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

We consider complex-valued solutions u_ε of the Ginzburg–Landau on a smooth bounded simply connected domain $\text{Ω}$ of $\text{R}{}^{\mathrm{N}}$, N⩾2 (here ε is a parameter between 0 and 1). We assume that u_ε=g_ε on $\text{∂Ω}$, where |g_ε|=1 and g_ε is uniformly bounded in $\text{H}{}^{\mathrm{1/2}}\text{(∂Ω)}$. We also assume that the Ginzburg–Landau energy E_ε(u_ε) is bounded by M₀|logε|, where M₀ is some given constant. We establish, for every 1⩽p<N/(N−1), uniform W^1,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.