Full Asymptotics and Laurent Series of Layer Potentials for Laplace's Equation on the Half-Space

TL;DR

This paper applies advanced distribution calculus to layer potentials for Laplace's equation on half-spaces, deriving full asymptotic expansions and new proofs of classical relations, with potential for broader geometric applications.

Contribution

It introduces a novel approach using conormal distributions to analyze layer potentials, providing full asymptotics and new proofs for classical jump relations.

Findings

Derived full asymptotic expansions for layer potential solutions.

Provided a new proof of classical jump relations.

Techniques can be extended to complex geometries.

Abstract

We probe the application of the calculus of conormal distributions, in particular the Pull-Back and Push-Forward Theorems, to the method of layer potentials to solve the Dirichlet and Neumann problems on half-spaces. We obtain full asymptotic expansions for the solutions (provided these exist for the boundary data) and a new proof of the classical jump relations as well as Siegel and Talvila's growth estimates, using techniques that can be generalised to geometrically more complex settings. This is intended to be a first step to understanding the method of layer potentials in the setting of certain non-Lipschitz singularities and to applying a matching asymptotics ansatz to singular perturbations of related problems.