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

Motion of discrete interfaces in low-contrast periodic media

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.
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.

[2]

F. Almgren, J. E. Taylor and L. Wang, Curvature driven flows: A variational approach,, SIAM J. Control Optim., 31 (1993), 387. doi: 10.1137/0331020.

[3]

A. Braides, Approximation of Free-Discontinuity Problems,, Lecture notes in Mathematics, (1694).

[4]

A. Braides, Local Minimization, Variational Evolution and $\Gamma$-Convergence,, Lecture Notes in Mathematics, (2094). 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. 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. 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). 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, 51 (1993), 417.

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.

[2]

F. Almgren, J. E. Taylor and L. Wang, Curvature driven flows: A variational approach,, SIAM J. Control Optim., 31 (1993), 387. doi: 10.1137/0331020.

[3]

A. Braides, Approximation of Free-Discontinuity Problems,, Lecture notes in Mathematics, (1694).

[4]

A. Braides, Local Minimization, Variational Evolution and $\Gamma$-Convergence,, Lecture Notes in Mathematics, (2094). 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. 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. 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). 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, 51 (1993), 417.

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