Multiscale homogenization of convex functionals with discontinuous integrand

This article is devoted to obtain the $\Gamma$-limit, as $\epsilon$ tends to zero, of the family of functionals $$F_{\epsilon}(u)=\int_{\Omega}f\Bigl(x,\frac{x}{\epsilon},..., \frac{x}{\epsilon^n},\nabla u(x)\Bigr)dx$$, where $f=f(x,y^1,...,y^n,z)$ is periodic in $y^1,...,y^n$, convex in $z$ and satisfies a very weak regularity assumption with respect to $x,y^1,...,y^n$. We approach the problem using the multiscale Young measures.Comment: 18 pages; a slight change in the title; to be published in J. Convex Anal. 14 (2007), No.

A variational approach to the local character of G-closure: the convex case

This article is devoted to characterize all possible effective behaviors of composite materials by means of periodic homogenization. This is known as a $G$-closure problem. Under convexity and $p$-growth conditions ($p>1$), it is proved that all such possible effective energy densities obtained by a $\Gamma$-convergence analysis, can be locally recovered by the pointwise limit of a sequence of periodic homogenized energy densities with prescribed volume fractions. A weaker locality result is also provided without any kind of convexity assumption and the zero level set of effective energy densities is characterized in terms of Young measures. A similar result is given for cell integrands which enables to propose new counter-examples to the validity of the cell formula in the nonconvex case and to the continuity of the determinant with respect to the two-scale convergence.Comment: 24 pages, 1 figur

Loss of polyconvexity by homogenization: a new example

This article is devoted to the study of the asymptotic behavior of the zero-energy deformations set of a periodic nonlinear composite material. We approach the problem using two-scale Young measures. We apply our analysis to show that polyconvex energies are not closed with respect to periodic homogenization. The counterexample is obtained through a rank-one laminated structure assembled by mixing two polyconvex functions with $p$-growth, where $p\geq2$ can be fixed arbitrarily.Comment: 12 pages, 1 figur

Γ\Gamma-limit of the cut functional on dense graph sequences

A sequence of graphs with diverging number of nodes is a dense graph sequence if the number of edges grows approximately as for complete graphs. To each such sequence a function, called graphon, can be associated, which contains information about the asymptotic behavior of the sequence. Here we show that the problem of subdividing a large graph in communities with a minimal amount of cuts can be approached in terms of graphons and the $\Gamma$-limit of the cut functional, and discuss the resulting variational principles on some examples. Since the limit cut functional is naturally defined on Young measures, in many instances the partition problem can be expressed in terms of the probability that a node belongs to one of the communities. Our approach can be used to obtain insights into the bisection problem for large graphs, which is known to be NP-complete.Comment: 25 pages, 5 figure

Liftings, Young measures, and lower semicontinuity

This work introduces liftings and their associated Young measures as new tools to study the asymptotic behaviour of sequences of pairs $(u_j,Du_j)j$ for $(u_j)_j \in \mathrm{BV}(\Omega;\mathbb{R}^m)$ under weak* convergence. These tools are then used to prove an integral representation theorem for the relaxation of the functional $$\mathcal{F}\colon u\to\int_\Omega f(x,u(x),\nabla u(x)) \;\mathrm{dx},\quad u\in\mathrm{W}^{1,1}({\Omega};\mathbb{R}^m),\quad {\Omega}\in\mathbb{R}^d\text{ open},$$ to the space $\mathrm{BV}(\Omega; \mathbb{R}^m)$. Lower semicontinuity results of this type were first obtained by Fonseca and M\"uller [Arch. Ration. Mech. Anal. 123 (1993), 1-49] and later improved by a number of authors, but our theorem is valid under more natural, essentially optimal, hypotheses than those currently present in the literature, requiring principally that $f$ be Carath\'eodory and quasiconvex in the final variable. The key idea is that liftings provide the right way of localising $\mathcal{F}$ in the $x$ and $u$ variables simultaneously under weak* convergence. As a consequence, we are able to implement an optimal measure-theoretic blow-up procedure.Comment: 75 pages. Updated to correct a series of minor typos/ inaccuracies. The statement and proof of Theorem have also been amended- subsequent steps relying upon the Theorem did not require updatin