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

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 $H_{\mathrm{\ell }oc}^1$ 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).