In this talk, we address the optimization of the product of the first Dirichlet eigenvalue of the Laplacian and the torsional rigidity under geometric constraints such as fixed perimeter or fixed volume. The problem is studied both in the general class of open sets and within the subclass of convex domains. We also present local results for related functionals involving the Dirichlet eigenvalue and powers of the torsional rigidity.
This is based on joint work with Carlo Nitsch, Cristina Trombetti, and Federico Villone.
