Serrin's overdetermined problem in rough domains

TL;DR

This paper extends Serrin's overdetermined problem to rough domains, including Lipschitz and sets with finite perimeter, showing the classical symmetry result holds beyond smooth boundaries.

Contribution

It proves Serrin's theorem applies to non-smooth domains such as Lipschitz and finite perimeter sets, resolving an open question in the field.

Findings

Serrin's overdetermined problem holds in Lipschitz domains.

The result extends to sets with finite perimeter and slit discontinuities.

Classical symmetry results are valid 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. This 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.