Geodesic
Minimisers of the Allen–Cahn equation and the asymptotic Plateau problem on hyperbolic groups
Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 3, pp. 687-711

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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.

DOI : 10.1016/j.anihpc.2017.07.004
Keywords: Analysis on metric spaces, Hyperbolic groups, Minimisers of variational nonlinear elliptic equations, Allen–Cahn equation, Asymptotic Plateau problem 
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     author = {Mramor, Bla\v{z}},
     title = {Minimisers of the {Allen{\textendash}Cahn} equation and the asymptotic {Plateau} problem on hyperbolic groups},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {687--711},
     publisher = {Elsevier},
     volume = {35},
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     year = {2018},
     doi = {10.1016/j.anihpc.2017.07.004},
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     zbl = {1420.35109},
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Mramor, Blaž. Minimisers of the Allen–Cahn equation and the asymptotic Plateau problem on hyperbolic groups. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 3, pp. 687-711. doi: 10.1016/j.anihpc.2017.07.004

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