On two functionals involving the maximum of the torsion function

TL;DR

This paper studies bounds for two shape functionals involving the maximum of the torsion function, focusing on convex sets, and explores their relationships with torsion and eigenvalues.

Contribution

It introduces new bounds for shape functionals involving the torsion function's maximum and analyzes their behavior, especially within convex sets.

Findings

Derived bounds for the functionals involving torsion and eigenvalues.

Identified special properties for convex sets.

Provided insights into the relationships between torsion, maximum, and eigenvalues.

Abstract

In this paper we investigate upper and lower bounds of two shape functionals involving the maximum of the torsion function. More precisely, we consider $T(\Omega)/(M(\Omega)|\Omega|)$ and $M(\Omega)\lambda_1(\Omega) $, where $\Omega$ is a bounded open set of $\mathbb{R}^d$ with finite Lebesgue measure $|\Omega|$, $M(\Omega)$ denotes the maximum of the torsion function, $T(\Omega)$ the torsion, and $\lambda_1(\Omega)$ the first Dirichlet eigenvalue. Particular attention is devoted to the subclass of convex sets.