New and Old Links Between PDEs and Voronoi Patterns

Abstract This paper presents a range of results in partial differential equations (PDEs) in which Voronoi patterns arise. Some are well-known and are presented here for the sake of completeness. Others are intuitive, but their proofs are not necessarily simple and are not found in the literature. We also introduce some new connections between Voronoi patterns and PDEs. We first investigate the connection between Voronoi tessellations and the solution to an elliptic equation and its probabilistic interpretation as a stochastic colonization game. An agent-based model is designed and implemented to generate Voronoi cells, motivated by experimental results with bacteria and chemical reactions, emulating this colonization game. We also consider the analytical solution to a related elliptic problem, which enables us to define what we call a harmonic Voronoi tessellation. We analyze parabolic equations in Riemannian manifolds, which have criticalapplications in chemical reactions and diffusive fronts. Using short-timeheat kernel estimates, we demonstrate that the interaction of $n$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> pointsources gives rise to a Voronoi tessellation. We recall some well-known results of wavefront interactions from point light sources and the Huygens principle. We apply results on the particular set of weak solutions to the eikonal equation to characterize Voronoi patterns arising in this context as rectifiable sets. Finally, we present an optimal transport problem and the corresponding Monge-Ampère equation, in which a uniform measure is transported to a sum of $n$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> Dirac masses with a cost given by the Euclidean distance. These problems are naturally linked to power sets, a generalization of Voronoi tessellations.