Borderline regularity in singular free boundary problems

TL;DR

This paper studies the regularity of local minimizers in singular free boundary problems, showing Log-Lipschitz continuity under minimal assumptions and $C^1$ regularity when the potential is continuous, revealing a sharp regularity threshold.

Contribution

It establishes the optimal regularity of minimizers under minimal conditions and identifies a precise threshold for differentiability based on the potential's regularity.

Findings

Sign-changing minimizers are Log-Lipschitz continuous with minimal assumptions.

Gradient bounds are established along free boundaries in the one-phase case.

Minimizers are $C^1$ along free boundaries if the potential is continuous.

Abstract

In this paper, we investigate the borderline regularity of local minimizers of energy functionals under minimal assumptions on the potential term $\sigma$. When $\sigma$ is merely bounded and measurable, we show that sign-changing minimizers are Log-Lipschitz continuous, which represents the optimal regularity in this general setting. In the one-phase case, however, we establish gradient bounds for minimizers along their free boundaries, revealing a structural gain in regularity. Most notably, we prove that if $\sigma$ is continuous, then minimizers are of class $C^1$ along the free boundary, thereby identifying a sharp threshold for differentiability in terms of the regularity of the potential.