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Pulsating wave for mean curvature flow in inhomogeneous medium

Published online by Cambridge University Press:  01 December 2008

N. DIRR ,
G. KARALI  and
N. K. YIP
N. DIRR
Affiliation:
Max-Planck Institute for Mathematics in the Sciences, D-04103 Leipzig, Germany email: [email protected]
G. KARALI
Affiliation:
Department of Applied Mathematics, University of Crete, GR 71409, Heraklion, Greece Institute for Applied and Computational Mathematics, FORTH, Crete, Greece email: [email protected]
N. K. YIP
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA email: [email protected]
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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.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 19 , Issue 6 , December 2008 , pp. 661 - 699
Copyright
Copyright © Cambridge University Press 2008

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