Overdetermined problems for fully nonlinear equations with constant Dirichlet boundary conditions in space forms
- Resource Type
- Working Paper
- Authors
- Gao, Shanze; Ma, Hui; Yang, Mingxuan
- Source
- Calc. Var. 62, 183 (2023)
- Subject
- Mathematics - Analysis of PDEs
Mathematics - Differential Geometry
35N25, 58J32, 53C40
- Language
We consider overdetermined problems for two classes of fully nonlinear equations with constant Dirichlet boundary conditions in a bounded domain in space forms. We prove that if the domain is star-shaped, then the solution to the Hessian quotient overdetermined problem is radially symmetric. By establishing a Rellich-Poho\v{z}aev type identity for the $k$-Hessian equation with constant Dirichlet boundary condition, we also show the radial symmetry of the solution to the $k$-Hessian overdetermined problem for some boundary value without star-shapedness assumption of the domain.
Comment: final version