*Published Paper*

**Inserted:** 4 apr 2000

**Last Updated:** 10 dec 2003

**Journal:** SIAM J. Math. Anal.

**Volume:** 32

**Number:** 6

**Pages:** 1198-1226

**Year:** 2001

**Abstract:**

To the aim of studying the homogenization of low-dimensional periodic structures,
we identify each of them to a periodic positive measure $\mu$ on $R^n$.
We introduce a new notion of two-scale convergence for a sequence of functions
$v_\epsilon \in L ^p_{\mu _\epsilon} (\Omega; R ^d)$, where $\Omega$ is an open bounded
subset of $R^n$, and the measures $\mu _\epsilon$ are the $\epsilon$-scalings of
$\mu$, namely, $\mu_\epsilon (B) := \epsilon ^n \mu (\epsilon ^ {-1}B)$. Enforcing
the concept of tangential calculus with respect to measures, and
related periodic Sobolev spaces, we prove a structure theorem for
all the possible two-scale limits reached by the sequences
$(u_\epsilon, \nabla u _\epsilon)$, when $\{u _\epsilon\} \subset {\cal C} ^1_0 (\Omega)$
satisfy the boundedness condition $\sup _\epsilon \int _{\Omega}

u_\epsilon

^p +

\nabla u_\epsilon

^p \, d \mu_\epsilon < + \infty$ and when the measure $\mu$
satisfies suitable connectedness properties. This leads to deduce
the homogenized density of a sequence of energies of the form
$\int _{\Omega} j ({x \over \epsilon}, \nabla u) \, d \mu_\epsilon$, where $j(y,z)$ is a
convex integrand, periodic in $y$, and satisfying a $p$-growth
condition. The case of two parameter integrals is also
investigated, in particular for what concerns the commutativity of
the limit process.

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