Overdetermined problems for fully nonlinear equations with constant Dirichlet boundary conditions in space forms

TL;DR

This paper investigates overdetermined boundary value problems for fully nonlinear equations in space forms, establishing conditions under which solutions are radially symmetric, including cases with and without star-shaped domains.

Contribution

It proves radial symmetry of solutions for Hessian quotient and $k$-Hessian overdetermined problems, extending symmetry results to broader boundary conditions and domain shapes.

Findings

Radial symmetry for Hessian quotient overdetermined problems in star-shaped domains.

Radial symmetry for $k$-Hessian problems without star-shapedness, using Rellich-Pohožaev identity.

Extension of symmetry results to space forms with constant boundary conditions.

Abstract

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.