Symmetry in Serrin-type overdetermined problems

TL;DR

This paper extends Serrin's symmetry results to overdetermined PDE problems in nonsmooth and ring-shaped domains using Steiner symmetrization and approximation techniques.

Contribution

It introduces new symmetry results for overdetermined problems with degenerate ellipticity in nonsmooth and ring-shaped domains.

Findings

Extended Serrin's symmetry to nonsmooth domains

Established symmetry in ring-shaped domains

Utilized Steiner symmetrization and approximation methods

Abstract

This paper investigates the geometric constraints imposed on a domain by overdetermined problems for partial differential equations. Serrin's symmetry results are extended to overdetermined problems with potentially degenerate ellipticity in nonsmooth bounded domains. Furthermore, analogous symmetry results are established for ring-shaped domains. The proof relies on continuous Steiner symmetrization, along with a carefully constructed approximation argument.