Stability of the second non trivial eigenvalue of the Neumann Laplacian

TL;DR

This paper establishes a stability estimate for the second non trivial eigenvalue of the Neumann Laplacian, showing how near-maximizers are close to the union of two equal balls, building on prior inequalities.

Contribution

It provides a stability estimate for Bucur and Henrot's inequality related to the Neumann Laplacian eigenvalues, extending previous results and offering quantitative bounds.

Findings

Proved a stability estimate for the second Neumann eigenvalue inequality.

Quantified how sets close to maximizers resemble unions of two equal balls.

Connected stability results with geometric configurations of domains.

Abstract

In this paper, building on the ideas of Brasco and Pratelli (Geom. Funct. Anal., 22 (2012), 107-135), we establish a stability estimate for Bucur and Henrot's inequality (Acta Math., 222 (2019), 337-361). Their inequality asserts that, among regular sets of given measure, the disjoint union of two balls with the same radius maximizes the second non trivial eigenvalue of the Neumann Laplacian. Last week I was informed of the same result of this paper has been published by Wang K, Wu H. A quantitative Bucur Henrot inequality. Mathematische Nachrichten, 2022, 295(12): 2436-2451. So thearticle has been deprecated.