39,089 research outputs found
Multiscale homogenization of convex functionals with discontinuous integrand
This article is devoted to obtain the -limit, as tends to
zero, of the family of functionals
, where is
periodic in , convex in and satisfies a very weak regularity
assumption with respect to . 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
-closure problem. Under convexity and -growth conditions (), it is
proved that all such possible effective energy densities obtained by a
-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 -growth, where
can be fixed arbitrarily.Comment: 12 pages, 1 figur
-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 -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 for
under weak* convergence. These
tools are then used to prove an integral representation theorem for the
relaxation of the functional
to the space . 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 be Carath\'eodory and quasiconvex in the final
variable. The key idea is that liftings provide the right way of localising
in the and 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
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