Serrin’s overdetermined problem for fully nonlinear nonelliptic equations

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