A Steklov version of the Torsional Rigidity

TL;DR

This paper introduces a new boundary torsional rigidity concept linked to Steklov eigenvalues, providing variational formulations, properties, and sharp geometric estimates for various sets.

Contribution

It proposes a novel boundary torsional rigidity related to Steklov eigenvalues, with variational characterizations and geometric bounds.

Findings

Derived equivalent variational formulations.

Analyzed properties of the state function.

Established sharp geometric estimates for specific sets.

Abstract

Motivated by the connection between the first eigenvalue of the Dirichlet-Laplacian and the torsional rigidity, the aim of this paper is to find a physically coherent and mathematically interesting new concept for boundary torsional rigidity, closely related to the Steklov eigenvalue. From a variational point of view, such a new object corresponds to the sharp constant for the trace embedding of $W^{1,2}(\Omega)$ into $L^1(\partial\Omega)$. We obtain various equivalent variational formulations, present some properties of the state function and obtain some sharp geometric estimates, both for planar simply connected sets and for convex sets in any dimension.