On Concavity of Solutions of the Nonlinear Poisson Equation

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.

References in corpus (4)

• A Microscopic Convexity Principle for Nonlinear Partial Differential Equations
• On two functionals involving the maximum of the torsion function
• The Stefan problem and concavity
• Localization of the first eigenfunction of a convex domain

Cited by in corpus (1)

• Concavity and perturbed concavity for $p$-Laplace equations