Abstract
In this paper, we aim to study the asymptotic behavior (when varepsilon ;rightarrow ; 0) of the solution of a quasilinear problem of the form -mathrm{{div}};(A^{varepsilon }(cdot ,u^{varepsilon }) nabla u^{varepsilon })=f given in a perforated domain Omega backslash T_{varepsilon } with a Neumann boundary condition on the holes T_{varepsilon } and a Dirichlet boundary condition on partial Omega . 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) 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) in the perforated domain. Once the H^{0}-limit A^{0}(cdot ,d) of the pair (A^{varepsilon },T^{varepsilon }) is determined, in the second step, we deduce that the solution u^{varepsilon } converges in some sense to the unique solution u^{0} in H^{1}_{0}(Omega ) of the quasilinear equation -mathrm{{div}};(A^{0}(cdot ,u^{0})nabla u )=chi ^{0}f (where chi ^{0} is L^{infty } 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.
Highlights
The main goal of this work is to give, in the framework of the H 0-convergence notion, a general homogenization result of a type of quasilinear equations with a mixed Neumann-Dirichlet boundary conditions, beyond the periodic setting
On ∂ Tε, where is a bounded open subset of Rn, {Tε} is sequence of compact subsets of, not necessarily periodically distributed, and where f ∈ L2( ), Aε : (x, d) ∈ (, R) −→ Aε(x, t) ∈ Rn×n is a sequence of Caratheodory functions uniformly coercive, uniformly bounded and uniformly equicontinuous matrix fields in the variable d
Under a suitable conditions on the equicontinuity modulus and L p-estimate assumption, there exists a subsequence of ε, a positive function χ 0 ∈ L∞( ) and a matrix field A0(·, ·) which satisfies the same properties as A (·, ·) such that χ ε χ 0 weakly in L∞( ), ( Aε(·, d), Tε) H0 A0(·, d) in, ∀d ∈ Rn, H
Summary
The main goal of this work is to give, in the framework of the H 0-convergence notion (the generalization of the H -convergence to perforated domains), a general homogenization result of a type of quasilinear equations with a mixed Neumann-Dirichlet boundary conditions, beyond the periodic setting. For periodically perforated domains, the same type of quasilinear equations was firstly studied in Bendib [2] and Bendib–Tcheugoué Teboué [3], with Lipschitz continuous coefficients and linear Robin conditions. After this Cabarrubias–Donato have studied in [7] this equation with a nonlinear Robin condition boundary of the holes and the module of equicontinuity satisfies a suitable assumption introduced by Chipot in [9], but not assumed to be Lipschitz continuous.
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