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Geometrical bounds for the torsion and the first eigenvalue of the Laplacian with Robin boundary condition

Rosa Barbato, Alba Lia Masiello, Rossano Sannipoli

Abstract

In this paper, we deal with functionals involving the torsion and the first eigenvalue of the Laplacian with Robin boundary conditions (to which we refer as Robin Torsion and Robin Eigenvalue), with other geometrical quantities, in the class of convex sets. Firstly, we prove an upper bound for the Robin Torsion in terms of the $L^1$ and $L^2$ norms of the distance function from the boundary, which allows us to prove a generalization of the Makai inequality involving the Robin Torsion, the Lebeasgue measure, and the inradius of a convex set. Subsequently, we prove quantitative estimates for the Robin Makai functional and for the Robin Pólya functionals, which link the Lebesgue measure and the perimeter with the Robin Torsion and the Robin Eigenvalue respectively. In particular, we prove that the optimal values of all these shape functionals are achieved by slab domains.

Geometrical bounds for the torsion and the first eigenvalue of the Laplacian with Robin boundary condition

Abstract

In this paper, we deal with functionals involving the torsion and the first eigenvalue of the Laplacian with Robin boundary conditions (to which we refer as Robin Torsion and Robin Eigenvalue), with other geometrical quantities, in the class of convex sets. Firstly, we prove an upper bound for the Robin Torsion in terms of the and norms of the distance function from the boundary, which allows us to prove a generalization of the Makai inequality involving the Robin Torsion, the Lebeasgue measure, and the inradius of a convex set. Subsequently, we prove quantitative estimates for the Robin Makai functional and for the Robin Pólya functionals, which link the Lebesgue measure and the perimeter with the Robin Torsion and the Robin Eigenvalue respectively. In particular, we prove that the optimal values of all these shape functionals are achieved by slab domains.

Paper Structure

This paper contains 11 sections, 15 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and basic facts
  4. Distance to the boundary and inner parallel sets
  5. Useful results about the remainder terms
  6. A one dimensional Laplacian eigenvalue problem
  7. Upper bound for the Robin Torsion
  8. On the optimal sets
  9. Quantitative estimates for the Robin Torsion
  10. Quantitative estimate for the Robin eigenvalue
  11. Quantitative inequalities involving $\mathcal{A}(\Omega)$

Key Result

Theorem 1.1

Let $\Omega$ be a bounded, open, nonempty and convex set of $\mathbb{R}^n$. Then

Theorems & Definitions (26)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.1
  • ...and 16 more