Vol. 4, No. 1, 2016

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ISSN: 2325-3444 (e-only)
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Discrete double-porosity models for spin systems

Andrea Braides, Valeria Chiadò Piat and Margherita Solci

Vol. 4 (2016), No. 1, 79–102
Abstract

We consider spin systems between a finite number N of “species” or “phases” partitioning a cubic lattice d. We suppose that interactions between points of the same phase are coercive while those between points of different phases (or possibly between points of an additional “weak phase”) are of lower order. Following a discrete-to-continuum approach, we characterize the limit as a continuum energy defined on N-tuples of sets (corresponding to the N strong phases) composed of a surface part, taking into account homogenization at the interface of each strong phase, and a bulk part that describes the combined effect of lower-order terms, weak interactions between phases, and possible oscillations in the weak phase.

Keywords
spin systems, lattice energies, double porosity, $\Gamma$-convergence, homogenization, discrete to continuum, high contrast, interfacial energies, multiphase materials
Mathematical Subject Classification 2010
Primary: 49M25, 39A12, 39A70, 35Q82
Milestones
Received: 1 December 2015
Revised: 21 January 2016
Accepted: 28 February 2016
Published: 28 June 2016

Communicated by Raffaele Esposito
Authors
Andrea Braides
Dipartimento di Matematica
Università di Roma Tor Vergata
via della ricerca scientifica 1
I-00133 Roma
Italy
Valeria Chiadò Piat
Dipartimento di Matematica
Politecnico di Torino
corso Duca degli Abruzzi 24
I-10129 Torino
Italy
Margherita Solci
Dipartimento di Architettura, Design, Urbanistica
Università di Sassari
piazza Duomo 6
I-07041 Alghero
Italy