Networks and Heterogeneous Media (NHM)

Motion of discrete interfaces in low-contrast periodic media

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

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Giovanni Scilla - Dipartimento di Matematica 'G. Castelnuovo', 'Sapienza' Università di Roma, piazzale Aldo Moro 5, 00185 Roma, Italy (email)

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.

Keywords:  Discrete systems, minimizing movements, motion by curvature, crystalline curvature, geometric motion, low-contrast media.
Mathematics Subject Classification:  Primary: 35B27; Secondary: 74Q10, 53C44, 49M25.

Received: June 2013;      Revised: February 2014;      Available Online: April 2014.