Serrin’s Overtermined Problem In Rough Domains

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.

In this talk, I will discuss a recent answer to this open question, which was given in a joint paper with Yi Zhang.

As we shall see, 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.