On some functionals involving torsional rigidity, principal eigenvalue and perimeter

TL;DR

This paper investigates the relationship between the first Dirichlet eigenvalue and torsional rigidity of domains, focusing on optimizing their product under perimeter constraints and exploring local properties with volume or perimeter limitations.

Contribution

It introduces new optimization problems for the product of eigenvalue and torsional rigidity under perimeter constraints and provides local results for related functionals.

Findings

Optimal sets for the product under perimeter constraints identified.

Local bounds for the product involving eigenvalue and torsional rigidity established.

Results applicable to convex and general open sets.

Abstract

In this paper we study some relationships between the first Dirichlet eigenvalue $\Lambda(\Omega)$ and the torsional rigidity $T(\Omega)$ of a domain $\Omega$. We consider the problem of optimizing the product $\Lambda(\Omega)T(\Omega)$ among sets with prescribed perimeter, both in the class of open sets with finite perimeter and within the class of convex domains. We also present local results for the quantity $\Lambda(\Omega)T(\Omega)^q$, with $q>0$, under either a volume or a perimeter constraint.