*Published Paper*

**Inserted:** 10 may 2005

**Last Updated:** 2 feb 2008

**Journal:** Appl. Math. Optim.

**Volume:** 55

**Pages:** 1-29

**Year:** 2007

**Abstract:**

This paper deals with the homogenization of a sequence of
non-linear conductivity energies in a
bounded open set $\Omega$ of $*R*^d$, for $d\geq 3$. The energy density
is of the same order as
$a_ \varepsilon({x\over \varepsilon})\,

Du(x)

^p$, where $ \varepsilon\to 0$, $a_ \varepsilon$ is
periodic, $u$ is vector-valued function in
$W^{1,p}(\Omega;*R*^m)$ and $p>1$. The conductivity $a_ \varepsilon$ is equal
to $1$ in the ``hard" phases composed by
$N\geq 2$ two by two disjoint-closure periodic sets while $a_ \varepsilon$
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 $\Gamma$-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
$\Gamma$-limit of the rescaled energy of density
$ \varepsilon^{-p}\,a_ \varepsilon(y)\,

Dv(y)

^p$ in the torus. Therefore, the
homogenization result is achieved by a double $\Ga$-convergence
procedure since the cell problem depends
on $ \varepsilon$.

**Keywords:**
Homogenization, double porosity, multi-phase limits

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