On Concavity of Solutions of the Nonlinear Poisson Equation

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.