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

Motivated by the Serrin problem, we study weak solutions of the generalised Alt-Caffarelli problem $-\mathrm{\Delta }u=f$ in $\mathrm{\Omega }$, $u=0$ on $\mathrm{\partial }\mathrm{\Omega }$, ${\mathrm{\partial }}_{\nu }u=Q$ on $\mathrm{\partial }\mathrm{\Omega }$. Our main result establishes that if $\mathrm{\Omega }$ is Lipschitz, then it is actually $C^{\mathrm{\infty }}$ (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.