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

Let u denote a solution to a rotationally invariant Hessian equation F(D2u) = 0 on a bounded simply connected domain Ω 2, with constant Dirichlet and Neumann data on 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 F0 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 but not real-analytic.

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overdetermined problems, fully nonlinear equations, Poincaré–Hopf index
Mathematical Subject Classification 2010
Primary: 35N25, 35M12, 53A10
Received: 5 February 2019
Accepted: 27 January 2020
Published: 22 August 2021
José A. Gálvez
Departamento de Geometría y Topología
Universidad de Granada
Pablo Mira
Departamento de Matemática Aplicada y Estadística
Universidad Politécnica de Cartagena