Motion by anisotropic mean curvature as sharp interface limit of an inhomogeneous and anisotropic Allen-Cahn equation

Matthieu Alfaro (1) , Harald Garcke (2) , Danielle Hilhorst (3) , Hiroshi Matano (4) , Reiner Schatzle (5)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 Naturwissenschaftliche Fakultat I - Mathematik
3 Laboratoire d'Analyse Numérique
4 GSMS - Graduate School of Mathematical Sciences
5 Mathematisches Institut, Arbeitsbereich Analysis

We study the singular limit of a spatially inhomogeneous and anisotropic reaction-diffusion equation. We use a Finsler metric related to the anisotropic diffusion term and work in relative geometry. We prove a weak comparison principle and perform an analysis of both the generation and the motion of interfaces. The limit problem involves motion by anisotropic mean curvature. We also prove an optimal estimate of the thickness of the transition layer.

Domaines: Equations aux dérivées partielles [math.AP]

Matthieu Alfaro, Harald Garcke, Danielle Hilhorst, Hiroshi Matano, Reiner Schatzle. Motion by anisotropic mean curvature as sharp interface limit of an inhomogeneous and anisotropic Allen-Cahn equation. Proceedings of the Royal Society of Edinburgh: Section A, Mathematics, 2010, 140 (4), pp.673--706. ⟨10.1017/S0308210508000541⟩. ⟨hal-00392029⟩