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An isoperimetric inequality for lower order Neumann eigenvalues in Gauss space

Yi Gao, Kui Wang

TL;DR

This paper proves a sharp isoperimetric inequality for the harmonic mean of the first $m-1$ nonzero Neumann eigenvalues in Gauss space. It combines Gaussian symmetrization with ball-based radial eigenfunctions to construct admissible test functions and compare domains via the variational principle. The main result states $\sum_{i=1}^{m-1} \frac{1}{\mu_i(\Omega)} \ge \frac{m-1}{\mu_1(B)}$ for origin-symmetric domains $\Omega$ with equal Gaussian volume to the ball $B$, with equality iff $\Omega=B$. This extends Szegö–Weinberger-type bounds to lower-order eigenvalues in Gaussian space and supports Ashbaugh–Benguria-type conjectures in this setting, highlighting the ball as the extremal domain under origin symmetry and fixed Gaussian volume.

Abstract

We prove a sharp isoperimetric inequality for the harmonic mean of the first $m-1$ nonzero Neumann eigenvalues for bounded Lipschitz domains symmetric about the origin in Gauss space. Our result generalizes the Szegö-Weinberger type inequality in Gauss space, as proved in [8, Theorem 4.1].

An isoperimetric inequality for lower order Neumann eigenvalues in Gauss space

TL;DR

This paper proves a sharp isoperimetric inequality for the harmonic mean of the first nonzero Neumann eigenvalues in Gauss space. It combines Gaussian symmetrization with ball-based radial eigenfunctions to construct admissible test functions and compare domains via the variational principle. The main result states for origin-symmetric domains with equal Gaussian volume to the ball , with equality iff . This extends Szegö–Weinberger-type bounds to lower-order eigenvalues in Gaussian space and supports Ashbaugh–Benguria-type conjectures in this setting, highlighting the ball as the extremal domain under origin symmetry and fixed Gaussian volume.

Abstract

We prove a sharp isoperimetric inequality for the harmonic mean of the first nonzero Neumann eigenvalues for bounded Lipschitz domains symmetric about the origin in Gauss space. Our result generalizes the Szegö-Weinberger type inequality in Gauss space, as proved in [8, Theorem 4.1].

Paper Structure

This paper contains 3 sections, 5 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Eigenvalue Problem for Balls In Gauss Space
  3. Proof of Main Theorem

Key Result

Theorem 1.1

Let $\Omega\subset \mathbb{R}^m$ be a Lipschitz domain (possibly unbounded) symmetric about the origin, and let $B \subset \mathbb{R}^m$ be the origin-centered ball with the same Gaussian volume as $\Omega$, i.e. $\int_{\Omega}\, d\gamma_m= \int_{B}\, d\gamma_m$. Let $\mu_i(\Omega)$ be the Neumann e Equality holds if and only if $\Omega=B$.

Theorems & Definitions (11)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.1
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • ...and 1 more