The Newtonian kernel at the intersection of two discs

TL;DR

This paper derives an exact closed-form expression for the Newtonian potential of two overlapping discs, useful in modeling nonlocal interactions in various physical systems, with detailed asymptotics for small overlaps.

Contribution

It provides the first explicit formula for the Newtonian potential of two overlapping discs, including asymptotic expansions for small overlaps.

Findings

Exact closed-form expression for the potential

Piecewise characterization based on geometry

Stable numerical implementation for small overlaps

Abstract

We present an exact, closed-form expression for the Newtonian potential of the characteristic function associated with two overlapping discs in the plane. This setting naturally arises when discretising nonlocal interaction terms present in models of phase separation, aggregation dynamics, and quantum systems. We characterise the convolution integral as a piecewise function on the distance between the disc centres, with transitions dictated by the geometry of the overlapping region. Additionally, we derive detailed asymptotic expansions for the small-overlap regime, which allows us to provide stable double-precision codes.