2019/10/01 by Braides, Andrea, Solci, Margherita
#35B27 #49J45 #49Q20 #82B20 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1910.00680
We consider energies on a periodic set \mathcal L of \mathbb Rd of the form ∑_i,j∈\mathcal L aεij|ui-uj|, defined on spin functions ui∈\0,1\, and we suppose that the typical range of the interactions is Rε with Rε→ +∞, i.e., if ‖i-j‖≤ Rε then aεij≥ c>0. In a discrete-to-continuum analysis, we prove that the overall behaviour as ε→ 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\mathcal L with equibounded energy a family of sets with equibounded perimeter. This agrees with the case of equibounded Rε 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 \mathcal L=\mathbb Zd.