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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{}
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\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 $00$ 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 $\alpha\ge0$, $00$
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 $r0$ 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 $00
\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 $\alpha>0$, $00$ 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
\end{document}