Consider the domain
$\varOmega_\varepsilon =\varOmega -T_\varepsilon$
obtained by removing a closed set
$T_\varepsilon$
of
$\varepsilon$
‐periodic holes of size
$\varepsilon$
from a bounded open set
$\varOmega$
. We study the homogenization of the nonlinear problem
$\cases{-{\rm div}(A({x/\varepsilon})Du_\varepsilon)+ \gamma u_\varepsilon = H( {x/\varepsilon}, u_\varepsilon , Du_\varepsilon )& $\hbox{in }\varOmega_\varepsilon ,$\cr (A({x/\varepsilon})Du_\varepsilon)\cdot\underline\nu =0 &$\hbox{on } \Ncurpartial T_\varepsilon ,$\cr u_\varepsilon =0&$\hbox{on } \Ncurpartial \varOmega,$\cr u_\varepsilon\in H^1(\varOmega_\varepsilon )\cap L^\infty(\varOmega_\varepsilon ),&\cr} $
where
$H( y,s,\xi)$
is
$]0,1[^n$
‐periodic in
$y$
and has quadratic growth with respect to
$\xi$
.
We prove that the linear part of the limit problem is the homogenized matrix of the linear problem and the nonlinear part is given by
$H^0( u, Du)$
, where
$H^0$
is defined by
$ H^0( s,\xi) =\int_{]0,1[^n-\overline T} H ( y,s,C(y)\xi)\, {\rm d}y \quad \forall( s,\xi)\in\NBbbR\times\NBbbR^n ,$
$C( {\cdot/\varepsilon})$
being the corrector matrices of the linear problem and
$T$
the reference hole.
