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

We consider variational energies of the form $$E_H(u)=\frac{1}{2}{\int }_{\mathrm{\Omega }}{H}^2(\mathrm{\nabla }u)dx$$ defined on the Sobolev space $H_0^1(\mathrm{\Omega })$, where $H$ is a general seminorm. Our primary objective is to investigate optimization problems associated with the first eigenvalue ${\lambda }_H(\mathrm{\Omega })$ and the torsional rigidity $T_H(\mathrm{\Omega })$ induced by the seminorm $H$. In particular, we focus on functionals of the type $$F_{q,\mathrm{\Omega }}(H)={\lambda }_H(\mathrm{\Omega })T_H^q(\mathrm{\Omega }),$$ 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 $F_{q,\mathrm{\Omega }}(H)$, as $H$ ranges over a suitable class of seminorms.