Calculus of Variations and Geometric Measure Theory

R. Alicandro - M. Cicalese

A General Integral Representation Result for Continuum Limits of Discrete Energies with superlinear growth

created on 31 Dec 2002
modified by cicalese on 08 Mar 2010


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


\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 $$ F{\varepsilon}(u)=\sum\limits{ {\scriptstyle \alpha, \beta \in \varepsilon ZN\cap\Omega} } g{\varepsilon}(\alpha,\beta,u(\alpha)-u(\beta)), $$ 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