W^1,p estimates for solutions to the Ginzburg–Landau equation with boundary data in H^1/2

We consider complex-valued solutions uε of the Ginzburg–Landau on a smooth bounded simply connected domain Ω of RN, N⩾2 (here ε is a parameter between 0 and 1). We assume that uε=gε on ∂Ω, where |gε|=1 and gε is uniformly bounded in H1/2(∂Ω). We also assume that the Ginzburg–Landau energy Eε(uε) is bounded by M0|logε|, where M0 is some given constant. We establish, for every 1⩽p<N/(N−1), uniform W1,p bounds for uε (independent of ε). These types of estimates play a central role in the asymptotic analysis of uε as ε→0.