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Lower bounds for the sum of the reciprocals of eigenvalues of bounded domains in $\mathbb{R}^n$, spheres, and closed orientable surfaces

Mehdi Eddaoudi

Abstract

We establish lower bounds for the sum of the reciprocals of eigenvalues of the Laplacian. For bounded domains, our result extends the upper bound provided by Bucur and Henrot on the second Neumann eigenvalue and is related to a result by Wang and Xia, which connects to a conjecture of Ashbaugh and Benguria. For spheres and surfaces, we extend known results on the first and second eigenvalues, and strengthen an analogous conjecture involving the conformal volume of Li and Yau.

Lower bounds for the sum of the reciprocals of eigenvalues of bounded domains in $\mathbb{R}^n$, spheres, and closed orientable surfaces

Abstract

We establish lower bounds for the sum of the reciprocals of eigenvalues of the Laplacian. For bounded domains, our result extends the upper bound provided by Bucur and Henrot on the second Neumann eigenvalue and is related to a result by Wang and Xia, which connects to a conjecture of Ashbaugh and Benguria. For spheres and surfaces, we extend known results on the first and second eigenvalues, and strengthen an analogous conjecture involving the conformal volume of Li and Yau.

Paper Structure

This paper contains 11 sections, 8 theorems, 100 equations.

Table of Contents

  1. Introduction and Main Results
  2. Neumann eigenvalues of bounded domain in $\mathbb{R}^n$.
  3. Eigenvalue estimates on closed manifolds
  4. The center of mass and folding via Bucur and Henrot's approach
  5. Center of mass argument: the Weinberger way
  6. Folding through hyperplanes
  7. Compatibility of Ashbaugh and Benguria's topological argument with the folding method
  8. Estimates on the sum of the reciprocals of eigenvalues for spheres and surfaces
  9. Hersch-Szegő center of mass of a Borel measure on a sphere
  10. Folding onto spherical caps
  11. The case of closed orientable surfaces and proof of Theorem \ref{['thmprincsurfaces']}

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^n$ be a regular bounded domain in $\mathbb{R}^n$. Then with equality achieved by a sequence of domains splitting into two balls of identical volume.

Theorems & Definitions (15)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['thmprncpalDomaine']}
  • Lemma 3.1: Hersch-Szegő center of mass
  • Remark 3.2
  • ...and 5 more