\Omega} \bigl|\nabla v(x)\bigr|\le M_1$.


\begin{Lemma}
\label{l2.3neu}
Let $\alpha>0$, $0<r_1<4r_2/3$, and $4r_2\le
R_0$. Then
  \[
   \log{
        h(2r_1)
        \over
        h(r_1)
       }
     \le
   \log{
        h(4r_2)
        \over
        h(r_2)
       }
     +
     C \left(
        \alpha+{
                dr_2^2\lambda
                \over
                \alpha+1
               }
       \right)
  \]
where $C$ depends only on $M_0$ and $M_1$, and where
$h(r)=\int_{B_{r}^{+}}\bigl|u(x)\bigr|^2\,dx$.
\end{Lemma}

\paragraph{Proof}
The proof is similar to that of Lemma~\ref{l2.3}; we only indicate
the main steps and point out the main differences. As before, we let
  \[
   H(r)=
   \int_{B_{r}^{+}}|u|^2\bigl(r^2-|x|^2\bigr)^{\alpha}\,dx\,.
  \]
Then (\ref{f04}) holds with (\ref{f20}) (where
$B_{r}\cap\Omega=B_{r}^{+}$). After a short computation, we get
  \begin{eqnarray*}
     I'(r)&=&
     {
      2\alpha+d
      \over
      r
     }
     I(r)
     - {2\alpha+d\over r}
     \int_{B_{r}^{+}}|u|^2vF'\bigl(|u|^2\bigr)
       \bigl(r^2-|x|^2\bigr)^{\alpha+1}\,dx \nonumber\\
     &&+{d\over r}\int_{B_{r}^{+}}
        v F\bigl(|u|^2\bigr)
        \bigl(r^2-|x|^2\bigr)^{\alpha+1}\,dx  \nonumber\\
     &&- {
        2(\alpha+1)
        \over
        r
       }
       \int_{B_{r}^{+}}|x|^2v
       F\bigl(|u|^2\bigr)
       \bigl(r^2-|x|^2\bigr)^{\alpha}\,dx \nonumber\\
     &&+
      {1\over r}
     \int_{B_{r}^{+}}x_m\partial_m v\,F\bigl(|u|^2\bigr)
     \bigl(r^2-|x|^2\bigr)^{\alpha+1}\,dx\\
     &&+{4(\alpha+1)\over r}
     \int_{B_{r}^{+}}
     x_j\partial_ju_k\,x_m\partial_mu_k\,
     \bigl(r^2-|x|^2\bigr)^{\alpha}\,dx
     \\
     &&+ 2(\alpha+1)r
     \int_{B_{r}^{+}}|u|^2vF'\bigl(|u|^2\bigr)
     \bigl(r^2-|x|^2\bigr)^{\alpha}\,dx 
  \end{eqnarray*}
in place of (\ref{f21}). From here, we get
  \begin{eqnarray*}
     I'(r)&\ge&
     {
      2\alpha+d
      \over
      r
     }
     I(r)
     - 4d\lambda r\max_{\Omega}v\,H(r) \nonumber\\
     &&+{4(\alpha+1)\over r}
     \int_{B_{r}^{+}}
     x_j\partial_ju_k\,x_m\partial_mu_k\,
     \bigl(r^2-|x|^2\bigr)^{\alpha}\,dx
     \\
     &&- \lambda (\max_{\Omega}|\nabla v|)
     \int_{B_{r}^{+}}|u|^2
     \bigl(r^2-|x|^2\bigr)^{\alpha+1}\,dx 
  \end{eqnarray*}
where we also used nonnegativity of $v$.
Instead of $N'(r)\ge-4d\lambda r$, which we had before, we now conclude
  \[
   N'(r)\ge
   -rd\lambda r\max_{\Omega}v
   -\lambda r^2\max_{\Omega}|\nabla v|\,.
  \]
The rest follows as before.
\hfill$\Box$

Now, we return to the Ginzburg-Landau equation (\ref{f22}) with the
Neumann boundary conditions (\ref{f23}), where $\Omega$ is a
$C^{3}$ bounded connected domain in ${\mathbb R}^2$.

\begin{Theorem}
\label{t3.1neu}
The order of vanishing of $u$ at
$x_0\in\overline\Omega$  is less than
(\ref{f12})
where $C$ is a constant which depends only on $\Omega$.
\end{Theorem}

\paragraph{Proof} (sketch) The proof of the theorem is analogous to
the proof os Theorem~\ref{t3.1}. The main difference is that we use a
conformal map to straighten the boundary. Namely, let
$x_0\in\partial\Omega$. Then, by the Riemann mapping theorem, there
exists $R_0>0$ and $r_0>0$ and a conformal map
  \[
   f\colon B_{r_0}(x_0)\cap\Omega\to B_{R_0}^{+}(0)
  \]
such that $f(x_0)=0$. The equation
(\ref{f22}) then transfers to
  \[
   \Delta u=-{1\over\epsilon^2}v
   \bigl(1-|u|^2\bigr)u
  \]
with $v=1/|f'|^2$. The boundary of $\Omega$ being $C^3$ guarantees
that $v$
and $\nabla v$ are bounded up to the lower boundary 
$\partial' B_{R_0}^{+}$
\cite{BK}. The rest is then established
as in the proof of Theorem~\ref{t3.1}, except that we use
Lemmas~\ref{l2.3neu} and~\ref{l2.3int} instead of Lemma~\ref{l2.3}.
\hfill$\Box$


\paragraph{Acknowledgement:} The author thanks Manuel Del~Pino
for numerous valuable discussions.
The work was
supported in part by the NSF grant DMS-0072662 while the author was
an Alfred P.~Sloan fellow.

\tolerance=100000

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\end{thebibliography} \medskip

\noindent {\sc Igor Kukavica}\\
Department of Mathematics\\
University of Southern California\\
Los Angeles, CA 90089\\
e-mail: kukavica@math.usc.edu

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