Sharp stability on the second Robin eigenvalue with negative boundary parameters

TL;DR

This paper refines the stability inequality for the second Robin eigenvalue with negative boundary parameters, establishing sharp exponents and extending previous results to boundary parameters near zero.

Contribution

It provides a quantitative stability estimate for the second Robin eigenvalue with negative boundary parameters, with sharp exponents for nearly spherical domains.

Findings

Sharp stability estimate for second Robin eigenvalue

Construction of nearly spherical domains demonstrating sharp exponents

Extension of previous inequalities to boundary parameters close to zero

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.