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

  • Blaž Mramor

    Freiburg Institute for Advance Sciences (FRIAS), University of Freiburg, Germany

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 Γ-convergence.

Cite this article

Blaž Mramor, Minimisers of the Allen–Cahn equation and the asymptotic Plateau problem on hyperbolic groups. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 3, pp. 687–711

DOI 10.1016/J.ANIHPC.2017.07.004