Sharp Convergence Rate of Eigenvalues in a Domain with a Shrinking Tube

TL;DR

This paper investigates how eigenvalues of a domain change when a cylindrical tube attached to it shrinks to zero, providing precise convergence rates using advanced mathematical tools.

Contribution

It introduces a sharp convergence rate analysis for eigenvalues in domains with shrinking tubes, employing Almgren-type monotonicity formulas and blow-up techniques.

Findings

Eigenvalues converge at a quantifiable rate as the tube shrinks.

The analysis uses Courant-Fischer Min-Max characterization.

Blow-up analysis elucidates eigenfunction behavior near the shrinking tube.

Abstract

In this paper we consider a class of singularly perturbed domains, obtained by attaching a cylindrical tube to a fixed bounded region and letting its section shrink to zero. We use an Almgren-type monotonicity formula to evaluate the sharp convergence rate of perturbed simple eigenvalues, via Courant-Fischer Min-Max characterization and blow-up analysis for scaled eigenfunctions.