Wolf-Keller theorem for Neumann eigenvalues

TL;DR

This paper investigates the maximization of Neumann eigenvalues in planar domains, revealing that unlike Dirichlet eigenvalues, they are not always maximized by disjoint unions of disks, challenging existing assumptions.

Contribution

It extends the Wolf-Keller theorem to Neumann eigenvalues, showing the limitations of disk unions as maximizers for these eigenvalues.

Findings

Neumann eigenvalues are not always maximized by disjoint unions of disks.

The result parallels Wolf and Keller's theorem for Dirichlet eigenvalues.

Disproves the conjecture that unions of disks always maximize Neumann eigenvalues.

Abstract

The classical Szego-Weinberger inequality states that among bounded planar domains of given area, the first nonzero Neumann eigenvalue is maximized by a disk. Recently, it was shown by Girouard, Nadirashvili and Polterovich that, for simply connected planar domains of given area, the second nonzero Neumann eigenvalue is maximized in the limit by a sequence of domains degenerating to a disjoint union of two identical disks. We prove that Neumann eigenvalues of planar domains of fixed area are not always maximized by a disjoint union of arbitrary disks. This is an analogue of a result by Wolf and Keller proved earlier for Dirichlet eigenvalues.