# Serrin's overdetermined problem for fully nonlinear non-elliptic equations

## TL;DR

This paper proves that solutions to certain fully nonlinear, rotationally invariant Hessian equations with constant boundary data are necessarily radial and the domain is a disk, assuming the solution is real analytic.

## Contribution

It establishes a Serrin-type overdetermined problem result for fully nonlinear, non-elliptic equations, extending classical symmetry results to broader operators.

## Key findings

- Solutions are radial if real analytic and satisfy boundary conditions
- The domain must be a disk under the given conditions
- The result fails without analyticity or simple connectivity

## Abstract

Let $u$ denote a solution to a rotationally invariant Hessian equation $F(D^2u)=0$ on a bounded simply connected domain $\Omega\subset R^2$, with constant Dirichlet and Neumann data on $\partial \Omega$. In this paper we prove that if $u$ is real analytic and not identically zero, then $u$ is radial and $\Omega$ is a disk. The fully nonlinear operator $F\not\equiv 0$ is of general type, and in particular, not assumed to be elliptic. We also show that the result is sharp, in the sense that it is not true if $\Omega$ is not simply connected, or if $u$ is $C^{\infty}$ but not real analytic.