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On some functionals involving torsional rigidity, principal eigenvalue and perimeter

Vincenzo Amato, Carlo Nitsch, Cristina Trombetti, Federico Villone

TL;DR

The paper investigates how domain geometry governs the interaction of torsional rigidity and the principal Laplacian eigenvalue under perimeter constraints by introducing the scale-invariant product $G(\Omega)=\dfrac{T(\Omega)\Lambda(\Omega)}{P(\Omega)^{\frac{m}{m-1}}}$ and relating it to the Pólya functional $F(\Omega)$. It establishes sharp bounds for $G$ on open and convex sets, proves the existence of maximizers in the convex class, and shows the ball is a local maximizer for nearly spherical domains for appropriate $q$-dependent functionals $F_q$ and $G_q$, with explicit coercivity ranges derived from spherical-harmonic expansions. A generalization $G_q(\Omega)=\dfrac{T(\Omega)^q\Lambda(\Omega)}{P(\Omega)^{\beta_q}}$ is proposed, and the paper delineates the corresponding extremal behavior across open, convex, and nearly spherical classes, highlighting thresholds where the ball ceases to be optimal and where local stability persists. The results deepen understanding of perimeter-constrained shape optimization for fundamental spectral/elastic quantities and provide precise local-optimality criteria via DL-type stability theorems.

Abstract

In this paper we study some relationships between the first Dirichlet eigenvalue $Λ(Ω)$ and the torsional rigidity $T(Ω)$ of a domain $Ω$. We consider the problem of optimizing the product $Λ(Ω)T(Ω)$ among sets with prescribed perimeter, both in the class of open sets with finite perimeter and within the class of convex domains. We also present local results for the quantity $Λ(Ω)T(Ω)^q$, with $q>0$, under either a volume or a perimeter constraint.

On some functionals involving torsional rigidity, principal eigenvalue and perimeter

TL;DR

The paper investigates how domain geometry governs the interaction of torsional rigidity and the principal Laplacian eigenvalue under perimeter constraints by introducing the scale-invariant product and relating it to the Pólya functional . It establishes sharp bounds for on open and convex sets, proves the existence of maximizers in the convex class, and shows the ball is a local maximizer for nearly spherical domains for appropriate -dependent functionals and , with explicit coercivity ranges derived from spherical-harmonic expansions. A generalization is proposed, and the paper delineates the corresponding extremal behavior across open, convex, and nearly spherical classes, highlighting thresholds where the ball ceases to be optimal and where local stability persists. The results deepen understanding of perimeter-constrained shape optimization for fundamental spectral/elastic quantities and provide precise local-optimality criteria via DL-type stability theorems.

Abstract

In this paper we study some relationships between the first Dirichlet eigenvalue and the torsional rigidity of a domain . We consider the problem of optimizing the product among sets with prescribed perimeter, both in the class of open sets with finite perimeter and within the class of convex domains. We also present local results for the quantity , with , under either a volume or a perimeter constraint.
Paper Structure (15 sections, 25 theorems, 188 equations, 2 figures, 3 tables)

This paper contains 15 sections, 25 theorems, 188 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Some Properties of Convex Sets
  4. Shape Derivatives and Local Optimality
  5. Optimization in the class of open sets
  6. Optimization in the class of convex sets
  7. Optimality of the ball for nearly spherical domains
  8. On the improved continuity condition
  9. Criticality of the ball
  10. On the coercivity conditions
  11. Proof of \ref{['Optimality of the ball']} and \ref{['teor F_q']}
  12. Possible generalization
  13. Optimization in the class of open sets
  14. Optimization in the class of convex sets
  15. Optimization in the class of nearly spherical sets

Key Result

Proposition 1

Let $G$ be the functional defined in function_G and $\mathcal{O}$ be the set defined in openperimeter. Then where $\omega_m$ denotes the volume of the unit ball in $\mathbb{R}^m$. Furthermore, neither of these extremal values is attained within the class $\mathcal{O}$.

Figures (2)

  • Figure 1: The set $\Omega_n$ in red
  • Figure 2: $T_\varepsilon$ in blue

Theorems & Definitions (54)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Definition 1
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1
  • Definition 2
  • Definition 3
  • Proposition 4
  • ...and 44 more