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An isoperimetric inequality for the second Robin eigenvalue of the Weighted Laplacian

Yi Gao, Kui Wang, Anqiang Zhu

TL;DR

This work addresses a shape-optimization problem for the second Robin eigenvalue $\lambda_{2,\alpha}(\Omega;\gamma_h)$ of the weighted Laplacian with density $d\gamma_h = e^{h(|x|)}\,dx$ on origin-symmetric bounded Lipschitz domains. Using Weinberger's trick and a detailed spectral analysis of balls in the weighted space, the authors prove that among domains with fixed $\gamma_h$-volume, the origin-centered ball maximizes $\lambda_{2,\alpha}$ for $\alpha \in [-\sigma_1(B;\gamma_h),0]$, with equality only for the ball; as a corollary, the first nonzero Steklov eigenvalue satisfies $\sigma_1(\Omega;\gamma_h) \le \sigma_1(B;\gamma_h)$. The results rely on monotonicity properties of the radial second eigenfunction on balls and a weighted rearrangement argument, and they connect to known Brock–Weinstock-type inequalities and to Neumann-type problems under Gaussian-like weights. This yields a unified isoperimetric framework for weighted second eigenvalues and highlights the ball as the extremal shape in a broad weighted setting.

Abstract

In this paper, we investigate a shape optimization problem for the second Robin eigenvalue of the weighted Laplacian on bounded Lipschitz domains symmetric about the origin. Our main theorem states that the ball centered at the origin maximizes the second Robin eigenvalue among all Lipschitz bounded domains of prescribed weighted measure and symmetric about the origin for a range of negative Robin parameters.

An isoperimetric inequality for the second Robin eigenvalue of the Weighted Laplacian

TL;DR

This work addresses a shape-optimization problem for the second Robin eigenvalue of the weighted Laplacian with density on origin-symmetric bounded Lipschitz domains. Using Weinberger's trick and a detailed spectral analysis of balls in the weighted space, the authors prove that among domains with fixed -volume, the origin-centered ball maximizes for , with equality only for the ball; as a corollary, the first nonzero Steklov eigenvalue satisfies . The results rely on monotonicity properties of the radial second eigenfunction on balls and a weighted rearrangement argument, and they connect to known Brock–Weinstock-type inequalities and to Neumann-type problems under Gaussian-like weights. This yields a unified isoperimetric framework for weighted second eigenvalues and highlights the ball as the extremal shape in a broad weighted setting.

Abstract

In this paper, we investigate a shape optimization problem for the second Robin eigenvalue of the weighted Laplacian on bounded Lipschitz domains symmetric about the origin. Our main theorem states that the ball centered at the origin maximizes the second Robin eigenvalue among all Lipschitz bounded domains of prescribed weighted measure and symmetric about the origin for a range of negative Robin parameters.
Paper Structure (3 sections, 8 theorems, 91 equations)

This paper contains 3 sections, 8 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Eigenvalue Problem for Balls In the Weighted Space
  3. Proof of Theorem \ref{['thm 1']}

Key Result

Theorem 1.1

Let $\Omega\subset \mathbb{R}^m$ be a bounded Lipschitz domain symmetric about the origin. Suppose $h(r)\in C^2([0, +\infty))$ satisfies $h'(r)\ge 0$ and $h"(r)\ge 0$ for $r\in (0, +\infty)$. Let $B \subset \mathbb{R}^m$ be the origin-centered round ball with the same $\gamma_h$-volume as $\Omega$, Equality holds if and only if $\Omega=B$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 6 more