Symmetry in Serrin-type overdetermined problems

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.