Uniform upper bounds on Courant sharp Neumann eigenvalues of chain domains

TL;DR

This paper establishes upper bounds on the number of Courant sharp Neumann eigenfunctions for chain domains, regardless of neck widths, advancing understanding of eigenfunction nodal properties in complex geometries.

Contribution

It introduces uniform upper bounds on Courant sharp eigenvalues for chain domains without requiring neck width constraints.

Findings

Upper bounds on the number of nodal domains for Laplace eigenfunctions.

Bounds are independent of the widths of the connecting necks.

Results apply to a broad class of chain domains with piecewise smooth boundaries.

Abstract

We obtain upper bounds on the number of nodal domains of Laplace eigenfunctions on chain domains with Neumann boundary conditions. The chain domains consist of a family of planar domains, with piecewise smooth boundary, that are joined by thin necks. Our work does not assume a lower bound on the width of the necks in the chain domain. As a consequence, we prove an upper bound on the number of Courant sharp eigenfunctions that is independent of the widths of the necks.