Neumann spectral problem in a domain with very corrugated boundary

TL;DR

This paper investigates how the spectrum of the Neumann Laplacian in a domain with highly corrugated boundary behaves as the boundary's protuberances become infinitely numerous and small, revealing convergence to a boundary value problem with nonlinear spectral conditions.

Contribution

It provides a detailed analysis of the spectral convergence for Neumann problems in domains with infinitely many small protuberances, extending classical results to complex geometries.

Findings

Spectrum converges to a boundary value problem with nonlinear spectral conditions.

Eigenvalues may accumulate at a finite point.

Describes spectral behavior in domains with periodic boundary corrugations.

Abstract

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. We perturb it to a domain $\Omega^\varepsilon$ attaching a family of small protuberances with "room-and-passage"-like geometry ($\varepsilon>0$ is a small parameter). Peculiar spectral properties of Neumann problems in so perturbed domains were observed for the first time by R. Courant and D. Hilbert. We study the case, when the number of protuberances tends to infinity as $\varepsilon\to 0$ and they are $\varepsilon$-periodically distributed along a part of $\partial\Omega$. Our goal is to describe the behaviour of the spectrum of the operator $\mathcal{A}^\varepsilon=-(\rho^\varepsilon)^{-1}\Delta_{\Omega^\varepsilon}$, where $\Delta_{\Omega^\varepsilon}$ is the Neumann Laplacian in $\Omega^\varepsilon$, and the positive function $\rho^\varepsilon$ is equal to $1$ in $\Omega$. We prove that the spectrum of $\mathcal{A}^\varepsilon$ converges as $\varepsilon\to 0$ to the "spectrum" of a certain boundary value problem for the Neumann Laplacian in $\Omega$ with boundary conditions containing the spectral parameter in a nonlinear manner. Its eigenvalues may accumulate to a finite point.