#### Vol. 14, No. 5, 2021

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Serrin's overdetermined problem for fully nonlinear nonelliptic equations

### José A. Gálvez and Pablo Mira

Vol. 14 (2021), No. 5, 1429–1442
##### 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 $\Omega \subset {ℝ}^{2}$, with constant Dirichlet and Neumann data on $\partial \Omega$. 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 $F\not\equiv 0$ 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.

##### Keywords
overdetermined problems, fully nonlinear equations, Poincaré–Hopf index
##### Mathematical Subject Classification 2010
Primary: 35N25, 35M12, 53A10