Serrin’s overdetermined problem for fully nonlinear nonelliptic equations

Gálvez, José A.; Mira, Pablo

doi:10.2140/apde.2021.14.1429

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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

DOI: 10.2140/apde.2021.14.1429

Abstract

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

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Keywords

overdetermined problems, fully nonlinear equations, Poincaré–Hopf index

Mathematical Subject Classification 2010

Primary: 35N25, 35M12, 53A10

Milestones

Received: 5 February 2019

Accepted: 27 January 2020

Published: 22 August 2021

Authors

José A. Gálvez
Departamento de Geometría y Topología Universidad de Granada Granada Spain

Pablo Mira
Departamento de Matemática Aplicada y Estadística Universidad Politécnica de Cartagena Cartagena Spain

Vol. 14, No. 5, 2021