Calculus of Variations and Geometric Measure Theory

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


Published Paper

Inserted: 29 jan 2002

Journal: C.R. Acad. Sc. Paris Serie 1
Volume: 333
Number: 12
Pages: 1069-1076
Year: 2001


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 $
=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
$, 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