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

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.

Minimisers of the Allen–Cahn equation on hyperbolic graphs

“…One step in the proofs that can be applied also to a graph like Γ is the following proposition, which states that shadows are "almost round". The proof is analogous to [25,Proposition 2.3].…”

“…A similar variational approach to the one used in this paper was developed by the author in [25] for solving the Allen-Cahn equation on a Cayley graph of a hyperbolic group. There, the equivalent statement to the existence theorem above is proved when (AC) has the form ρ∆g(x) − V ′ (xg) = 0 with asymptotically small ρ (see [25,Theorem 4.14]). In Theorem 3.7 of that paper it has been shown that for small enough ρ, all global minimisers converge to either c0 or c1, which gives additional motivation for studying the Dirichlet problem in the form presented in theorem A above.…”

“…The following two lemmas are standard for elliptic difference operators. For proofs, see [25], [8] or [22]. In particular, any two global minimisers x, y with xg ≤ yg for all g ∈ G and such that x = y, are totally ordered: xg < yg for all g ∈ G.…”

“…A similar variational approach to the one used in this paper was developed by the author in [25] for solving the Allen-Cahn equation on a Cayley graph of a hyperbolic group. There, the equivalent statement to the existence theorem above is proved when (AC) has the form ρ∆g(x) − V ′ (xg) = 0 with asymptotically small ρ (see [25,Theorem 4.14]).…”

“…In Theorem 3.7 of that paper it has been shown that for small enough ρ, all global minimisers converge to either c0 or c1, which gives additional motivation for studying the Dirichlet problem in the form presented in theorem A above. Even though a part of the construction of the proofs in this paper is similar to those in [25], a couple of serious technical difficulties needed to be overcome to prove the statements in this text. The first and technically the most difficult problem was to move away from the perturbative case, that is, to drop the constant ρ from the equation.…”