Inserted: 12 dec 2002
Journal: ESAIM, C.O.C.V.
We consider complex-valued solutions $u_\varepsilon$ of
the Ginzburg-Landau equation on a smooth bounded simply connected
domain $\Omega$ of $*R*^N$, $N\ge 2$, where $\varepsilon>0$ is a small parameter. We assume that the
Ginzburg-Landau energy $E_\varepsilon(u_\varepsilon)$ verifies the bound (natural in the context)
$, where $M_0$ is some given constant. We also make several assumptions on the boundary data. An important step in the asymptotic analysis of $u_\varepsilon$, as $\varepsilon\to 0$, is to establish uniform $L^p$ bounds for the gradient, for some $p>1$. We review some recent techniques developed in the elliptic case in 7, discuss some variants, and extend the methods to the associated parabolic equation.
Keywords: parabolic equations, Hodge decomposition, Ginzburg-Landau, Jacobians