Two symmetry problems in potential theory

TL;DR

This paper investigates two elliptic overdetermined boundary value problems, extending classical results by Serrin, and demonstrates that solutions enforce the domain to be a Euclidean ball.

Contribution

It introduces variants of Serrin's classical problems, providing new conditions under which the domain must be spherical.

Findings

Domains are necessarily Euclidean balls under the given conditions

Extends classical symmetry results to new elliptic boundary problems

Provides conditions for domain shape based on overdetermined problems

Abstract

We consider two eliiptic overdetermined boundary value problems. There are variants on J. Serrin's 1971 classical results and having the same conclusion that the domains should be forcibly Euclidean balls.