On Concavity of Solutions of the Nonlinear Poisson Equation

TL;DR

This paper proves that solutions to certain nonlinear Poisson equations are concave under specific conditions and extends classical isoperimetric inequalities to these nonlinear cases.

Contribution

It establishes concavity of solutions for nonlinear Poisson equations with monotone, concave nonlinearities and extends the Saint Venant conjecture to these nonlinear scenarios.

Findings

Solutions are concave when boundary Hessian is negative semi-definite and $f$ is monotone and concave.

The maximum of solutions occurs in the ball among domains of fixed measure for certain nonlinear equations.

Extension of the Saint Venant conjecture to nonlinear Poisson equations with small Lipschitz constant.

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.