Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Courant-like bound for Neumann domain count

Aleksei Kislitsyn

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 $Ω^n$ such that the first Dirichlet eigenfunction for $Ω^n$ 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.

On Courant-like bound for Neumann domain count

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 such that the first Dirichlet eigenfunction for has at least 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.

Paper Structure

This paper contains 16 sections, 15 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Neumann domains
  3. Definition of the Neumann domains
  4. On a motivation of the definition of the Neumann domains
  5. Supplementary statements from anoop2024neumanndomainsplanaranalytic
  6. Supplementary statements
  7. Supplementary statements for a proof of the Theorem \ref{['th_uniq']}
  8. Supplementary statements for a proof of the Theorem \ref{['th2']}
  9. Proof of the Theorems
  10. Proof of the Theorem \ref{['th_uniq']}
  11. Proof of the Corollary \ref{['col_1']}
  12. Proof of the Corollary \ref{['col_2']}
  13. Proof of the Theorem \ref{['th1']}
  14. Proof of the statement (1) of the Theorem \ref{['th2']}
  15. Proof of the statement (2) of the Theorem \ref{['th2']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $u$ be an analytic solution of the problem (pr1) and $\Omega_i$ be a Neumann domain of the function $u$. Let $x_1, \, x_2 \in \overline{\Omega}_i$ be local maxima of the function $u$. Then there exists a curve of local maxima $\theta$ such that $x_1, \, x_2 \in \theta$.

Figures (2)

  • Figure 1: The Neumann domains (A) and the nodal domains (B) of the second eigenfunction of the Dirichlet problem for the unit circle. Notations: black circle is a local maximum, white circle is a local minimum, cross is a saddle point.
  • Figure 2: (A) Neumann domains of the first eigenfunction of a square. (B) The domain $\Omega_{6, 1/3}$ and the set $L_6$.

Theorems & Definitions (42)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2: anoop2024neumanndomainsplanaranalytic
  • Remark 2.3
  • Remark 2.4
  • ...and 32 more