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

Abstract

Let $u$ denote a solution to a rotationally invariant Hessian equation $F(D^{2}u)=0$ 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.

References in corpus (1)

• Serrin's Overdetermined Problem and Constant Mean Curvature Surfaces