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

Giuseppe Buttazzo, Raul Fernandes Horta

Abstract

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.

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

Abstract

We consider variational energies of the form defined on the Sobolev space , where is a general seminorm. Our primary objective is to investigate optimization problems associated with the first eigenvalue and the torsional rigidity induced by the seminorm . In particular, we focus on functionals of the type where is a fixed real parameter. The optimization is performed with respect to the control ; we analyze both minimization and maximization problems for , as ranges over a suitable class of seminorms.
Paper Structure (6 sections, 26 theorems, 155 equations, 1 figure, 4 tables)

This paper contains 6 sections, 26 theorems, 155 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Continuity Properties
  4. On the existence of optimal seminorms
  5. Remarks on some inequalities
  6. Final remarks and questions

Key Result

Proposition 2.2

Let $\Omega$ be a bounded domain, $A$ an invertible matrix and $H_{A}(\xi) = H(A\xi)$. Then

Figures (1)

  • Figure 1: Domain for which $H\mapsto\lambda_{H}(\Omega)$ is discontinuous

Theorems & Definitions (47)

  • Definition 2.1: Generalized First Eigenvalue and Torsion
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Corollary 2.4
  • proof
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • Definition 2.7
  • ...and 37 more