Some inequalities involving perimeter and torsional rigidity

TL;DR

This paper investigates inequalities involving the perimeter and torsional rigidity of shapes, analyzing the extremal properties of a combined functional across various classes of domains.

Contribution

It introduces and studies shape functionals combining perimeter and torsional rigidity, providing new inequalities and extremal domain characterizations.

Findings

Identifies extremal shapes for the functional in different domain classes.

Establishes bounds and inequalities relating perimeter and torsional rigidity.

Provides insights into shape optimization problems involving these geometric quantities.

Abstract

We consider shape functionals of the form $F_q(\Omega)=P(\Omega)T^q(\Omega)$ on the class of open sets of prescribed Lebesgue measure. Here $q>0$ is fixed, $P(\Omega)$ denotes the perimeter of $\Omega$ and $T(\Omega)$ is the torsional rigidity of $\Omega$. The minimization and maximization of $F_q(\Omega)$ is considered on various classes of admissible domains $\Omega$: in the class $\mathcal{A}_{all}$ of all domains, in the class $\mathcal{A}_{convex}$ of convex domains, and in the class $\mathcal{A}_{thin}$ of thin domains.