AbstractIn this paper, we aim to study the asymptotic behavior (when $$\varepsilon \;\rightarrow \; 0$$
ε
→
0
) of the solution of a quasilinear problem of the form $$-\mathrm{{div}}\;(A^{\varepsilon }(\cdot ,u^{\varepsilon }) \nabla u^{\varepsilon })=f$$
-
div
(
A
ε
(
·
,
u
ε
)
∇
u
ε
)
=
f
given in a perforated domain $$\Omega \backslash T_{\varepsilon }$$
Ω
\
T
ε
with a Neumann boundary condition on the holes $$T_{\varepsilon }$$
T
ε
and a Dirichlet boundary condition on $$\partial \Omega $$
∂
Ω
. We show that, if the holes are admissible in certain sense (without any periodicity condition) and if the family of matrices $$(x,d)\mapsto A^{\varepsilon }(x,d)$$
(
x
,
d
)
↦
A
ε
(
x
,
d
)
is uniformly coercive, uniformly bounded and uniformly equicontinuous in the real variable d, the homogenization of the problem considered can be done in two steps. First, we fix the variable d and we homogenize the linear problem associated to $$A^{\varepsilon }(\cdot ,d)$$
A
ε
(
·
,
d
)
in the perforated domain. Once the $$H^{0}$$
H
0
-limit $$A^{0}(\cdot ,d)$$
A
0
(
·
,
d
)
of the pair $$(A^{\varepsilon },T^{\varepsilon })$$
(
A
ε
,
T
ε
)
is determined, in the second step, we deduce that the solution $$u^{\varepsilon }$$
u
ε
converges in some sense to the unique solution $$u^{0}$$
u
0
in $$H^{1}_{0}(\Omega )$$
H
0
1
(
Ω
)
of the quasilinear equation $$-\mathrm{{div}}\;(A^{0}(\cdot ,u^{0})\nabla u )=\chi ^{0}f$$
-
div
(
A
0
(
·
,
u
0
)
∇
u
)
=
χ
0
f
(where $$ \chi ^{0}$$
χ
0
is $$L^{\infty }$$
L
∞
weak $$^{\star }$$
⋆
limit of the characteristic function of the perforated domain). We complete our study by giving two applications, one to the classical periodic case and the second one to a non-periodic one.
Existence results for a nonlinear nonautonomous transmission problem via domain perturbation
