中科院数学与系统科学研究院

数学研究所

学术报告

偏微分方程研讨班

报告人：杨明轩博士 （清华大学）

题  目：The overdetermined problems in space forms and a convex cone

时  间：2023.11.16（星期四）14:00-16:00

地  点：思源楼S415

摘 要：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 will introduce two cases of overdetermined problems. The first case considers 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.  The second case considers a partially overdetermined problem for the $p$-Laplace equation in a convex cone $\mathcal{C}$ intersected with the exterior of a smooth bounded domain $\overline{\Omega}$ in $\mathbb{R}^n$ ($n\geq2$).  Based on some properties of a capacitary potential, a rigidity result is obtained under the assumption of orthogonal intersection, using $P$-function, the isoperimetric inequality, and a Heintze-Karcher type inequality in a convex cone.

This talk is based on the recent joint works with Prof. Hui Ma, Dr. Shanze Gao and Dr. Jiabin Yin.