On Some Functionals Involving Torsional Rigidity, Principal Eigenvalue and Perimeter

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