Localization of Neumann Eigenfunctions near Irregular Boundaries
Analysis of PDEs
2019-02-20 v1 Disordered Systems and Neural Networks
Mathematical Physics
math.MP
Spectral Theory
Abstract
It has been empirically observed that eigenfunctions of Laplace's equation 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.
Cite
@article{arxiv.1809.01108,
title = {Localization of Neumann Eigenfunctions near Irregular Boundaries},
author = {Peter W. Jones and Stefan Steinerberger},
journal= {arXiv preprint arXiv:1809.01108},
year = {2019}
}