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Compactness by Coarse-Graining in Long-Range Lattice Systems

Authors
  • Braides, Andrea1
  • Solci, Margherita2
  • 1 Università di Roma Tor Vergata, via della ricerca scientifica 1, 00133 , (Italy)
  • 2 Università di Sassari, piazza Duomo 6, 07041, (SS) , (Italy)
Type
Published Article
Journal
Advanced Nonlinear Studies
Publisher
De Gruyter
Publication Date
Sep 08, 2020
Volume
20
Issue
4
Pages
783–794
Identifiers
DOI: 10.1515/ans-2020-2100
Source
De Gruyter
Keywords
License
Yellow

Abstract

We consider energies on a periodic set ℒ{\mathcal{L}} of the form ∑i,j∈ℒai⁢jε⁢|ui-uj|{\sum_{i,j\in\mathcal{L}}a^{\varepsilon}_{ij}\lvert u_{i}-u_{j}\rvert}, defined on spin functions ui∈{0,1}{u_{i}\in\{0,1\}}, and we suppose that the typical range of the interactions is Rε{R_{\varepsilon}} with Rε→+∞{R_{\varepsilon}\to+\infty}, i.e., if |i-j|≤Rε{\lvert i-j\rvert\leq R_{\varepsilon}}, then ai⁢jε≥c>0{a^{\varepsilon}_{ij}\geq c>0}. In a discrete-to-continuum analysis, we prove that the overall behavior as ε→0{\varepsilon\to 0} of such functionals is that of an interfacial energy. The proof is performed using a coarse-graining procedure which associates to scaled functions defined on ε⁢ℒ{\varepsilon\mathcal{L}} with equibounded energy a family of sets with equibounded perimeter. This agrees with the case of equibounded Rε{R_{\varepsilon}} and can be seen as an extension of coerciveness result for short-range interactions, but is different from that of other long-range interaction energies, whose limit exits the class of surface energies. A computation of the limit energy is performed in the case ℒ=ℤd{\mathcal{L}=\mathbb{Z}^{d}}.

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