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Estimates for the lowest Neumann eigenvalues of parallelograms and domains of constant width

Corentin Léna, Jonathan Rohleder

Abstract

We prove sharp upper bounds for the first and second non-trivial eigenvalues of the Neumann Laplacian in two classes of domains: parallelograms and domains of constant width. This gives in particular a new proof of an isoperimetric inequality for parallelograms recently obtained by A. Henrot, A. Lemenant and I. Lucardesi.

Estimates for the lowest Neumann eigenvalues of parallelograms and domains of constant width

Abstract

We prove sharp upper bounds for the first and second non-trivial eigenvalues of the Neumann Laplacian in two classes of domains: parallelograms and domains of constant width. This gives in particular a new proof of an isoperimetric inequality for parallelograms recently obtained by A. Henrot, A. Lemenant and I. Lucardesi.
Paper Structure (5 sections, 13 theorems, 127 equations, 1 figure)

This paper contains 5 sections, 13 theorems, 127 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bounds for low eigenvalues of parallelograms
  4. Domains of constant width
  5. Perturbation approach

Key Result

Proposition 2.1

With the above hypotheses, the quadratic form $\mathfrak t_{A, \Omega}$ in $L^2_{1/f} (\Omega)$ given by with is symmetric, non-negative (hence semi-bounded below) and closed.

Figures (1)

  • Figure 1: A domain of constant width.

Theorems & Definitions (25)

  • Conjecture 1.1
  • Proposition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • Theorem 3.1
  • Remark 3.2
  • proof
  • ...and 15 more