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

TL;DR

This paper explores the relationship between fundamental frequency and torsional rigidity in anisotropic energies, analyzing optimization problems for eigenvalues and torsional rigidity induced by general seminorms.

Contribution

It introduces a framework for optimizing eigenvalues and torsional rigidity in anisotropic settings using general seminorms, extending classical spectral theory.

Findings

Characterization of extremal seminorms for eigenvalue and torsional rigidity optimization.

Analysis of the behavior of the functional $F_{q, olinebreak \\Omega}(H)$ under various seminorm controls.

Insights into the interplay between anisotropic energies and spectral properties.

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.