Periodic layer potentials and domain perturbations

TL;DR

This paper reviews the construction of periodic layer potentials for various differential operators and demonstrates their application in analyzing how solutions behave under domain perturbations, including in unbounded and perforated domains.

Contribution

It introduces a systematic approach to constructing periodic layer potentials and applies them to study the asymptotic behavior of solutions under domain perturbations.

Findings

Asymptotic analysis of quasi-periodic solutions in perforated domains

Dependence of periodic heat solutions on domain perturbations

Construction of fundamental solutions for periodic operators

Abstract

In this paper, we review the construction of periodic fundamental solutions and periodic layer potentials for various differential operators. Specifically, we focus on the Laplace equation, the Helmholtz equation, the Lam\'e system, and the heat equation. We then describe how these layer potentials can be applied to analyze domain perturbation problems. In particular, we present applications to the asymptotic behavior of quasi-periodic solutions for a Dirichlet problem for the Helmholtz equation in an unbounded domain with small periodic perforations. Additionally, we investigate the dependence of spatially periodic solutions of an initial value Dirichlet problem for the heat equation on regular perturbations of the base of a parabolic cylinder.