The role of the curvature of a surface in the shape of the solutions to elliptic equations
Abstract
Abstract We prove the uniqueness and nondegeneracy of the critical point of positive, semistable solutions of with Dirichlet boundary conditions for a class of star‐shaped domains on the sphere and in the hyperbolic plane satisfying a geometric condition. In the spherical case, this condition is weaker than convexity, while in the hyperbolic case, it is weaker than horoconvexity. Finally, we construct examples showing that this geometric condition is indeed optimal.
Fields of Study & Subjects
Nonlinear Partial Differential Equations
Geometric Analysis and Curvature Flows
Navier-Stokes equation solutions
Publication details
- Journal / source
- Proceedings of the London Mathematical Society
- Volume
- 133
- Issue
- 1
- Publication date
- 1 July 2026
- Language
- EN