On Boundary -- Value Problems for the Laplacian in Boundedand in Unbounded Domains Withperforated Boundaries

TL;DR

This paper investigates boundary-value problems for the Laplacian in complex domains with perforations, classifies their homogenized limits based on geometric parameters, and explores analogues of Helmholtz resonators in such settings.

Contribution

It provides a comprehensive classification of homogenized problems for Laplacian boundary conditions in perforated domains, considering various geometric ratios.

Findings

Classification of limit problems depending on boundary feature ratios

Analysis of Helmholtz resonator analogues in perforated domains

Results on homogenization in complex boundary geometries

Abstract

In the paper we consider boundary -- value problems with rapidly alternating type of boundary conditions, including problems in domains with perforated boundaries. We present the classification of homogenized (limit) problems depending on the ratio of small parameters, which characterize the diameter of parts of the boundary with different types of boundary conditions or on the ratio of small parameters, which characterize the diameter and the distance between holes. Also we studied the analogue of the Helmholtz resonator for domains with perforated boundary.