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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: ndirr@mis.mpg.de
G. KARALI
Affiliation:
Department of Applied Mathematics, University of Crete, GR 71409, Heraklion, Greece Institute for Applied and Computational Mathematics, FORTH, Crete, Greece email: gkarali@tem.uoc.gr
N. K. YIP
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA email: yip@math.purdue.edu
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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.

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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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References

[1]Allen, S. & Cahn, J. (1979) A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1084–1095.CrossRefGoogle Scholar
[2]Bhattacharya, K. & Craciun, B. (2003) Homogenization of a Hamilton–Jacobi equation associated with the geometric motion of an interface. Proc. Royal. Soc. Edin. 133A, 773–805.Google Scholar
[3]Bhattacharya, K. & Craciun, B. (2004) Effective motion of a curvature-sensitive interface through a heterogeneous medium. Interfaces Free Bound. 6, 151–173.Google Scholar
[4]Caffarelli, L. & De la Llave, R. (2001) Plane-like minimizers in periodic media. Comm. Pure Appl. Math. 54 (12), 1403–1441.CrossRefGoogle Scholar
[5]Cahn, J. W., Mallet-Paret, J. & Van Vleck, E. S. (1999) Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice. SIAM J. Appl. Math. 59, 455–493.Google Scholar
[6]Cardaliaguet, P., Lions, P. L. & Souganidis, P. E. (2007) A discussion about the homogenization of moving interfaces, preprint.Google Scholar
[7]Chen, Y. G., Giga, Y. & Goto, S. (1991) Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Diff. Geom. 33, 749–786.Google Scholar
[8]Crandall, M. G., Ishii, H. & Lions, P.-L. (1992) User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27, 1–67.CrossRefGoogle Scholar
[9]Cardaliaguet, P., Da Lio, F., Forcadel, N. & Monneau, R. (2007) Dislocation dynamics: A non-local moving boundary. In Free Boundary Problems, Internat. Ser. Numer. Math., Vol. 154, Birkhäuser, Basel, pp. 125–135.CrossRefGoogle Scholar
[10]Dirr, N., Lucia, M. & Novaga, M. (2006) Γ-convergence of the Allen–Cahn energy with an oscillating forcing term. Interfaces Free Bound. 8 (1), 47–78.Google Scholar
[11]Dirr, N. & Yip, N. K. (2006) Pinning and de-pinning phenomena in front propagation in heterogeneous media. Interfaces Free Bound. 8, 79–109.CrossRefGoogle Scholar
[12]Ecker, K. & Huisken, G. (1989) Mean curvature evolution of entire graphs. Ann. Math. 130, 453–471.CrossRefGoogle Scholar
[13]Ecker, K. & Huisken, G. (1991) Interior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105, 547–569.CrossRefGoogle Scholar
[14]Evans, L. C. & Spruck, J. (1991) Motion of level sets by mean curvature. I. J. Differential Geom. 33, 635–681.CrossRefGoogle Scholar
[15]Evans, L. C. & Spruck, J. (1992) Motion of level sets by mean curvature. III. J. Geom. Anal. 2, 121–150.CrossRefGoogle Scholar
[16]Friedman, A. (1958) Remarks on the maximum principle for parabolic equations and its applications. Pacific J. Math. 8, 201–211.CrossRefGoogle Scholar
[17]Huisken, G. (1984) Flow by mean curvature of convex surfaces into spheres. J. Diff. Geom. 20, 237–266.Google Scholar
[18]Korevaar, N. (1986) An easy proof of the interior gradient bound for solutions to the prescribed mean curvature equation, Nonlinear functional analysis and its applications, Part 2. Proc. Sympos. Pure Math. Part 2, Amer. Math. Soc. 45, 81–89.CrossRefGoogle Scholar
[19]Lions, P.-L. & Souganidis, P. E. (2005) Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 22, 667–677.CrossRefGoogle Scholar
[20]Namah, G. & Roquejoffre, J. M. (1997) Convergence to periodic fronts in a class of semi-linear parabolic equations. Nonlinear Diff. Eq. Appl. 4, 521–536.CrossRefGoogle Scholar
[21]Phillips, R. (2001) Crystals, Defects and Microstructures, Cambridge, UK, Cambridge University Press.CrossRefGoogle Scholar
[22]Sandier, E. & Serfaty, S. (2004) Gamma-convergence of gradient flows with applications to Ginzburg–Landau. Comm. Pure Appl. Math. 57, 1627–1672.CrossRefGoogle Scholar