## F. Bethuel - J. Bourgain - H. Brezis - G. Orlandi

# $W^{1,p}$ estimates for solutions to the Ginzburg-Landau equation with boundary data in $H^{1/2}$

created on 29 Jan 2002

[

BibTeX]

*Published Paper*

**Inserted:** 29 jan 2002

**Journal:** C.R. Acad. Sc. Paris Serie 1

**Volume:** 333

**Number:** 12

**Pages:** 1069-1076

**Year:** 2001

**Abstract:**

We consider complex-valued solutions $u_\varepsilon$ of
the Ginzburg-Landau on a smooth bounded simply connected
domain $\Omega$ of $R^N$, $N\ge 2$ (here $\varepsilon$ is a parameter between $0$ and $1$). We assume that
$u_\varepsilon=g_\varepsilon$ on $\partial\Omega$, where
$

g_\varepsilon

=1$ and $g_\varepsilon$ is uniformly bounded in
$H^{1/2}(\partial\Omega)$. We also assume that the
Ginzburg-Landau energy $E_\varepsilon(u_\varepsilon)$ is bounded by
$M_0

\log\varepsilon

$, where $M_0$ is some given constant. We
establish, for every $1\le p<N/(N-1)$, uniform $W^{1,p}$
bounds for $u_\varepsilon$ (independent of $\varepsilon$). These types of
estimates play a central role in the asymptotic analysis
of $u_\varepsilon$ as $\varepsilon\to 0$.

**Keywords:**
Hodge decomposition, Ginzburg-Landau equation, Jacobians

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