Concavity properties of solutions to Robin problems

Abstract

We prove that the Robin ground state and the Robin torsion function are respectively log-concave and $\frac{1}{2}$-concave on an uniformly convex domain $\Omega \subset {\mathbb{R}}^N$ of class ${\mathcal{C}}^m$, with $[m-\frac{N}{2}]\ge 4$, provided the Robin parameter exceeds a critical threshold. Such threshold depends on $N$, $m$, and on the geometry of $\Omega$, precisely on the diameter and on the boundary curvatures up to order $m$.

24 pages