Uniqueness of Blow-ups for the Superconductivity Free Boundary Problem

We study the free-boundary equation $$\Delta u=\chi_{\{|\nabla u|>0\}}$$ near the origin. We prove that, at a singular point of $\partial\{|\nabla u|>0\}$, the quadratic blow-up is unique. As noted in \cite[Notes to Chapter 7]{PSU2012}, little is known about the singular set for this problem. The usual Weiss--Monneau monotonicity argument does not seem to apply directly, because the inactive set is determined by the vanishing of $\nabla u$, rather than by a sign condition on $u$. The proof follows the quadratic part of the rescalings. Projecting onto the trace-free quadratic harmonics yields a finite-dimensional differential equation for the quadratic coefficient. Together with a Lyapunov identity and estimates on dyadic annuli, this implies convergence of the quadratic coefficient, and hence uniqueness of the blow-up.