Overdetermined elliptic problems on model Riemannian manifolds

2025/05/31 by João Marcos Ó, Ó, João Marcos do, de Lima, Jaqueline +2
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2506.00697

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.

Citations

• Serrin's type Problems in convex cones in Riemannian manifolds
• Radial symmetry and partially overdetermined problems in a convex cone
• Serrin's type overdetermined problems in convex cones
• Overdetermined problems and constant mean curvature surfaces in cones
• A Serrin-type symmetry result on model manifolds: an extension of the\n Weinberger argument
• On Serrin's overdetermined problem in space forms
• On Serrin’s overdetermined problem in space forms
• Serrin's overdetermined problem on the sphere
• Serrin’s result for hyperbolic space and sphere
• A symmetry problem in potential theory

Related

• Splitting theorems, symmetry results and overdetermined problems for\n Riemannian manifolds
• Non-symmetric solutions to an overdetermined problem for the Helmholtz equation in the plane
• Overdetermined elliptic problems in topological disks
• Overdetermined elliptic problems in nontrivial contractible domains of the sphere
• A rigidity result for overdetermined elliptic problems in the plane