Overdetermined problems for fully non linear operators

TL;DR

This paper investigates overdetermined boundary value problems for fully nonlinear elliptic operators, demonstrating that solutions with constant sign imply the domain must be spherical, extending classical symmetry results.

Contribution

It extends Serrin's classical symmetry result to fully nonlinear singular or degenerate elliptic operators in bounded smooth domains.

Findings

Existence of solutions implies the domain is a ball

Results extend classical symmetry theorems to nonlinear operators

Provides conditions for overdetermined problems in nonlinear settings

Abstract

In this paper, we consider the overdetermined problem for fully non linear singular or degenerate elliptic operators in bounded smooth domains with both Dirichlet and Neumann condition, as in the classical result of Serrin we prove that the existence of nontrivial constant sign solution imply that the domain is a ball.