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On Concavity of Solutions of the Nonlinear Poisson Equation

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Abstract

We consider the nonlinear Poisson equation \(-\Delta u = f(u)\) in domains \(\Omega \subset {\mathbb {R}}^n\) with Dirichlet boundary conditions on \(\partial \Omega \). We show (for monotonically increasing concave f with small Lipschitz constant) that if \(D^2 u\) is negative semi-definite on the boundary, then u is concave. A conjecture of Saint Venant from 1856 (proven by Polya in 1948) is that among all domains \(\Omega \) of fixed measure, the solution of \(-\Delta u =1\) assumes its largest maximum when \(\Omega \) is a ball. We extend this to \(-\Delta u =f(u)\) for monotonically increasing f with small Lipschitz constant.

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Authors and Affiliations

  1. Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

    Stefan Steinerberger

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  1. Stefan Steinerberger
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Correspondence to Stefan Steinerberger.

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Communicated by P.-L. Lions.

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S.S. is supported by the NSF (DMS-2123224) and the Alfred P. Sloan Foundation.

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Steinerberger, S. On Concavity of Solutions of the Nonlinear Poisson Equation. Arch Rational Mech Anal 244, 209–224 (2022). https://doi.org/10.1007/s00205-022-01759-3

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