Global Uniform Estimate for the Modulus of Two-Dimensional Ginzburg-Landau Vortexless Solutions with Asymptotically Infinite Boundary Energy

For ? > 0, let u? : ? ? R 2 be a solution of the Ginzburg-Landau system ??u? = 1 ? 2 u?(1 ? |u?| 2) in a Lipschitz bounded domain ?. In an energy regime that excludes interior vortices, we prove that 1 ? |u?| is uniformly estimated by a positive power of ? globally in ? provided that the energy of u? at the boundary ?? does not grow faster than ? ?? with ? ? (0, 1)