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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

A. Braides, Approximation of Free-Discontinuity Problems, Lecture notes in Mathematics, 1694, Springer-Verlag, Berlin, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

G. W. Milton, The Theory of Composites, Cambridge University Press, 2002. doi: 10.1017/CBO9780511613357.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

A. Braides, Approximation of Free-Discontinuity Problems, Lecture notes in Mathematics, 1694, Springer-Verlag, Berlin, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

G. W. Milton, The Theory of Composites, Cambridge University Press, 2002. doi: 10.1017/CBO9780511613357.  Google Scholar

[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.  Google Scholar

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