Inequalities between torsional rigidity and principal eigenvalue of the $p$-Laplacian

TL;DR

This paper investigates inequalities relating torsional rigidity and principal eigenvalues of the p-Laplacian, providing bounds and analyzing special cases like p=1 and p=∞, connecting to Cheeger constants and boundary distance functionals.

Contribution

It establishes new bounds for products of torsional rigidity and eigenvalues across different domain classes, including limit cases p=1 and p=∞, linking to Cheeger constants.

Findings

Derived bounds for torsional rigidity and eigenvalues in various domains.

Analyzed limit cases p=1 and p=∞, connecting to Cheeger constants.

Provided insights into inequalities involving the p-Laplacian.

Abstract

We consider the torsional rigidity and the principal eigenvalue related to the $p$-Laplace operator. The goal is to find upper and lower bounds to products of suitable powers of the quantities above in various classes of domains. The limit cases $p=1$ and $p=\infty$ are also analyzed, which amount to consider the Cheeger constant of a domain and functionals involving the distance function from the boundary.