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

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.