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Upper bounds for Courant-sharp Neumann and Robin eigenvalues

Katie Gittins, Corentin Léna

Abstract

We consider the eigenvalues of the Laplacian on an open, bounded, connected set in $\mathbb{R}^n$ with $C^2$ boundary, with a Neumann boundary condition or a Robin boundary condition. We obtain upper bounds for those eigenvalues that have a corresponding eigenfunction which achieves equality in Courant's Nodal Domain theorem. In the case where the set is also assumed to be convex, we obtain explicit upper bounds in terms of some of the geometric quantities of the set. Corrigendum. A previous version of this work was accepted and published by the "Bulletin de la Société Mathématique de France" (see [2] in the bibliography of Appendix B). It contained a gap: the classical (Euclidean) Faber-Krahn inequality was applied in a setting where it might not hold. This version reproduces the previous one with the addition of a corrigendum in Appendix B that addresses the issue. All the results in Sections 2--8 and most of those in Section 9 are thus preserved.

Upper bounds for Courant-sharp Neumann and Robin eigenvalues

Abstract

We consider the eigenvalues of the Laplacian on an open, bounded, connected set in with boundary, with a Neumann boundary condition or a Robin boundary condition. We obtain upper bounds for those eigenvalues that have a corresponding eigenfunction which achieves equality in Courant's Nodal Domain theorem. In the case where the set is also assumed to be convex, we obtain explicit upper bounds in terms of some of the geometric quantities of the set. Corrigendum. A previous version of this work was accepted and published by the "Bulletin de la Société Mathématique de France" (see [2] in the bibliography of Appendix B). It contained a gap: the classical (Euclidean) Faber-Krahn inequality was applied in a setting where it might not hold. This version reproduces the previous one with the addition of a corrigendum in Appendix B that addresses the issue. All the results in Sections 2--8 and most of those in Section 9 are thus preserved.

Paper Structure

This paper contains 27 sections, 17 theorems, 176 equations, 2 tables.

Table of Contents

  1. MSC classification (2010):
  2. Keywords:
  3. Introduction
  4. Statement of the problem
  5. Goal of the article
  6. Organisation of the article
  7. Preliminaries
  8. Strategy for Courant-sharp Neumann eigenvalues
  9. Application to Courant-sharp Robin eigenvalues
  10. Tubular coordinates in 2D
  11. Cut-off functions
  12. Estimates of the nodal count
  13. Straightening of the boundary
  14. Number of nodal domains
  15. Geometry of the domain
  16. ...and 12 more sections

Key Result

Proposition 1.1

Let $\Omega$ be an open, bounded, convex set in $\mathbb{R}^2$ with $C^2$ boundary. There exist constants $C>0$ and $C'>0$, that do not depend on $\Omega$, such that for any Courant-sharp eigenvalue $\mu_k(\Omega)$, and

Theorems & Definitions (29)

  • Proposition 1.1
  • Proposition 3.1
  • proof
  • Lemma 5.1
  • proof
  • Corollary 5.2
  • Corollary 5.3
  • Proposition 5.4
  • proof
  • Proposition 5.5
  • ...and 19 more