Some inequalities involving perimeter and torsional rigidity

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.

References in corpus (1)

• Sharp estimates for the anisotropic torsional rigidity and the principal frequency