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arXiv · 2508.14736

Borderline regularity in singular free boundary problems

Damião J. Araújo·Aelson Sobral·Eduardo V. Teixeira·José Miguel Urbano
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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.

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Damião J. Araújo, Aelson Sobral, Eduardo V. Teixeira, José Miguel Urbano. 2025-08-20. Borderline regularity in singular free boundary problems. https://arxiv.org/abs/2508.14736

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