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

TL;DR

This work analyzes the Robin-Laplacian torsional rigidity $\tau_\alpha(\Omega)$ for negative boundary parameters and establishes sharp optimality results under isoperimetric and isochoric constraints. By formulating the problem in an operator-theoretic framework and proving a key lower bound $\tau_\alpha(\Omega) \ge \tau_D(\Omega) + \frac{|\Omega|^2}{\alpha|\partial\Omega|}$ when $\alpha\in(-\sigma_1(\Omega),0)$, the authors reduce global optimization to comparison with the Dirichlet torsion on spherical shells and explicit radial models. Extending the parallel-coordinates method to all $d\ge3$, they prove that, for $|\alpha|$ small, the ball minimizes $\tau_\alpha(\Omega)$ (equivalently, maximizes under negative sign) among convex domains with fixed volume or fixed perimeter, with planar analogues for simply connected planar sets under perimeter constraint. The results rely on the Steklov spectrum bound $\sigma_1(\Omega)\le 1/R$ and provide quantitative isoperimetric/isochoric inequalities, resolving open questions by Bandle and Wagner in higher dimensions and for perimeter constraints.

Abstract

Motivated by pioneering works of Bandle and Wagner, given a bounded Lipschitz domain $Ω\subset \mathbb R^d$ with $d\ge3$, we consider the Robin-Laplacian torsional rigidity $τ_α(Ω)$ with negative boundary parameter $α$ and we show that sharp inequalities for $τ_α(Ω)$ hold if $|α|$ is small enough. In particular, we prove that, if $|α|$ is smaller than the first non-trivial Steklov-Laplacian eigenvalue, then the ball maximises $τ_α(Ω)$ 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.

Theorems & Definitions (19)

• Theorem 1: Bucur and Giacomini Bucur-Giacomini_2015b
• Theorem 2: Bandle and Wagner BW-sign
• Theorem 3
• Theorem 4
• Lemma 1
• Lemma 2
• Remark 1
• Lemma 3: Bandle and Wagner BW-sign
• Proposition 1: Payne and Weinberger Payne-Weinberger_1961
• proof : Proof of Theorem \ref{['Thm.2D']}