The Functional Analytic Approach for quasi-periodic boundary value problems for the Helmholtz equation

TL;DR

This paper develops a functional analytic framework for solving quasi-periodic boundary value problems of the Helmholtz equation, introducing fundamental solutions and layer potentials, and analyzing their behavior under domain perturbations.

Contribution

It introduces a quasi-periodic fundamental solution and layer potentials, providing a new approach to analyze quasi-periodic Helmholtz problems and their perturbations.

Findings

Constructed quasi-periodic fundamental solutions and layer potentials.

Analyzed the behavior of solutions under domain perturbations.

Applied the framework to a nonlinear Robin problem with shrinking holes.

Abstract

We lay down the preliminary work to apply the Functional Analytic Approach to quasi-periodic boundary value problems for the Helmholtz equation. This consists in introducing a quasi-periodic fundamental solution and the related layer potentials, showing how they are used to construct the solutions of quasi-periodic boundary value problems, and how they behave when we perform a singular perturbation of the domain. To show an application, we study a nonlinear quasi-periodic Robin problem in a domain with a set of holes that shrink to points.