\(\Gamma\)-convergence analysis for discrete topological singularities: the anisotropic triangular lattice and the long range interaction energy (Q2805926)

The author considers \(2D\) discrete systems governed by periodic interaction potentials \(\{g_{i,j}\}_{i,j\in \Lambda}\) acting on pairs of atoms of the lattice, and the energy associated to a scalar field \(u:\varepsilon\Lambda \cap \Omega \rightarrow \mathbb{R} \) is defined as NEWLINE\[NEWLINEF_{\varepsilon, \Lambda}(u,\Omega):=\sum_{\varepsilon i,\varepsilon j\in \varepsilon\Lambda \cap \Omega} g_{i,j}\left(u(\varepsilon i)-u(\varepsilon j)\right).NEWLINE\]NEWLINE In [\textit{R. Alicandro} et al., Arch. Ration. Mech. Anal. 214, No. 1, 269--330 (2014; Zbl 1305.82013)], the asymptotic expansion, as \(\varepsilon\rightarrow 0\), of the energy \(F_{\varepsilon, \Lambda}\) has been rigorously derived in terms of \(\Gamma\)-convergence for \(\Lambda=\mathbb{Z}^2\) and assuming that \(g_{i,i+e_1}=g_{i,i+e_2}\) and \(g_{i,j}=0\) otherwise. In this paper the author presents some generalizations of these results for energies accounting for isotropic long range interactions in the square lattice and anisotropic nearest-neighbors interactions on the hexagonal lattice.