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arXiv · 2412.13001

Periodic layer potentials and domain perturbations

Roberto Bramati·Matteo Dalla Riva·Paolo Luzzini·Paolo Musolino
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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.

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Roberto Bramati, Matteo Dalla Riva, Paolo Luzzini, Paolo Musolino. 2024-12-17. Periodic layer potentials and domain perturbations. https://arxiv.org/abs/2412.13001

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