On Some Functionals Involving Torsional Rigidity, Principal Eigenvalue and Perimeter

In this paper we study some relationships between the first Dirichlet Laplacian eigenvalue \Lambda(\Omega) and the torsional rigidity T(\Omega) of a domain . 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 characterize the values of q > 0 for which the ball is a stable local minimizer or maximizer of the quantity \Lambda(\Omega)T(\Omega)^q , under either a volume or a perimeter constraint.