On the relations between fundamental frequency and torsional rigidity in the case of anisotropic energies

We consider variational energies of the form \[E_H(u)=\frac12\int_ΩH^2(\nabla u)\,dx\] defined on the Sobolev space $H^1_0(Ω)$, where $H$ is a general seminorm. Our primary objective is to investigate optimization problems associated with the first eigenvalue $λ_H(Ω)$ and the torsional rigidity $T_H(Ω)$ induced by the seminorm $H$. In particular, we focus on functionals of the type \[F_{q,Ω}(H)=λ_H(Ω)\,T_H^q(Ω),\] where $q>0$ is a fixed real parameter. The optimization is performed with respect to the control $H$; we analyze both minimization and maximization problems for $F_{q,Ω}(H)$, as $H$ ranges over a suitable class of seminorms.