On Courant-like bound for Neumann domain count

TL;DR

This paper demonstrates that a Courant-like bound does not generally exist for Neumann domain counts by constructing specific domain sequences, but identifies conditions where such bounds may hold for convex domains.

Contribution

It shows the non-existence of a universal Courant-like bound for Neumann domains and provides conditions under which such bounds could exist for convex domains.

Findings

Constructed domains with at least n Neumann domains for the first eigenfunction

Established non-existence of a general Courant-like bound for Neumann domains

Identified conditions for possible Courant-like bounds in convex domains

Abstract

In this work we show that in general there is no Courant-like bound for Neumann domain count. In order to do that we construct a sequence of domains $\Omega^n$ such that the first Dirichlet eigenfunction for $\Omega^n$ has at least $n$ Neumann domains. Also a special case of convex domains is considered and sufficient conditions for existence of Courant-like bound for small eigenvalues are found.