Serrin's overdetermined problem for fully nonlinear nonelliptic equations

Abstract

Let $u$ denote a solution to a rotationally invariant Hessian equation $F\left(D^{2}u\right)=0$ on a bounded simply connected domain $\mathrm{\Omega }\subset {ℝ}^2$, with constant Dirichlet and Neumann data on $\partial \mathrm{\Omega }$. We prove that if $u$ is real analytic and not identically zero, then $u$ is radial and $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ is not simply connected, or if $u$ is $C^{\infty }$ but not real-analytic.