Calculus of Variations and Geometric Measure Theory

A. Braides - M. Solci

Compactness by coarse-graining in long-range lattice systems

created by braidesa on 01 Oct 2019
modified on 30 Dec 2020


Published Paper

Inserted: 1 oct 2019
Last Updated: 30 dec 2020

Journal: Adv. Nonlin. Studies
Volume: 20
Pages: 783–794
Year: 2020
Doi: 10.1515/ans-2020-2100


We consider energies on a periodic set ${\mathcal L}$ of ${\mathbb R}^d$ of the form $\sum_{i,j\in{\mathcal L}} a^\varepsilon_{ij}\
$, defined on spin functions $u_i\in\{0,1\}$, and we suppose that the typical range of the interactions is $R_\varepsilon$ with $R_\varepsilon\to +\infty$, i.e., if $\|i-j\|\le R_\varepsilon$ then $a^\varepsilon_{ij}\ge c>0$. In a discrete-to-continuum analysis, we prove that the overall behaviour as $\varepsilon\to 0$ 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 $\varepsilon{\mathcal L}$ with equibounded energy a family of sets with equibounded perimeter. This agrees with the case of equibounded $R_\varepsilon$ 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 ${\mathcal L}={\mathbb Z}^d$.