Motion of discrete interfaces in low-contrast periodic media

Giovanni Scilla; Giovanni Scilla

doi:10.3934/nhm.2014.9.169

Networks and Heterogeneous Media

2014, Volume 9, Issue 1: 169-189. doi: 10.3934/nhm.2014.9.169

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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

Received: 01 June 2013 Revised: 01 February 2014
Primary: 35B27; Secondary: 74Q10, 53C44, 49M25.

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.
- Discrete systems,
- minimizing movements,
- motion by curvature,
- crystalline curvature,
- geometric motion,
- low-contrast media
Citation: Giovanni Scilla. Motion of discrete interfaces in low-contrast periodic media[J]. Networks and Heterogeneous Media, 2014, 9(1): 169-189. doi: 10.3934/nhm.2014.9.169

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Abstract

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.

References

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