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

TL;DR

This paper proves that solutions to a two-phase obstacle-like problem with a logarithmic singularity have optimal regularity, specifically that their gradients are log-Lipschitz, using a monotonicity formula approach.

Contribution

It establishes the optimal regularity of solutions to a complex semilinear problem with logarithmic singularities, overcoming challenges related to scaling invariance and energy integrability.

Findings

Solutions have log-Lipschitz regularity.

The method addresses non-invariance and non-integrability issues.

Provides a regularity result for a novel class of problems.

Abstract

We consider the semilinear problem \[ \Delta u = \lambda_+ \left(-\log u^+\right) 1_{\{u > 0\}} - \lambda_- \left(-\log u^- \right) 1_{\{u < 0\}} \qquad \hbox{ in } B_1, \] where $B_1$ is the unit ball in $\mathbb{R}^n$ and assume $\lambda_+, \lambda_- > 0$. Using a monotonicity formula argument, we prove an optimal regularity result for solutions: $\nabla u$ 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.