Sharp estimates for the Laplacian torsional rigidity with negative Robin boundary conditions

TL;DR

This paper establishes sharp bounds for the Laplacian torsional rigidity with negative Robin boundary conditions, showing the ball maximizes this quantity under certain constraints when the boundary parameter is small.

Contribution

It proves the maximization of Robin-Laplacian torsional rigidity by the ball for small negative boundary parameters, solving an open problem by Bandle and Wagner.

Findings

Ball maximizes torsional rigidity for small negative Robin parameters.

Sharp inequalities hold under perimeter or volume constraints.

Results extend to planar simply connected sets.

Abstract

Motivated by pioneering works of Bandle and Wagner, given a bounded Lipschitz domain $\Omega \subset \mathbb R^d$ with $d\ge3$, we consider the Robin-Laplacian torsional rigidity $\tau_\alpha(\Omega)$ with negative boundary parameter $\alpha$ and we show that sharp inequalities for $\tau_\alpha(\Omega)$ hold if $|\alpha|$ is small enough. In particular, we prove that, if $|\alpha|$ is smaller than the first non-trivial Steklov-Laplacian eigenvalue, then the ball maximises $\tau_\alpha(\Omega)$ among all convex domains under perimeter or volume constraints.This solves an open problem raised by Bandle and Wagner. We also prove the result in the planar case among simply connected sets and under perimeter constraint.