Uniqueness of Blow-ups for the Superconductivity Free Boundary Problem

We study the free-boundary equation $$\mathrm{\Delta }u={\chi }_{\{\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }>0\}}$$ near the origin. We prove that, at a singular point of $\mathrm{\partial }\{\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }>0\}$, 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 $\mathrm{\nabla }u$, 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.