Overdetermined elliptic problems on model Riemannian manifolds
João Marcos Ó, de Lima, Jaqueline, Márcio S. Santos · arXiv (Cornell University) · 2025
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.