Motion of discrete interfaces in low-contrast periodic media

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March 2014, 9(1): 169-189. doi: 10.3934/nhm.2014.9.169

Motion of discrete interfaces in low-contrast periodic media

Giovanni Scilla ^1,

1.	Dipartimento di Matematica 'G. Castelnuovo', 'Sapienza' Università di Roma, piazzale Aldo Moro 5, 00185 Roma, Italy

Received June 2013 Revised February 2014 Published April 2014

Abstract
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We study the motion of discrete interfaces driven by ferromagnetic interactions in a two-dimensional low-contrast periodic environment, by coupling the minimizing movements approach by Almgren, Taylor and Wang and a discrete-to-continuum analysis. As in a recent paper by Braides and Scilla dealing with high-contrast periodic media, we give an example showing that in general the effective motion does not depend only on the $\Gamma$-limit, but also on geometrical features that are not detected in the static description. We show that there exists a critical value $\widetilde{\delta}$ of the contrast parameter $\delta$ above which the discrete motion is constrained and coincides with the high-contrast case. If $\delta<\widetilde{\delta}$ we have a new pinning threshold and a new effective velocity both depending on $\delta$. We also consider the case of non-uniform inclusions distributed into periodic uniform layers.

Keywords: crystalline curvature, geometric motion, minimizing movements, low-contrast media., Discrete systems, motion by curvature.

Mathematics Subject Classification: Primary: 35B27; Secondary: 74Q10, 53C44, 49M2.

Citation: Giovanni Scilla. Motion of discrete interfaces in low-contrast periodic media. Networks and Heterogeneous Media, 2014, 9 (1) : 169-189. doi: 10.3934/nhm.2014.9.169

References:

[1]	F. Almgren and J. E. Taylor, Flat flow is motion by crystalline curvature for curves with crystalline energies, J. Diff. Geom., 42 (1995), 1-22.
[2]	F. Almgren, J. E. Taylor and L. Wang, Curvature driven flows: A variational approach, SIAM J. Control Optim., 31 (1993), 387-438. doi: 10.1137/0331020.
[3]	A. Braides, Approximation of Free-Discontinuity Problems, Lecture notes in Mathematics, 1694, Springer-Verlag, Berlin, 1998.
[4]	A. Braides, Local Minimization, Variational Evolution and $\Gamma$-Convergence, Lecture Notes in Mathematics, 2094, Springer Verlag, Berlin, 2014. doi: 10.1007/978-3-319-01982-6.
[5]	A. Braides, M. S. Gelli and M. Novaga, Motion and pinning of discrete interfaces, Arch. Ration. Mech. Anal., 195 (2010), 469-498. doi: 10.1007/s00205-009-0215-z.
[6]	A. Braides and G. Scilla, Motion of discrete interfaces in periodic media, Interfaces Free Bound., 15 (2013), 451-476. doi: 10.4171/IFB/310.
[7]	C. Conca, J. San Martín, L. Smaranda and M. Vanninathan, On Burnett coefficients in periodic media in low contrast regime, J. Math. Phys., 49 (2008), 053514, 23 pp. doi: 10.1063/1.2919066.
[8]	G. W. Milton, The Theory of Composites, Cambridge University Press, 2002. doi: 10.1017/CBO9780511613357.
[9]	J. E. Taylor, Motion of curves by crystalline curvature, including triple junctions and boundary points, Differential Geometry, Proceedings of Symposia in Pure Math., 51 (1993), 417-438.

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References:

[1]	F. Almgren and J. E. Taylor, Flat flow is motion by crystalline curvature for curves with crystalline energies, J. Diff. Geom., 42 (1995), 1-22.
[2]	F. Almgren, J. E. Taylor and L. Wang, Curvature driven flows: A variational approach, SIAM J. Control Optim., 31 (1993), 387-438. doi: 10.1137/0331020.
[3]	A. Braides, Approximation of Free-Discontinuity Problems, Lecture notes in Mathematics, 1694, Springer-Verlag, Berlin, 1998.
[4]	A. Braides, Local Minimization, Variational Evolution and $\Gamma$-Convergence, Lecture Notes in Mathematics, 2094, Springer Verlag, Berlin, 2014. doi: 10.1007/978-3-319-01982-6.
[5]	A. Braides, M. S. Gelli and M. Novaga, Motion and pinning of discrete interfaces, Arch. Ration. Mech. Anal., 195 (2010), 469-498. doi: 10.1007/s00205-009-0215-z.
[6]	A. Braides and G. Scilla, Motion of discrete interfaces in periodic media, Interfaces Free Bound., 15 (2013), 451-476. doi: 10.4171/IFB/310.
[7]	C. Conca, J. San Martín, L. Smaranda and M. Vanninathan, On Burnett coefficients in periodic media in low contrast regime, J. Math. Phys., 49 (2008), 053514, 23 pp. doi: 10.1063/1.2919066.
[8]	G. W. Milton, The Theory of Composites, Cambridge University Press, 2002. doi: 10.1017/CBO9780511613357.
[9]	J. E. Taylor, Motion of curves by crystalline curvature, including triple junctions and boundary points, Differential Geometry, Proceedings of Symposia in Pure Math., 51 (1993), 417-438.

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