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

Bethuel, Fabrice; Bourgain, Jean; Brezis, Haïm; Orlandi, Giandomenico

C. R. Acad. Sci., Paris, Sér. I, Math. 333, No. 12, 1069-1076 (2001).

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 \(\varepsilon\to 0\).

MSC:

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