Relations between principal eigenvalue and torsional rigidity with Robin boundary conditions

TL;DR

This paper investigates bounds relating the principal eigenvalue and torsional rigidity for Laplace operators with Robin boundary conditions, revealing differences from Dirichlet cases and explicitly determining the critical exponent.

Contribution

It provides explicit bounds and identifies the threshold exponent for Robin boundary conditions, highlighting differences from Dirichlet cases in Lipschitz domains.

Findings

Threshold exponent for Robin case explicitly determined

Robin boundary conditions have a smaller threshold exponent than Dirichlet

Bounds on products of eigenvalue and torsional rigidity established

Abstract

We consider the torsional rigidity and the principal eigenvalue related to the Laplace operator with Dirichlet and Robin boundary conditions. The goal is to find upper and lower bounds to products of suitable powers of the quantities above in the class of Lipschitz domains. The threshold exponent for the Robin case is explicitly recovered and shown to be strictly smaller than in the Dirichlet one.