\(W^{1,p}\) estimates for solutions to the Ginzburg-Landau equation with boundary data in \(H^{1/2}\). (English. Abridged French version) Zbl 1080.35020

Summary: We consider complex-valued solutions \(u_\varepsilon\) of the Ginzburg-Landau on a smooth bounded simply connected domain \(\Omega\) of \(\mathbb R^N\), \(N\geq 2\) (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 \(1\leq p<N/(N-1)\), 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 \(\varepsilon\to 0\).


35J60 Nonlinear elliptic equations
35B25 Singular perturbations in context of PDEs
35J20 Variational methods for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
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