A General Integral Representation Result for Continuum Limits of Discrete Energies with Superlinear Growth
Roberto Alicandro, Marco Cicalese · SIAM Journal on Mathematical Analysis · 2004
We study the asymptotic behavior, as the mesh size $\varepsilon$ tends to zero, of a general class of discrete energies defined on functions $u:\alpha\in\varepsilon\mathbb Z^N\cap\ \Omega\mapsto u(\alpha)\in{{\mathbb R}^d}$ of the form \begin{eqnarray*} F_{\varepsilon}(u)=\sum\limits_{ \begin{array}{ll} {\scriptstyle \alpha, \beta \in \varepsilon\mathbb Z^N} \\ \scriptstyle \symbol{91}\alpha ,\beta \symbol{93} \subset\Omega \end{array} }\hskip-0.3cm g_{\varepsilon}(\alpha,\beta,u(\alpha)-u(\beta)) \end{eqnarray*} and satisfying superlinear growth conditions. We show that all the possible varia\-tional limits are defined on $W^{1,p}(\Omega;{{\mathbb R}^d})$ of the local type $$ \int_\Omega f(x, abla u)\, dx. $$ We show that, in general, f may be a quasi-convex nonconvex 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).