Square integrable surface potentials on non-smooth domains and application to the Laplace equation in $L^2$

TL;DR

This paper extends the theory of surface potentials to locally square integrable functions on non-smooth domains, facilitating numerical simulations in fluid dynamics involving polygonal meshes.

Contribution

It develops a new framework for single and double layer potentials applicable to $L^2$ functions on Lipschitz and polygonal domains, expanding existing theories beyond $H^1_{loc}$ regularity.

Findings

Extended layer potential theory to $L^2$ functions on non-smooth domains.

Applicable to numerical simulations with polygonal meshes.

Facilitates analysis of the Laplace equation in $L^2$ contexts.

Abstract

Motivated by applications in fluid dynamics involving the harmonic Bergman projection we aim at extending the theory of single and double layer potentials (well documented for functions with $H^1_{\ell oc}$ regularity) to locally square integrable functions. Having in mind numerical simulations in which functions are usually defined on a polygonal mesh, we wish this theory to cover the cases of non-smooth domains (i.e.with Lipschitz continuous or polygonal boundaries).