NONLOCAL ELLIPTIC AND PARABOLIC PROBLEMS BANACH CENTER PUBLICATIONS, VOLUME 66 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2004 INTERFACE DYNAMICS FOR AN ANISOTROPIC ALLEN-CAHN EQUATION MICHAL BENE ˇ S Department of Mathematics, Czech Technical University of Prague Trojanova 13, 120 00 Prague, Czech Republic E-mail: Benes@km1.fjfi.cvut.cz DANIELLE HILHORST Laboratoire de Math´ ematique, Universit´ e de Paris Sud 91405 Orsay Cedex, France E-mail: Danielle.Hilhorst@math.u-psud.fr R ´ EMI WEIDENFELD Ecole Centrale de Lyon, D´ epartement Math-Info 36, Avenue Guy de Collongue, 69134 Ecully Cedex, France E-mail: Remi.Weidenfeld@ec-lyon.fr 1. Introduction. Main results. The purpose of this work is to study the limiting behavior as ǫ → 0 of the solution u ǫ of Problem (P ǫ ), (P ǫ )        u t = div  Φ 0 (∇u)Φ 0 ξ (∇u)  − F ˜ Φ 0 (∇u)+ 1 ǫ 2 f (u) in Ω × (0, ∞), ∇ Φ u.ν =0 on ∂Ω × (0, ∞), u(x, 0) = u 0 (x) for all x ∈ Ω, where Ω is a smooth bounded domain of R N (N ≥ 2), ν is the Euclidean unit normal vector exterior to ∂Ω, f (s)=2s(1 − s 2 ), F = F (x,t) is a smooth function on ¯ Ω × (0, ∞), Φ and ˜ Φ are two Finsler metrics with dual functions Φ 0 and ˜ Φ 0 respectively and where Φ 0 ξ denotes the gradient of Φ 0 . For a precise definition of Finsler metric and Finsler geometry, we refer to [2] or to [3]. 2000 Mathematics Subject Classification : Primary 35B25, 82C26; Secondary 35K55, 80A22, 65N40, 53C80. The paper is in final form and no version of it will be published elsewhere. [39]