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Design of lattice surface energies. (English) Zbl 1397.35014

Summary: We provide a general framework for the design of surface energies on lattices. We prove sharp bounds for the homogenization of discrete systems describing mixtures of ferromagnetic interactions by constructing optimal microgeometries, and we prove a localization principle which allows to reduce to the periodic setting in the general nonperiodic case.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
74Q20 Bounds on effective properties in solid mechanics
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
49J45 Methods involving semicontinuity and convergence; relaxation
49Q20 Variational problems in a geometric measure-theoretic setting
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