We consider energy functionals, or Dirichlet forms,
$ J_\varOmega^\varepsilon(u)= \int_\varOmega (A^\varepsilon \nabla u, \nabla u) \,\mathrm{d}x= \sum^N_{i,j=1} \, \int_\varOmega a^\varepsilon_{ij} \, \frac{\curpartial u}{\curpartial x_i} \, \frac{\curpartial u}{\curpartial x_j} \,\mathrm{d}x, $
for a class
$\mathcal{G}$
of bounded domains
$\varOmega\subset\mathbb{R}^N$
, with
$\varepsilon >0$
a fine structure parameter and with symmetric conductivity matrices
$A^\varepsilon=(a^\varepsilon_{ij})\in L_\mathrm{loc}^\infty(\mathbb{R})^{N\times N}$
which are functions only of the first coordinate
$x_1$
and which are locally uniformly elliptic for each fixed
$\varepsilon>0$
. We show that if the functions (of
$x_1$
)
$b^\varepsilon_{11}={1}/{a^\varepsilon_{11}}$
,
$b^\varepsilon_{1j}={a^\varepsilon_{1j}}/{a^\varepsilon_{11}}\ (j\geq 2)$
,
$b^\varepsilon_{ij}= a^\varepsilon_{ij} - {a^\varepsilon_{i1}a^\varepsilon_{1j}}/{a^\varepsilon_{11}} \ (i, j\geq 2)$
converge weakly* as measures towards corresponding limit measures
$b_{ij}$
as
$\varepsilon\to 0$
, if the
$(1,1)$
‐coefficient
$m_{11}^\varepsilon$
of
$(A^\varepsilon)^{-1}$
is bounded in
$L_\mathrm{loc}^1(\mathbb{R})$
and if none of its weak* cluster measures has atoms in common with
$b_{ii}$
,
$i\geq 2$
, then the family
$J^\varepsilon=\{J_\varOmega^\varepsilon \}_{\varOmega\in \mathcal{G}}$
$\varGamma$
‐converges in a local sense towards a naturally defined limit family
$J=\{J_\varOmega \}_{\varOmega\in \mathcal{G}}$
as
$\varepsilon\to 0$
. An alternative way of formulating the conclusion is to say that the energy densities
$(A^\varepsilon\nabla u,\nabla u)$
$\varGamma$
‐converge in a distributional sense towards the corresponding limit density.
Writing
$J_\varOmega^\varepsilon$
in terms of
$B^\varepsilon=(b_{ij}^\varepsilon)$
it becomes
$ J_\varOmega^\varepsilon(u) = \int_\varOmega \biggl(\frac{\curpartial u}{\curpartial x_1} + \sum^N_{j=2} b^\varepsilon_{1j} \,\frac{\curpartial u}{\curpartial x_j}\biggr)^2 \frac{1}{b^\varepsilon_{11}}\,\mathrm{d}x+ \sum^N_{i,j=2} \,\int_\varOmega \, \frac{\curpartial u}{\curpartial x_i} \, \frac{\curpartial u}{\curpartial x_j}\, b^\varepsilon_{ij} \,\mathrm{d}x, $
and the definition of
$J_\varOmega$
and the limit density
$(A\nabla u,\nabla u)$
is obtained by properly replacing the
$b^\varepsilon_{ij}\in L_\mathrm{loc}^\infty (\mathbb{R})$
by the limit measures
$b_{ij}$
and making sense to everything for
$u$
in a certain linear subspace of
$L_\mathrm{loc}^2(\mathbb{R}^N)$
.
