Motion of discrete interfaces in periodic media

Abstract

We study the motion of discrete interfaces driven by ferromagnetic interactions in a two-dimensional periodic environment by coupling the minimizing movements approach by Almgren, Taylor and Wang and a discrete-to-continuous analysis. The case of a homogeneous environment has been recently treated by Braides, Gelli and Novaga, showing that the effective continuous motion is a flat motion related to the crystalline perimeter obtained by Γ-convergence from the ferromagnetic energies, with an additional discontinuous dependence on the curvature, giving in particular a pinning threshold. In this paper we give an example showing that in general the motion does not depend only on the Γ-limit, but also on geometrical features that are not detected in the static description. In particular we show how the pinning threshold is influenced by the microstructure and that the effective motion is described by a new homogenized velocity. arXiv:1407.7117v1 [math.AP]

Local Minimization, Variational Evolution and Γ-Convergence

“…The incremental problems for the minimizing-movement scheme for F in (8.16) are of the form 17) where for technical reasons we consider the ∞-distance…”

“…Geometric motions with a non-trivial homogenized velocity are described in the paper by Braides and Scilla [17].…”

Motion of discrete interfaces in low-contrast random environments

“…Remark 3.5. In contrast to the deterministic environments considered in [16,19] in our setting the effective velocity of two opposite sides can be different. However this is not due to random effects but can already be caused by a slightly more complex periodic structure as shown in the following example.…”

“…Before we state our next theorem, let us derive a suitable expression for the velocity. We remark that due to Proposition 3.2 the argument is similar to the deterministic case treated in [16]. To reduce notation, we set µ k = E[c eff k,| ] and λ k = E[c eff k,− ] and identify the indices modulo m whenever necessary.…”

Crystalline evolutions in chessboard-like microstructures

“…In the case of geometric motions, a general understanding of the effects of microstructure is still missing. Recently, some results have been obtained for two-dimensional crystalline energies, for which a simpler description can be given in terms of a system of ODEs (see for instance [9,11,8,12,9,10]).…”