Minimisers of the Allen–Cahn equation on hyperbolic graphs

  • Blaž Mramor


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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.University of FreiburgFreiburgGermany

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