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Phase transition solutions to an Allen-Cahn model equation

Authors
Keywords
  • 35-Xx Partial Differential Equations
Disciplines
  • Mathematics

Abstract

Phase transition solutions to an Allen-Cahn model equation Jaeyoung Byeon KAIST, Republic of Korea We consider the following equation (0.1) −∆u+ A(x)G′(u) = 0 in Rn where A(x) = A(x+ i) for x ∈ Rn, i ∈ Zn, G(u+ j) = G(u) for u ∈ R, j ∈ Z, G(u) > 0 for u ∈ R \Z, G(j) = 0, G′′(j) > 0 for j ∈ Z and G(t) = G(−t) for t ∈ R. Several authors have studied Allen-Cahn phase transition models in which the spatial phase transition manifests itself as a heteroclinic or homoclinic solution of the corresponding partial differential equation. See e.g. [1]-[4], [16]-[17] and [20]. Similar type of solutions in more general settings arise in extension’s of Moser’s work [14] on developing an Aubry-Mather theory for PDE’s: [5], [6]-[7], [9]-[11], [18]-[19], [21], and [23]. In all of these sources the transition type solutions are unidirectional in the sense that they change in a particular direction. In this talk, I would like to introduce some recent works with Paul Ra- binowitz about some multidirectional solutions. For autonomous problem, that is, A = constant, there have been many works [22], [12],[13] and ref- erences therein which show that the bounded solutions of (0.1) are closely related to the minimal surfaces in Rn. We believe that there would be a generic condition for existence of multi directional solutions like gap conditions for construction of multi-transition (unidirectional) solutions(see [21] and references therein). We do not know how to do this yet for (0.1), but will present a certain type of non-autonomous term A for which such solutions can be found. To describe our results, let x = (x1, · · · , xn) ∈ Rn and A ∈ C1(Rn) be a nonnegative function that is 1-periodic in xi, 1 ≤ i ≤ n. Set Ω ≡ 1 {x ∈ (0, 1)n | A(x) > 0}. We assume 2d∗ = |∂Ω − ∂[0, 1]n| > 0 and ∂Ω is a smooth manifold. Set Ωd ≡ {x ∈ Ω | |x − ∂Ω| > d}. Then for sufficiently small d ∈ (0, d∗), ∂Ωd is diffeomorphic to ∂Ω. Fixing such a small d, let ε > 0 and Aε = 1 + 1 ε A. Our model equation is (0.2) −∆u+ AεG′(u

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