Serrin’s overdetermined problem in rough domains

Abstract

The classical Serrin’s overdetermined theorem states that a $C^2$ bounded domain, which admits a function with constant Laplacian that satisfies both constant Dirichlet and Neumann boundary conditions, must necessarily be a ball. While extensions of this theorem to non-smooth domains have been explored since the 1990s, the applicability of Serrin’s theorem to Lipschitz domains remained unresolved. Our paper answers this open question affirmatively. Actually, our approach shows that the result holds for domains that are sets of finite perimeter with a uniform upper bound on the density, and it also allows for slit discontinuities.