European Journal of Applied Mathematics

Papers

Pulsating wave for mean curvature flow in inhomogeneous medium

N. DIRRa1 p1, G. KARALIa2a3 and N. K. YIPa4

a1 Max-Planck Institute for Mathematics in the Sciences, D-04103 Leipzig, Germany email: ndirr@mis.mpg.de

a2 Department of Applied Mathematics, University of Crete, GR 71409, Heraklion, Greece

a3 Institute for Applied and Computational Mathematics, FORTH, Crete, Greece email: gkarali@tem.uoc.gr

a4 Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA email: yip@math.purdue.edu

Abstract

We prove the existence and uniqueness of pulsating waves for the motion by mean curvature of an n-dimensional hypersurface in an inhomogeneous medium, represented by a periodic forcing. The main difficulty is caused by the degeneracy of the equation and the fact the forcing is allowed to change sign. Under the assumption of weak inhomogeneity, we obtain uniform oscillation and gradient bounds so that the evolving surface can be written as a graph over a reference hyperplane. The existence of an effective speed of propagation is established for any normal direction. We further prove the Lipschitz continuity of the speed with respect to the normal and various stability properties of the pulsating wave. The results are related to the homogenisation of mean curvature flow with forcing.

(Received August 28 2007)

(Revised June 11 2008)

(Online publication August 07 2008)

Correspondence

p1 Current address: Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK. Email: N.Dirr@maths.bath.ac.uk