Calculus of Variations and Geometric Measure Theory

R. Alicandro - M. Cicalese - A. Gloria

Integral representation of the bulk limit of a general class of energies for bounded and unbounded spin systems

created by cicalese on 20 Jul 2007
modified on 25 Jul 2021


Published Paper

Inserted: 20 jul 2007
Last Updated: 25 jul 2021

Journal: Nonlinearity
Volume: 21
Number: 8
Pages: 1881-1910
Year: 2008


We study the asymptotic behaviour of a general class of discrete energies defined on functions $u:\alpha\in\varepsilon Z^N\cap\Omega\mapsto u(\alpha)\in R^m$ of the form $E_\varepsilon(u)=\sum_{\alpha,\beta \in \varepsilon Z^N\cap\Omega} \varepsilon^N g_\varepsilon(\alpha,\beta,u(\alpha),u(\beta))$, as the mesh size $\varepsilon$ goes to $0$. We prove that under general assumptions, that cover the case of bounded and unbounded spin system in the thermodynamic limit, the variational limit of $E_\varepsilon$ has the form $E(u)=\int_{\Omega}g(x,u(x))dx$. The case of homogenization and that of non-pairwise interacting systems (e.g. multiple-exchange spin-systems) is also discussed.

Keywords: Homogenization, $\Gamma$-convergence, spin systems