W1,p estimates for solutions to the Ginzburg-Landau equation with boundary data in H1/2

We consider complex-valued solutions u(epsilon) of the Ginzburg-Landau on a smooth bounded simply connected domain Omega of R-N, N greater than or equal to 2 (here epsilon is a parameter between 0 and 1). We assume that u(epsilon) = g(epsilon) on partial derivativeOmega, where \g(epsilon)\ = 1 and g(epsilon) is uniformly bounded in H-1/2(partial derivativeOmega). We also assume that the Ginzburg-Landau energy E-epsilon (u(epsilon)) is bounded by M-o\log epsilon\, where M-o is some given constant. We establish, for every 1 less than or equal to p < N/ (N - 1), uniform W-1,W-p bounds for u(ε), (independent of ε). These types of estimates play a central role in the asymptotic analysis of u(ε) as ε --> 0. (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.