Concavity properties of solutions to Robin problems

TL;DR

This paper establishes concavity properties of solutions to Robin boundary problems, showing that the ground state is log-concave and the torsion function is 1/2-concave under certain convexity and regularity conditions.

Contribution

It proves the concavity properties of Robin problem solutions on convex domains with regular boundaries, depending on geometric parameters and a critical Robin parameter threshold.

Findings

Robin ground state is log-concave.

Robin torsion function is 1/2-concave.

Concavity holds under specific geometric and regularity conditions.

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