Concavity properties of solutions to Robin problems

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 $[m -\frac{ N}{2}]\geq 4$, 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 $m$.