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

Abstract

We consider variational energies of the form \[E_H(u)=\frac12\int_\Omega H^2(\nabla u)\,dx\] defined on the Sobolev space $H^1_0(\Omega)$, where $H$ is a general seminorm. Our primary objective is to investigate optimization problems associated with the first eigenvalue $\lambda_H(\Omega)$ and the torsional rigidity $T_H(\Omega)$ induced by the seminorm $H$. In particular, we focus on functionals of the type \[F_{q,\Omega}(H)=\lambda_H(\Omega)\,T_H^q(\Omega),\] 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 $F_{q,\Omega}(H)$, as $H$ ranges over a suitable class of seminorms.