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arXiv · 2601.20493

Regularity of Lipschitz free boundaries for weak solutions of Alt-Caffarelli type problems

Joan Domingo-Pasarin·Xavier Ros-Oton
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Abstract

Motivated by the Serrin problem, we study weak solutions of the generalised Alt-Caffarelli problem $-\Delta u = f$ in $\Omega$, $u = 0$ on $\partial\Omega$, $\partial_\nu u = Q$ on $\partial\Omega$. Our main result establishes that if $\Omega$ is Lipschitz, then it is actually $C^{\infty}$ (provided that $f$ and $Q$ are smooth). This was known before only for viscosity solutions. As a corollary, we obtain an alternative solution of Serrin's problem in the case of Lipschitz domains. We also discuss the characterisation of the regularity of Lipschitz domains in terms of their Poisson kernel.

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Joan Domingo-Pasarin, Xavier Ros-Oton. 2026-01-28. Regularity of Lipschitz free boundaries for weak solutions of Alt-Caffarelli type problems. https://arxiv.org/abs/2601.20493

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