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

TL;DR

This paper proves that weak solutions to a generalized Alt-Caffarelli problem in Lipschitz domains are actually smooth if the data are smooth, extending known results from viscosity solutions and providing new insights into domain regularity.

Contribution

It establishes the smoothness of weak solutions in Lipschitz domains for the first time, generalizing previous viscosity solution results and offering an alternative approach to Serrin's problem.

Findings

Weak solutions are $C^{ abla}$ in Lipschitz domains with smooth data.

Provides an alternative solution to Serrin's problem for Lipschitz domains.

Characterizes Lipschitz domain regularity via the Poisson kernel.

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.