On some functionals involving torsional rigidity, principal eigenvalue and perimeter

Abstract

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

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