On some functionals involving torsional rigidity, principal eigenvalue and perimeter

In this paper we study some relationships between the first Dirichlet eigenvalue $Λ(Ω)$ and the torsional rigidity $T(Ω)$ of a domain $Ω$. We consider the problem of optimizing the product $Λ(Ω)T(Ω)$ 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 $Λ(Ω)T(Ω)^q$, with $q>0$, under either a volume or a perimeter constraint.