Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Geometric inequalities between Dirichlet and Neumann eigenvalues

Lawford Hatcher

TL;DR

This work investigates how the isoperimetric ratio governs the count $N(\Omega)$ of Neumann Laplacian eigenvalues not exceeding the first Dirichlet eigenvalue. For bounded convex domains, it establishes explicit two-sided bounds on $N(\Omega)$ in terms of $|\partial\Omega|^n/|\Omega|^{n-1}$ and, via cross-sections $A_{n-1}(\Omega)$, provides alternative geometric controls. It also shows that such bounds fail for general non-convex domains, yet persist for polygonal domains and certain fiber-bundle families. In the fiber-bundle setting, the paper derives a sharp asymptotic as the fibers shrink, linking spectral data of the fiber and base and yielding a universal scaling in $\epsilon$. Together, these results illuminate the delicate interplay between geometry (isoperimetric data) and spectral behaviour under convexity, polygonal structure, and degenerations to lower dimensions, with implications for domain monotonicity and bracketing techniques in spectral geometry.

Abstract

Comparing Neumann and Dirichlet eigenvalues of the Laplacian on a bounded domain $Ω\subseteq\Rbb^n$ is a topic that goes back at least to the work of Pólya \cite{polya}. We study the effect of the isoperimetric ratio of $Ω$ on the number $N(Ω)$ of Neumann eigenvalues that do not exceed the first Dirichlet eigenvalue, proving that $N(Ω)$ is bounded above and below by a constant multiple of the isoperimetric ratio in the case of convex domains. We also show that these estimates do not hold in the non-convex setting, addressing questions of Cox-MacLachlan-Steeves \cite{coxetal} and Freitas \cite{freitas}. Despite these counterexamples, we find similar estimates for polygonal domains in $\Rbb^2$ as well as certain families of fiber bundles that asymptotically collapse onto their base spaces, the motivating examples being tubular neighborhoods of submanifolds.

Geometric inequalities between Dirichlet and Neumann eigenvalues

TL;DR

This work investigates how the isoperimetric ratio governs the count of Neumann Laplacian eigenvalues not exceeding the first Dirichlet eigenvalue. For bounded convex domains, it establishes explicit two-sided bounds on in terms of and, via cross-sections , provides alternative geometric controls. It also shows that such bounds fail for general non-convex domains, yet persist for polygonal domains and certain fiber-bundle families. In the fiber-bundle setting, the paper derives a sharp asymptotic as the fibers shrink, linking spectral data of the fiber and base and yielding a universal scaling in . Together, these results illuminate the delicate interplay between geometry (isoperimetric data) and spectral behaviour under convexity, polygonal structure, and degenerations to lower dimensions, with implications for domain monotonicity and bracketing techniques in spectral geometry.

Abstract

Comparing Neumann and Dirichlet eigenvalues of the Laplacian on a bounded domain is a topic that goes back at least to the work of Pólya \cite{polya}. We study the effect of the isoperimetric ratio of on the number of Neumann eigenvalues that do not exceed the first Dirichlet eigenvalue, proving that is bounded above and below by a constant multiple of the isoperimetric ratio in the case of convex domains. We also show that these estimates do not hold in the non-convex setting, addressing questions of Cox-MacLachlan-Steeves \cite{coxetal} and Freitas \cite{freitas}. Despite these counterexamples, we find similar estimates for polygonal domains in as well as certain families of fiber bundles that asymptotically collapse onto their base spaces, the motivating examples being tubular neighborhoods of submanifolds.

Paper Structure

This paper contains 10 sections, 23 theorems, 110 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Bounded convex domains
  3. Non-convex domains
  4. Fiber bundles
  5. Outlines of proofs
  6. Convex geometry
  7. Proof of Theorem \ref{['mainthmconvex']}
  8. Non-convex counterexamples
  9. Polygonal domains
  10. Fiber bundles with shrinking fibers

Key Result

Theorem 1.2

For each $n\geq 2$, there exist constants $C_1,C_2>0$ depending only on $n$ such that for any bounded convex domain $\Omega\subseteq {\mathbb R}^n$, Moreover, it suffices to take where $\omega_n$ is the volume of the unit ball in ${\mathbb R}^n$.

Figures (4)

  • Figure 1: The domain $\Omega_k$ constructed in the proof of Theorem \ref{['counterexs']} with $k=4$. Illustrated in Desmos (2025).
  • Figure 2: The domain $\Omega_k$ constructed in the proof of Theorem \ref{['diamcounterex']} with $k=3$. Illustrated in Desmos (2025).
  • Figure 3: The domain $\Omega_k$ constructed in the proof of Theorem \ref{['uppercounterex']} with $k=3$. Illustrated in Desmos (2025).
  • Figure 4: The domain $\Omega_k$ constructed in Example \ref{['polygoncounterex']} with $k=5$. Illustrated in Desmos (2025).

Theorems & Definitions (50)

  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9: cf. Theorem 1.1 funano
  • Lemma 2.1
  • proof
  • ...and 40 more