Serrin's overdetermined problem and sharp harmonic quadrature identities in the plane

TL;DR

This paper investigates Serrin's overdetermined boundary value problem in planar Jordan domains, establishing conditions under which the domain must be a disk and demonstrating the sharpness of regularity assumptions.

Contribution

It proves that harmonic quadrature identities imply the domain is a disk within certain regularity classes and constructs counterexamples showing the necessity of these conditions.

Findings

Harmonic quadrature identity implies the domain is a disk in Smirnov domains.

Constructs non-Smirnov Jordan domains satisfying the same quadrature identity.

Demonstrates the sharpness of regularity assumptions in Serrin's problem.

Abstract

We study a weak formulation of Serrin's overdetermined boundary value problem in planar Jordan domains with rectifiable boundary. Our first result establishes that, within the class of rectifiable Jordan Smirnov domains, the corresponding harmonic quadrature identity, equivalent to Serrin's overdetermined problem, necessarily implies that the domain is a disk. Subsequently, we construct a family of rectifiable, non-Smirnov Jordan domains that nonetheless satisfy the same quadrature identity, thereby demonstrating the sharpness of the Smirnov regularity assumption. Consequently, there exists a nontrivial Jordan domain with rectifiable boundary satisfying the weak formulation of Serrin's overdetermined system in $\mathbb R^2$.