On some functionals involving torsional rigidity, principal eigenvalue and perimeter

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