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Spectral stability under removal of small segments

Xiang He

Abstract

In the present paper we deepen the works of L. Abatangelo, V. Felli, L. Hillairet and C. Lena on the asymptotic estimates of the eigenvalue variation under removal of segments from the domain in R2. We get a sharp asymptotic estimate when the eigenvalue is simple and the removed segment is tangent to a nodal line of the associated eigenfunction. Moreover, we extend their results to the case when the eigenvalue is not simple.

Spectral stability under removal of small segments

Abstract

In the present paper we deepen the works of L. Abatangelo, V. Felli, L. Hillairet and C. Lena on the asymptotic estimates of the eigenvalue variation under removal of segments from the domain in R2. We get a sharp asymptotic estimate when the eigenvalue is simple and the removed segment is tangent to a nodal line of the associated eigenfunction. Moreover, we extend their results to the case when the eigenvalue is not simple.
Paper Structure (4 sections, 5 theorems, 60 equations)

This paper contains 4 sections, 5 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. preparation
  3. The Proof of Theorem \ref{['tangent simple']}
  4. multiple case

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^2$ be a bounded domain containing the origin $(0,0)$. Suppose $\lambda_N(\Omega)$ is a simple Dirichlet eigenvalue of $\Omega$, with $L^2$-normalized eigenfunction $u_N$, and suppose that $(0,0)$ is a zero point of $u_N$ of order $k$ and the segment $s_\varepsilon=[-\v

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1: Proposition 2.7 in AFHL18
  • Definition 4.1: Definition 1.7 in ALM22
  • Lemma 4.2
  • proof
  • Lemma 4.3: Proposition 3.1 in ALM22