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Pinning of interfaces in a random elastic medium and logarithmic lattice embeddings in percolation

Published online by Cambridge University Press:  03 June 2015

Patrick W. Dondl ,
Michael Scheutzow  and
Sebastian Throm
Patrick W. Dondl
Affiliation:
Department of Mathematical Sciences, Durham University, Science Laboratories, South Road, Durham DH1 3LE, UK, (patrick.dondl@durham.ac.uk)
Michael Scheutzow
Affiliation:
Fakultät II, Institut für Mathematik, Sekr. MA 7-5, Technische Universität Berlin, Strasse des 17 Juni 136, 10623 Berlin, Germany, (ms@math.tu-berlin.de)
Sebastian Throm
Affiliation:
Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany, (throm@iam.uni-bonn.de)
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Abstract

For a model of a driven interface in an elastic medium with random obstacles we prove the existence of a stationary positive supersolution at non-vanishing driving force. This shows the emergence of a rate-independent hysteresis through the interaction of the interface with the obstacles despite a linear (force = velocity) microscopic kinetic relation. We also prove a percolation result, namely, the possibility to embed the graph of an only logarithmically growing function in a next-nearest neighbour site percolation cluster at a non-trivial percolation threshold.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 3 , June 2015 , pp. 481 - 512
Copyright
Copyright © Royal Society of Edinburgh 2015 

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References

1 Briani, A. and Monneau, R.. Time-homogenization of a first order system arising in the modelling of the dynamics of dislocation densities. C. R. Math. 347 (2009), 231–236.Google Scholar
2 Caffarelli, L. and Silvestre, L.. An extension problem related to the fractional Laplacian. Commun. PDEs 32 (2007), 1245–1260.Google Scholar
3 Nezza, E. Di, Palatucci, G. and Valdinoci, E.. Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573.Google Scholar
4 Dirr, N., Dondl, P. W., Grimmett, G. R., Holroyd, A. E. and Scheutzow, M.. Lipschitz percolation. Electron. Commun. Prob. 15 (2010), 14–21.Google Scholar
5 Dirr, N., Dondl, P. W. and Scheutzow, M.. Pinning of interfaces in random media. Interfaces Free Boundaries 13 (2011), 411–321.Google Scholar
6 Droniou, J. and Imbert, C.. Fractal first-order partial differential equations. Arch. Ration. Mech. Analysis 182 (2006), 299–331.Google Scholar
7 Ertas, D. and Kardar, M.. Critical dynamics of contact line depinning. Phys. Rev. E 49 (1994), R2532–R2535.Google Scholar
8 Forcadel, N., Imbert, C. and Monneau, R.. Homogenization of some particle systems with two-body interactions and of the dislocation dynamics. Discrete Contin. Dynam. Syst. A 23 (2009), 785–826.Google Scholar
9 Gao, H. and Rice, J. R.. A first-order perturbation analysis of crack trapping by arrays of obstacles. J. Appl. Mech. 56 (1989), 828–836.Google Scholar
10 Grimmett, G. R. and Holroyd, A. E.. Geometry of Lipschitz percolation. Annales Inst. H. Poincaré B48 (2012), 309–326.Google Scholar
11 Imbert, C.. A non-local regularization of first order Hamilton–Jacobi equations. J. Diff. Eqns 211 (2005), 218–246.Google Scholar
12 Joanny, J. F. and Gennes, P. G. de. A model for contact angle hysteresis. J. Chem. Phys. 81 (1984), 552–562.Google Scholar
13 Moulinet, S., Guthmann, C. and Rolley, E.. Roughness and dynamics of a contact line of a viscous fluid on a disordered substrate. Eur. Phys. J. E 8 (2002), 437–443.Google Scholar
14 Schmittbuhl, J., Delaplace, A., Maloy, K. J., Perfettini, H. and Vilotte, J. P.. Slow crack propagation and slip correlations. Pure Appl. Geophys. 160 (2003), 961–976.Google Scholar
15 Tanguy, A. and Vettorel, T.. From weak to strong pinning. I. A finite size study. Eur. Phys. J. B38 (2004), 71–82.Google Scholar