Appl Math Optim 55:1–29 (2007)
2006 Springer Science+Business Media, Inc.
Homogenization of Non-Linear Variational Problems
with Thin Low-Conducting Layers
and Marc Briane
Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”,
Via della Ricerca Scientiﬁca, 00133 Roma, Italy
Centre de Math´ematiques, I.N.S.A. de Rennes & I.R.M.A.R.,
20 avenue des Buttes de Co¨esmes, 35043 Rennes Cedex, France
Communicated by D. Kinderlehrer
Abstract. This paper deals with the homogenization of a sequence of non-linear
conductivity energies in a bounded open set of R
, for d ≥ 3. The energy density
is of the same order as a
, where ε → 0, a
is periodic, u is a
vector-valued function in W
) and p > 1. The conductivity a
to 1 in the “hard” phases composed by N ≥ 2 two by two disjoint-closure periodic
sets while a
tends uniformly to 0 in the “soft” phases composed by periodic thin
layers which separate the hard phases. We prove that the limit energy, according to
-convergence, is a multi-phase functional equal to the sum of the homogenized
energies (of order 1) induced by the hard phases plus an interaction energy (of
order 0) due to the soft phases. The number of limit phases is less than or equal
to N and is obtained by evaluating the -limit of the rescaled energy of density
in the torus. Therefore, the homogenization result is achieved
by a double -convergence procedure since the cell problem depends on ε.
Key Words. Homogenization, -Convergence, Non-linear functionals, Multi-
phase limits, Double-porosity.
AMS Classiﬁcation. 35B27, 49J45, 74Q05.
This work is a contribution to the study of the homogenization of non-linear and non-
uniformly coercive problems with a complicated underlying microstructure. Problems