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Periodic layer potentials and domain perturbations

Roberto Bramati, Matteo Dalla Riva, Paolo Luzzini, Paolo Musolino

TL;DR

This work develops a comprehensive, periodic potential theory framework for elliptic and parabolic PDEs in periodically structured domains, and uses a Functional Analytic Approach to quantify how solutions and layer potentials depend on regular and singular domain perturbations. It provides explicit constructions of periodic and quasi-periodic fundamental solutions, along with layer potentials for the Laplace, Helmholtz, Lamé, and heat operators, and derives their jump relations and mapping properties in Schauder spaces. The paper then applies these tools to two perturbation scenarios: small periodic perforations in Helmholtz Dirichlet problems and regularly perturbed domains for the heat equation, obtaining density-based representations of solutions and real-analytic or smooth dependence on perturbation parameters, including asymptotic expansions. The results offer rigorous methods for asymptotic analysis, dilute-material modeling, and shape optimization in periodic media and extend potential theory to vector-valued systems and parabolic equations.

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 Lamé 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.

Periodic layer potentials and domain perturbations

TL;DR

This work develops a comprehensive, periodic potential theory framework for elliptic and parabolic PDEs in periodically structured domains, and uses a Functional Analytic Approach to quantify how solutions and layer potentials depend on regular and singular domain perturbations. It provides explicit constructions of periodic and quasi-periodic fundamental solutions, along with layer potentials for the Laplace, Helmholtz, Lamé, and heat operators, and derives their jump relations and mapping properties in Schauder spaces. The paper then applies these tools to two perturbation scenarios: small periodic perforations in Helmholtz Dirichlet problems and regularly perturbed domains for the heat equation, obtaining density-based representations of solutions and real-analytic or smooth dependence on perturbation parameters, including asymptotic expansions. The results offer rigorous methods for asymptotic analysis, dilute-material modeling, and shape optimization in periodic media and extend potential theory to vector-valued systems and parabolic equations.

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 Lamé 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.

Paper Structure

This paper contains 7 sections, 19 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Periodic layer potentials for the Laplace equation
  3. Quasi-periodic layer potentials for the Helmholtz equation
  4. A singularly perturbed boundary value problem
  5. The case of systems: the Lamé equations
  6. The extension to parabolic equations: space-periodic layer potentials for the heat equation
  7. A regularly perturbed boundary value problem

Key Result

Theorem 2.1

The generalized series defines a tempered distribution in ${\mathbb{R}}^{n}$ such that $S_{q,n}$ is $Q$-periodic and such that where $\delta_{qz}$ denotes the Dirac measure with mass at $qz$, for all $z\in {\mathbb{Z}}^{n}$. Moreover, $S_{q,n}$ is real analytic in ${\mathbb{R}}^{n}\setminus q{\mathbb{Z}}^{n}$, $S_{q,n}\in L^{1}_{ {\mathrm{loc}} }({\mathbb{R}}^{n})$, and $S_{q,n}(x)=S_{q,n}(-x)$

Theorems & Definitions (29)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Remark 3.4
  • Proposition 3.5
  • ...and 19 more