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Isoperimetric inequalities for Neumann eigenvalues on bounded domains in rank-1 symmetric spaces

Yifeng Meng, Kui Wang

Abstract

In this paper, we prove sharp isoperimetric inequalities for lower order eigenvalues of Neumann Laplacian on bounded domains in both compact and noncompact rank-1 symmetric spaces. Our results generalize the work of Wang and Xia for bounded domains in the hyperbolic space [13], and Szegö-Weinberger inequality in rank-1 symmetric spaces obtained by Aithal and Santhanam [1].

Isoperimetric inequalities for Neumann eigenvalues on bounded domains in rank-1 symmetric spaces

Abstract

In this paper, we prove sharp isoperimetric inequalities for lower order eigenvalues of Neumann Laplacian on bounded domains in both compact and noncompact rank-1 symmetric spaces. Our results generalize the work of Wang and Xia for bounded domains in the hyperbolic space [13], and Szegö-Weinberger inequality in rank-1 symmetric spaces obtained by Aithal and Santhanam [1].
Paper Structure (7 sections, 6 theorems, 69 equations)

This paper contains 7 sections, 6 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Geometry of Rank-1 Symmetric spaces
  3. Some mathematical tools needed
  4. Properties of eigenfunctions for geodesic balls
  5. Trial functions for lower order Neumann eigenvalues
  6. proof of theorem \ref{['thm1']}
  7. proof of theorem \ref{['thm2']}

Key Result

Theorem 1.2

For any bounded domain $\Omega$ in $\mathbb{R}^n$ ($n\ge 3$), it holds where $B$ is a round ball in $\mathbb{R}^n$ having the same volume as $\Omega$. Moreover the equality occurs if and only if $\Omega$ is a round ball.

Theorems & Definitions (11)

  • Conjecture 1.1
  • Theorem 1.2: WX23
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 4.1: AS96
  • Lemma 4.2
  • proof
  • proof : Proof of Theorem \ref{['thm1']}
  • Lemma 5.1
  • proof
  • ...and 1 more