Sharp stability on the second Robin eigenvalue with negative boundary parameters

TL;DR

This work establishes a sharp, quantitative stability estimate for the second Robin eigenvalue under negative boundary parameters, refining the Freitas-Laugesen isoperimetric inequality by linking the eigenvalue gap to the Fraenkel asymmetry with a quadratic rate. The authors derive an explicit stability constant γ(n,α,R) and prove that the quadratic exponent is optimal by constructing a family of nearly spherical domains and performing a detailed eigenfunction approximation analysis. A corollary yields a quantitative Szegö-Weinberger-type inequality as α approaches 0−, connecting to Neumann eigenvalue stability. The analysis combines test-function constructions based on ball eigenfunctions, careful boundary perturbation arguments, and deep elliptic estimates to achieve sharpness results in the Robin setting.

Abstract

In this paper, we prove a quantitative refinement of the isoperimetric type inequality for the second Robin eigenvalue with negative boundary parameters established by Freitas and Laugesen [Amer.J.Math.143 (2021), no.3, 969-994].Such new stability estimate is proved when the boundary parameter is not too far from 0.By constructing a suitable family of nearly spherical domains, we prove that the exponent for the Fraenkel asymmetry in this quantitative type inequality is sharp.

Theorems & Definitions (28)

• Theorem 1.1
• Corollary 1.2
• Remark 1.3
• Remark 1.4
• Theorem 1.5
• Proposition 2.1: First Robin eigenfunction of the ball
• Proposition 2.2: Second Robin eigenfunctions of the ball
• proof : Proof of Theorem \ref{['main result one']}
• Lemma 3.1
• proof : Proof of Lemma \ref{['keypoint']}