*Published Paper*

**Inserted:** 31 dec 2002

**Last Updated:** 8 mar 2010

**Journal:** SIAM J. Math. Anal.

**Volume:** 36

**Number:** 1

**Pages:** 1-37

**Year:** 2004

**Abstract:**

\begin{document}
We study the asymptotic behavior, as the mesh size $\varepsilon$ tends to zero, of a general class of discrete energies defined on functions $u:\alpha\in\varepsilon*Z*^N\cap\ \Omega\mapsto u(\alpha)\in*R*^d$, of the form
$$
F_{{\varepsilon}}(u)=\sum\limits_{{
}
{\scriptstyle \alpha, \beta \in \varepsilon **Z**^{N\cap\Omega}
}
} g_{{\varepsilon}}(\alpha,\beta,u(\alpha)-u(\beta)),
$$
and satisfying superlinear growth conditions. We show that all the possible variational limits are defined on $W^{1,p}(\Omega;*R*^d)$ of the local type
$$
\int_{\Omega} f(x,\nabla u)\, dx.
$$
We show that in general $f$ may be a quasiconvex non convex function even if very simple interactions are considered. We also treat the case of homogenization giving a general asymptotic formula that can be simplified in many situations (e.g. in the case of nearest neighbor interactions or under convexity hypotheses).
\end{document}

**Keywords:**
Homogenization, $\Gamma$-convergence, discrete systems

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