Two disks maximize the third Robin eigenvalue: positive parameters

TL;DR

This paper establishes a sharp upper bound for the third Robin eigenvalue on simply-connected planar domains with fixed area, extending previous results to positive Robin parameters using a novel degree-theoretic approach.

Contribution

It extends the known eigenvalue inequality to positive Robin parameters by developing a new degree-theoretic method to handle non-monotonic eigenfunctions.

Findings

The third Robin eigenvalue is maximized by two equal disks for parameters in (0,4π].

The inequality previously known for negative parameters is now extended to positive parameters.

A new proof technique overcomes the non-monotonicity issue of eigenfunctions on disks.

Abstract

The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the corresponding eigenvalue of a disjoint union of two equal disks, for Robin parameters in $[-4\pi,4\pi]$. This sharp inequality was known previously only for negative parameters in $[-4\pi,0]$, by Girouard and Laugesen. Their proof fails for positive Robin parameters because the second eigenfunction on a disk has non-monotonic radial part. This difficulty is overcome for parameters in $(0,4\pi]$ by means of a degree-theoretic approach suggested by Karpukhin and Stern that yields suitably orthogonal trial functions.