Fourier-based potential theory without an explicit Green's function

TL;DR

This paper develops a Fourier-based potential theory approach that does not require explicit Green's functions, enabling solutions for PDEs and systems where such functions are unavailable.

Contribution

It introduces a Green's function-free formulation of potential theory using Fourier symbols and asymptotic expansions for localized components.

Findings

Derives explicit asymptotic expansions for potentials in powers of a length scale.

Applicable to Poisson equations in 2D and 3D and certain coupled elliptic systems.

Provides a Fourier domain method for potential theory without explicit Green's functions.

Abstract

Integral equation methods provide an effective framework for solving partial differential equations, but their applicability typically relies on the availability of explicit free-space Green's functions. For coupled systems arising in multiphysics applications, such Green's functions are generally not available, limiting the scope of classical potential theory-based approaches. In this work, we introduce a formulation of potential theory that avoids explicit use of Green's functions entirely, relying instead on the Fourier symbol of the governing operator. The central idea is a parabolic regularization of the symbol, which yields a decomposition of the solution into a smooth, nonlocal component and a spatially localized residual. For the localized component, we derive explicit asymptotic expansions for volume, single layer, and double layer potentials in powers of a length scale parameter $\varepsilon$. The coefficients are expressed in terms of local geometric quantities and derivatives of the source data. The derivation is carried out entirely in the Fourier domain and applies to the Poisson equation in two and three dimensions, as well as to a class of coupled strongly elliptic systems.