Serrin's overdetermined problem for fully nonlinear nonelliptic 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 of R-2, with constant Dirichlet and Neumann data on partial derivative Omega. 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 equivalent to 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-infinity but not real-analytic.