The overdetermined problem in space forms

Abstract

The study of overdetermined problems was initially motivated by specific problems in mathematical physics. However, it has evolved into a rich field of mathematical research at the intersection of analysis and geometry. In this talk, we consider overdetermined problems for a class of fully nonlinear equations with constant Dirichlet boundary conditions in a bounded domain in space forms. It is proven that if the domain is star-shaped or the Dirichlet boundary condition is restricted, then the solution to the Hessian quotient overdetermined problem is radially symmetric. This talk is based on the recent joint works with Prof. Hui Ma, Dr. Shanze Gao.