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On Concavity of Solutions of the Nonlinear Poisson Equation

We consider the nonlinear Poisson equation $-Δu = f(u)$ in domains $Ω\subset \mathbb{R}^n$ with Dirichlet boundary conditions on $\partial Ω$. 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 $Ω$ of fixed measure, the solution of $-Δu =1$ assumes its largest maximum when $Ω$ is a ball. We extend this to $-Δu =f(u)$ for monotonically increasing $f$ with small Lipschitz constant.