Minimisers of the Allen–Cahn equation on hyperbolic graphs



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.

Mathematics Subject Classification

58J05 35J20 53C23 



I would like to thank Prof. V. Bangert for the helpful conversations and for his valuable comments.


  1. 1.
    Allen, S., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1084–1095 (1979)CrossRefGoogle Scholar
  2. 2.
    Alonso, J.M., Brady, T., Cooper, D., Ferlini, V., Lustig, M., Mihalik, M., Shapiro, M., Short, H.: Notes on word hyperbolic groups, Group theory from a geometrical viewpoint (Trieste, 1990). World Sci. Publ., River Edge (1991)Google Scholar
  3. 3.
    Ancona, A.: Théorie du potentiel sur les graphes et les variétés, École d’été de Probabilités de Saint-Flour XVIII—1988, Lecture Notes in Math., vol. 1427. Springer, Berlin, pp. 1–112 (1990)Google Scholar
  4. 4.
    Anderson, M.T.: The Dirichlet problem at infinity for manifolds of negative curvature. J. Differ. Geom. 18(4), 701–721 (1984)Google Scholar
  5. 5.
    Birindelli, I., Mazzeo, R.: Symmetry for solutions of two-phase semilinear elliptic equations on hyperbolic space. Indiana Univ. Math. J. 58(5), 2347–2368 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bonk, M., Schramm, O.: Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal. 10(2), 266–306 (2000)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Calegari, D.: The ergodic theory of hyperbolic groups, Geometry and topology down under, Contemp. Math., vol. 597. Amer. Math. Soc., Providence, pp. 15–52 (2013)Google Scholar
  8. 8.
    Candel, A., de la Llave, R.: On the Aubry–Mather theory in statistical mechanics. Commun. Math. Phys. 192(3), 649–669 (1998)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Choi, H.I.: Asymptotic Dirichlet problems for harmonic functions on Riemannian manifolds. Trans. Am. Math. Soc. 281(2), 691–716 (1984)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Coornaert, M.: Mesures de Patterson–Sullivan sur le bord d’un espace hyperbolique au sens de Gromov. Pac. J. Math. 159(2), 241–270 (1993)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Coornaert, M., Delzant, T., Papadopoulos, A.: Géométrie et théorie des groupes. Lecture Notes in Mathematics, vol. 1441. Springer, Berlin (1990)Google Scholar
  12. 12.
    Coulhon, T.: Random walks and geometry on infinite graphs, Lecture notes on analysis in metric spaces (Trento, 1999), Appunti Corsi Tenuti Docenti Sc., Scuola Norm. Sup., Pisa, pp. 5–36 (2000)Google Scholar
  13. 13.
    De Giorgi, E.: Convergence problems for functionals and operators. In: Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), pp. 131–188 (1979)Google Scholar
  14. 14.
    de la Harpe, P.: Topics in geometric group theory, Chicago Lectures in Mathematics. University of Chicago Press, Chicago (2000)Google Scholar
  15. 15.
    del Pino, M., Kowalczyk, M., Wei, J.: Entire solutions of the Allen–Cahn equation and complete embedded minimal surfaces of finite total curvature in \(\mathbb{R}^3\). J. Differ. Geom. 93(1), 67–131 (2013)MathSciNetMATHGoogle Scholar
  16. 16.
    Del Pino, Manuel, Kowalczyk, Michal, Wei, Juncheng: On De Giorgi’s conjecture and beyond. Proc. Natl. Acad. Sci. USA 109(18), 6845–6850 (2012)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Gromov, M.: Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8. Springer, New York, pp. 75–263 (1987)Google Scholar
  18. 18.
    Gromov, M.: Asymptotic invariants of infinite groups, Geometric group theory, vol. 2 (Sussex, 1991), London Math. Soc. Lecture Note Ser., vol. 182. Cambridge Univ. Press, Cambridge, pp. 1–295 (1993)Google Scholar
  19. 19.
    Heinonen, J.: Lectures on Analysis on Metric Spaces, Universitext. Springer, New York (2001)CrossRefMATHGoogle Scholar
  20. 20.
    Holopainen, I., Lang, U., Vähäkangas, A.: Dirichlet problem at infinity on Gromov hyperbolic metric measure spaces. Math. Ann. 339(1), 101–134 (2007)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kapovich, I., Benakli, N., Boundaries of hyperbolic groups, Combinatorial and geometric group theory (New York, 2000, Hoboken, NJ, 2001), Contemp. Math., vol. 296. Amer. Math. Soc. Providence, pp. 39–93 (2002)Google Scholar
  22. 22.
    Koch, H., de la Llave, R., Radin, C.: Aubry–Mather theory for functions on lattices. Discrete Contin. Dyn. Syst. 3(1), 135–151 (1997)MathSciNetMATHGoogle Scholar
  23. 23.
    Mazzeo, R., Saez, M.: Multiple-layer solutions to the Allen–Cahn equation on hyperbolic space. Proc. Am. Math. Soc. 142(8), 2859–2869 (2014)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Modica, L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98(2), 123–142 (1987)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Mramor, B.: Minimisers of the Allen–Cahn equation and the asymptotic plateau problem on hyperbolic groups (2015, preprint)Google Scholar
  26. 26.
    Pisante, A., Ponsiglione, M.: Phase transitions and minimal hypersurfaces in hyperbolic space. Commun. Partial Differ. Equ. 36(5), 819–849 (2011)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Savin, O.: Regularity of flat level sets in phase transitions. Ann. Math. (2) 169(1), 41–78 (2009)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Savin, O.: Phase transitions, minimal surfaces and a conjecture of de Giorgi. Curr. Dev. Math. 2010, 59–113 (2009)CrossRefMATHGoogle Scholar
  29. 29.
    Sullivan, D.: The density at infinity of a discrete group of hyperbolic motions. Inst. Hautes Études Sci. Publ. Math., vol. 50, pp. 171–202 (1979)Google Scholar
  30. 30.
    Sullivan, D.: The Dirichlet problem at infinity for a negatively curved manifold. J. Differ. Geom. 18(4), 723–732 (1984)MathSciNetMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.University of FreiburgFreiburgGermany

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