Overdetermined problems for fully nonlinear elliptic equations

TL;DR

This paper proves that for fully nonlinear elliptic equations, having both Dirichlet and Neumann boundary conditions constant forces the domain to be a ball, generalizing Serrin's classical symmetry result.

Contribution

It extends Serrin's symmetry result to fully nonlinear elliptic equations with overdetermined boundary conditions.

Findings

Domains with overdetermined conditions are necessarily balls.

Generalization of classical symmetry results to nonlinear equations.

Provides new techniques for fully nonlinear elliptic boundary value problems.

Abstract

We prove that the existence of a solution to a fully nonlinear elliptic equation in a bounded domain $\Omega$ with an overdetermined boundary condition prescribing both Dirichlet and Neumann constant data forces the domain $\Omega$ to be a ball. This is a generalization of Serrin's classical result from 1971.