Some inequalities involving perimeter and torsional rigidity

Abstract

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