New links between PDE's and Voronoi patterns

TL;DR

This paper explores the mathematical and computational connections between PDEs and Voronoi patterns, including analytical solutions, stochastic models, and applications in physics and optimal transport, revealing new insights into pattern formation.

Contribution

It introduces the concept of harmonic Voronoi tessellations, links PDE solutions to Voronoi patterns, and applies these ideas to various physical and mathematical models, including heat kernels and optimal transport.

Findings

Voronoi patterns emerge from PDE interactions in Riemannian manifolds.

Analytical solutions define harmonic Voronoi tessellations.

Heat kernel estimates relate point sources to Voronoi tessellations.

Abstract

This paper presents a range of results in partial differential equations (PDEs) in which Voronoi patterns arise. We investigate the connection between 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 the Voronoi cells simulating experimental results with bacteria. We also consider the analytical solution to the problem, which enables us to define what we call a harmonic Voronoi tessellation. We analyze parabolic equations in Riemannian manifolds, which have important applications in chemical reactions and diffusive fronts. By utilizing short-time heat kernel estimates, we demonstrate that the interaction of $n$ point sources gives rise to a Voronoi tessellation. We recall some well-known results of wavefronts interactions from point light sources and the Huygens principle. We apply results about 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-Amp\`ere equation, in which a uniform measure is transported to a sum of $n$ Dirac masses with a cost given by the Euclidean distance. These problems are naturally linked to power sets, a generalization of Voronoi tessellations.