Minimisers of the Allen-Cahn equation and the asymptotic Plateau problem on hyperbolic groups

TL;DR

This paper studies minimal solutions of the Allen-Cahn equation on hyperbolic groups, showing existence under certain conditions and connecting to the asymptotic Plateau problem via Gamma-convergence.

Contribution

It establishes the existence of minimal solutions with prescribed asymptotic behaviors and links phase transition solutions to the asymptotic Plateau problem in hyperbolic groups.

Findings

Existence of minimal solutions with prescribed asymptotics when the Laplace term is small.

Solutions describe phase transitions converging towards given asymptotic directions.

Limit solutions solve the asymptotic Plateau problem via Gamma-convergence.

Abstract

We investigate the existence of non-constant uniformly-bounded minimal solutions of the Allen-Cahn equation on a Gromov-hyperbolic group. We show that whenever the Laplace term in the Allen-Cahn equation is small enough, there exist minimal solutions satisfying a large class of prescribed asymptotic behaviours. For a phase field model on a hyperbolic group, such solutions describe phase transitions that asymptotically converge towards prescribed phases, given by asymptotic directions. In the spirit of de Giorgi's conjecture, we then fix an asymptotic behaviour and let the Laplace term go to zero. In the limit we obtain a solution to a corresponding asymptotic Plateau problem by $\Gamma$-convergence.