Minimisers of the Allen-Cahn equation on hyperbolic graphs

TL;DR

This paper studies minimal solutions of the Allen-Cahn equation on hyperbolic graphs, demonstrating the existence of bounded solutions with specific asymptotic behaviors that model energy-minimizing phase transitions.

Contribution

It establishes the existence of non-constant minimal solutions with prescribed asymptotics on Gromov-hyperbolic graphs, linking geometric properties to phase transition models.

Findings

Existence of non-constant bounded minimal solutions with prescribed asymptotics

Solutions describe energy-minimizing steady states in hyperbolic graph models

Connections between hyperbolic geometry and phase transition behavior

Abstract

We investigate minimal solutions of the Allen-Cahn equation on a Gromov-hyperbolic graph. Under some natural conditions on the graph, we show the existence of non-constant uniformly-bounded minimal solutions with prescribed asymptotic behaviours. For a phase field model on a hyperbolic graph, such solutions describe energy-minimising steady-state phase transitions that converge towards prescribed phases given by the asymptotic directions on the graph.