Overdetermined elliptic problems on model Riemannian manifolds

TL;DR

This paper proves that in certain Riemannian manifolds, domains with solutions to overdetermined elliptic problems must be spherical sectors, revealing strong geometric restrictions due to boundary conditions.

Contribution

It extends classical rigidity results to overdetermined elliptic problems on model Riemannian manifolds, characterizing domains as spherical sectors.

Findings

Domains with solutions are spherical sectors.

Solutions exhibit radial symmetry.

Boundary conditions impose geometric constraints.

Abstract

We establish a rigidity theorem for annular sector-like domains in the setting of overdetermined elliptic problems on model Riemannian manifolds. Specifically, if such a domain admits a solution to the inhomogeneous Helmholtz equation satisfying both constant Dirichlet and constant Neumann boundary conditions, then the domain must be a spherical sector, and the solution must be radially symmetric. This result underscores the strong geometric constraints imposed by overdetermined boundary conditions, extending classical rigidity phenomena to this more general framework.