Localization of Neumann Eigenfunctions near Irregular Boundaries

TL;DR

This paper investigates how Neumann eigenfunctions of Laplace's equation tend to localize near irregular, rough boundaries, explaining the mathematical mechanism and quantifying the phenomenon's strength with examples.

Contribution

It provides a mathematical explanation for boundary localization of Neumann eigenfunctions on rough domains and quantifies this effect through specific examples.

Findings

Eigenfunctions tend to localize near rough boundaries.

The paper quantifies the localization strength for certain domain examples.

Implications for acoustics and noise attenuation are discussed.

Abstract

It has been empirically observed that eigenfunctions of Laplace's equation $-\Delta \phi = \lambda \phi$ with Neumann boundary conditions sometimes localize near the boundary of the domain if that boundary is rough (say, fractal). This has some nontrivial implications in acoustics that has been put to real-life use (sound attenuation by noise-protective walls); this short paper describes the mathematical mechanism responsible for this and describes the quantitative strength of the phenomenon for some examples.