no. 3
Motion of discrete interfaces in low-contrast random environments
ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 3, pp. 1275-1301.

We study the asymptotic behavior of a discrete-in-time minimizing movement scheme for square lattice interfaces when both the lattice spacing and the time step vanish. The motion is assumed to be driven by minimization of a weighted random perimeter functional with an additional deterministic dissipation term. We consider rectangular initial sets and lower order random perturbations of the perimeter functional. In case of stationary, α-mixing perturbations we prove a stochastic homogenization result for the interface velocity. We also provide an example which indicates that only stationary, ergodic perturbations might not yield a spatially homogenized limit velocity for this minimizing movement scheme.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2017067
Classification : 53C44, 49J55, 49J45
Mots clés : Minimizing movement, discrete interface motion, crystalline curvature, stochastic homogenization
Ruf, Matthias 1

1 
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     title = {Motion of discrete interfaces in low-contrast random environments},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1275--1301},
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     volume = {24},
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Ruf, Matthias. Motion of discrete interfaces in low-contrast random environments. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 3, pp. 1275-1301. doi : 10.1051/cocv/2017067. http://www.numdam.org/articles/10.1051/cocv/2017067/

[1] R. Alicandro, A. Braides and M. Cicalese, Phase and anti-phase boundaries in binary discrete systems: a variational viewpoint. Netw. Heterog. Media 1 (2006) 85–107 | DOI | MR | Zbl

[2] R. Alicandro, M. Cicalese and A. Gloria, Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity. Arch. Ration. Mech. Anal. 200 (2011) 881–943 | DOI | MR | Zbl

[3] R. Alicandro, M. Cicalese and M. Ruf, Domain formation in magnetic polymer composites: an approach via stochastic homogenization. Arch. Ration. Mech. Anal. 218 (2015) 945–984 | DOI | MR | Zbl

[4] F. Almgren and J.E. Taylor, Flat flow is motion by crystalline curvature for curves with crystalline energies. J. Differ. Geom. 42 (1995) 1–22 | DOI | MR | Zbl

[5] F. Almgren, J.E. Taylor and L. Wang, Curvature-driven flows: a variational approach. SIAM J. Control Optimiz. 31 (1993) 387–438 | DOI | MR | Zbl

[6] L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, New York (2000) | MR | Zbl

[7] G. Bellettini, V. Caselles, A. Chambolle and M. Novaga, Crystalline mean curvature flow for convex sets. Arch. Ration. Mech. Anal. 179 (2006) 109–152 | DOI | MR | Zbl

[8] H. Berbee, Convergence rates in the strong law for bounded mixing sequences. Probab. Theory Related Fields 74 (1987) 255–270 | DOI | MR | Zbl

[9] R.C. Bradley, A caution on mixing conditions for random fields. Statist. Probab. Lett. 8 (1989) 489–491 | DOI | MR | Zbl

[10] A. Braides, Γ-convergence for beginners. In Vol. 22 of Oxford Lect. Series Mathe. Appl. Oxford University Press, Oxford (2002) | MR | Zbl

[11] A. Braides, Local minimization, variational evolution and Γ-convergence. Springer (2014) | DOI | MR | Zbl

[12] A. Braides, M. Cicalese and M. Ruf, Continuum limit and stochastic homogenization of discrete ferromagnetic thin films. Anal. & PDE 11 (2018) 499–553 | DOI | MR | Zbl

[13] A. Braides, M. Cicalese and N.K. Yip, Crystalline motion of interfaces between patterns. J. Stat. Phys. 165 (2016) 274–319 | DOI | MR | Zbl

[14] A. Braides, M.S. Gelli and M. Novaga, Motion and pinning of discrete interfaces. Arch. Ration. Mech. Anal. 195 (2010) 469–498 | DOI | MR | Zbl

[15] A. Braides and A. Piatnitski, Homogenization of surface and length energies for spin systems. J. Funct. Anal. 264 (2013) 1296–1328 | DOI | MR | Zbl

[16] A. Braides and G. Scilla, Motion of discrete interfaces in periodic media. Interfaces Free Bound. 15 (2013) 187–207 | DOI | MR | Zbl

[17] A. Braides and M. Solci, Motion of discrete interfaces through mushy layers. J. Nonlin. Sci. 26 (2016) 1031–1053 | DOI | MR | Zbl

[18] A. Chambolle, M. Morini and M. Ponsiglione, Existence and uniqueness for a crystalline mean curvature flow. Commun. Pure Appl. Math. 70 (2016) 1084–1114 | DOI | MR | Zbl

[19] G. Scilla, Motion of discrete interfaces in low contrast periodic media. Netw. Heterog. Media 9 (2014) 169–189 | DOI | MR | Zbl

[20] J.E. Taylor, Motion of curves by crystalline curvature, including triple junctions and boundary points. Proc. Sympos. Pure Math. 54(1993) 417–438 | DOI | MR | Zbl

[21] N.K. Yip, Stochastic motion by mean curvature. Arch. Ration. Mech. Anal. 144 (1998) 313–355 | DOI | MR | Zbl

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