On some functionals involving torsional rigidity, principal eigenvalue and perimeter

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.