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On localisation of eigenfunctions of the Laplace operator

Michiel van den Berg, Dorin Bucur

TL;DR

This work addresses localisation of Laplacian eigenfunctions on bounded domains, introducing the notion of $\kappa$- localisation for sequences of first Dirichlet eigenfunctions. It provides a simple geometric condition under a uniform strong Hardy inequality that guarantees localisation, and constructs a broad class of elongating horn-shaped domains where localisation occurs without convexity assumptions. The authors also give explicit planar domain sequences for which the first Dirichlet eigenfunction (and, in a Neumann setting, the first nonzero eigenfunction) $L^2$-localises, including a Neumann example based on a two-piece geometry. Collectively, the results illuminate how domain geometry, Hardy-type bounds, and cross-sectional remodeling drive spectral concentration, with implications for spectral geometry and domain design.

Abstract

We prove (i) a simple sufficient geometric condition for localisation of a sequence of first Dirichlet eigenfunctions provided the corresponding Dirichlet Laplacians satisfy a uniform Hardy inequality, and (ii) localisation of a sequence of first Dirichlet eigenfunctions for a wide class of elongating horn-shaped domains. We give examples of sequences of simply connected, planar, polygonal domains for which the corresponding sequence of first eigenfunctions with either Dirichlet, or Neumann, boundary conditions $κ$-localise in $L^2$.

On localisation of eigenfunctions of the Laplace operator

TL;DR

This work addresses localisation of Laplacian eigenfunctions on bounded domains, introducing the notion of - localisation for sequences of first Dirichlet eigenfunctions. It provides a simple geometric condition under a uniform strong Hardy inequality that guarantees localisation, and constructs a broad class of elongating horn-shaped domains where localisation occurs without convexity assumptions. The authors also give explicit planar domain sequences for which the first Dirichlet eigenfunction (and, in a Neumann setting, the first nonzero eigenfunction) -localises, including a Neumann example based on a two-piece geometry. Collectively, the results illuminate how domain geometry, Hardy-type bounds, and cross-sectional remodeling drive spectral concentration, with implications for spectral geometry and domain design.

Abstract

We prove (i) a simple sufficient geometric condition for localisation of a sequence of first Dirichlet eigenfunctions provided the corresponding Dirichlet Laplacians satisfy a uniform Hardy inequality, and (ii) localisation of a sequence of first Dirichlet eigenfunctions for a wide class of elongating horn-shaped domains. We give examples of sequences of simply connected, planar, polygonal domains for which the corresponding sequence of first eigenfunctions with either Dirichlet, or Neumann, boundary conditions -localise in .
Paper Structure (5 sections, 9 theorems, 120 equations, 5 figures)

This paper contains 5 sections, 9 theorems, 120 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Example of $\kappa$-localisation for Dirichlet eigenfunctions
  3. Localisation of the first Dirichlet eigenfunction and Hardy's inequality
  4. Localisation of the first Dirichlet eigenfunction for elongated horn-shaped regions
  5. Example of $\kappa$-localisation for Neumann eigenfunctions

Key Result

Lemma 1

For $n\in \mathbb{N}$, let $f_n\in L^2(\Omega_n)$ with $\|f_{n}\|_2>0,$ and $|\Omega_n|<\infty$. Then $(f_{n})$ localises in $L^2$ if and only if

Figures (5)

  • Figure 1: $\Omega_{\varepsilon,\theta,\delta}=R_{\varepsilon}\cup T_{\theta}\cup S_{\delta}$
  • Figure 2: The mass distribution of $u_1$ when perturbing the size of the square on the right: $\varepsilon=0.4$, $\theta=0.2$, $\delta=\frac{\sqrt{2} \varepsilon}{\sqrt{1+\varepsilon^4}}-c$, for $c=0.00281$, $c=0.00286$, $c=0.00287$, $c=0.00292$, respectively.
  • Figure 3: $\Omega_{n,\alpha,d}$ with $n-1$ parallel vertical line segments at distance $n^{-1}$ of length $1-dn^{-\alpha}$ in the open unit square $Q$.
  • Figure 4: $S\cup R_{\delta,\theta}$
  • Figure 5: The graph of $u^1_{\delta, \theta}$, from localisation to non localisation, when perturbing the length of the thin rectangle: $\theta=0.02$ and $\delta=-0.039$, $\delta=-0.04491$, $\delta=-0.05$, respectively.

Theorems & Definitions (20)

  • Definition 1
  • Lemma 1
  • Theorem 2
  • proof
  • Definition 2
  • Theorem 3
  • proof
  • Example 4
  • Definition 3
  • Theorem 5
  • ...and 10 more