Optimal regularity for a two-phase obstacle-like problem with logarithmic singularity

Abstract

We consider the semilinear problem where B 1 is the unit ball in and assume Using a monotonicity formula argument, we prove an optimal regularity result for solutions: is a log-Lipschitz function. This problem introduces two main difficulties. The first is the lack of invariance in the scaling and blow-up of the problem. The other (more serious) issue is a term in the Weiss energy which is potentially non-integrable unless one already knows the optimal regularity of the solution: this puts us in a catch-22 situation.