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Laplacian eigenvalues for large negative Robin parameters on domains with outward peaks

Konstantin Pankrashkin, Firoj Sk, Marco Vogel

TL;DR

The paper analyzes the leading-order behavior of individual Robin Laplacian eigenvalues on domains with outward peaks as the Robin parameter becomes large. By reducing the problem to a cylindrical peak model and exploiting a transverse decomposition with the first cross-sectional Robin eigenfunction, the authors derive a sharp asymptotic expansion $\lambda_j(R^\Omega_\alpha)=A_\omega^{2/(2-q)}\lambda_j(L_1)\alpha^{2/(2-q)}+O(\alpha^{2/(2-q)-(q-1)})$, where $A_\omega=\mathcal{H}^{d-2}(\partial\omega)/\mathcal{H}^{d-1}(\omega)$ and $L_1$ is the one-dimensional operator $L_1 f= -f''+\big( {q^2(d-1)^2-2q(d-1) \over 4s^2}-1/s^q\big)f$. An Agmon-type estimate shows localization of eigenfunctions near the peak tip, and the analysis extends previous ball-cross-section results to general cross-sections $\omega$, illuminating how sharpness $q$ and geometry enter the asymptotics. The methods rely on variational principles, a cylindrical coordinate reduction, and careful truncation arguments to isolate peak contributions, with potential for generalizations to multiple peaks and non-isotropic settings.

Abstract

We study the asymptotic behavior of individual eigenvalues of the Laplacian in domains with outward peaks for large negative Robin parameters. A large class of cross-sections is allowed, and the resulting asymptotic expansions reflect both the sharpness of the peak and the geometric shape of its cross-section. The results are an extension of previous works dealing with peaks whose cross-sections are balls.

Laplacian eigenvalues for large negative Robin parameters on domains with outward peaks

TL;DR

The paper analyzes the leading-order behavior of individual Robin Laplacian eigenvalues on domains with outward peaks as the Robin parameter becomes large. By reducing the problem to a cylindrical peak model and exploiting a transverse decomposition with the first cross-sectional Robin eigenfunction, the authors derive a sharp asymptotic expansion , where and is the one-dimensional operator . An Agmon-type estimate shows localization of eigenfunctions near the peak tip, and the analysis extends previous ball-cross-section results to general cross-sections , illuminating how sharpness and geometry enter the asymptotics. The methods rely on variational principles, a cylindrical coordinate reduction, and careful truncation arguments to isolate peak contributions, with potential for generalizations to multiple peaks and non-isotropic settings.

Abstract

We study the asymptotic behavior of individual eigenvalues of the Laplacian in domains with outward peaks for large negative Robin parameters. A large class of cross-sections is allowed, and the resulting asymptotic expansions reflect both the sharpness of the peak and the geometric shape of its cross-section. The results are an extension of previous works dealing with peaks whose cross-sections are balls.

Paper Structure

This paper contains 14 sections, 22 theorems, 211 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preparations
  3. Robin Laplacians on bounded Lipschitz domains
  4. A one-dimensional operator
  5. Finite peaks and related operators
  6. Reduction to cylindrical domains
  7. A lower bound outside the peak
  8. Spectral asymptotics near the peak
  9. Model operators near the peak
  10. Upper bound
  11. Lower bound
  12. Truncations and the proof of Theorem \ref{['thmain']} (eigenvalue asymptotics)
  13. Proof of Theorem \ref{['thmeig']} (localization of eigenfunctions)
  14. Spectral properties of the effective one-dimensional Schrödinger operator $L_\mu$

Key Result

Theorem 2

Let $\Omega\subset\mathbb{R}^d$ be as in Definition defin1 with some bounded Lipschitz cross-section $\omega\subset\mathbb{R}^{d-1}$ and sharpness order $q\in(1,2)$, then for any fixed $j\in\mathbb{N}$ one has where $L_1$ is the differential operator in $L^2(0,\infty)$ defined first by and then extended using the Friedrichs extension.

Figures (2)

  • Figure 1: An example of a domain $\Omega$ with an outward peak with cross-section $\omega$.
  • Figure 2: An example of the model peak domain $V_{\varepsilon,I}$.

Theorems & Definitions (38)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Remark 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • proof
  • Lemma 9
  • ...and 28 more