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.

本文旨在研究形式为\（-\ mathrm {{div}} \;（）的拟线性问题解的渐近行为（当\（\ varepsilon \; \ rightarrow \; 0 \）时）甲^ {\ varepsilon}（\ CDOT中，u ^ {\ varepsilon}）\ nablaÚ^ {\ varepsilon}）= F \）在穿孔域给出\（\欧米茄\反斜杠Ť_ {\ varepsilon} \）与孔\（T _ {\ varepsilon} \）的Neumann边界条件和\（\ partial \ Omega \）的Dirichlet边界条件。我们证明，如果孔在某种意义上是允许的（没有任何周期性条件），并且矩阵族\（（x，d）\ mapsto A ^ {\ varepsilon}（x，d）\）在实变量d中是一致矫顽的，一致有界的和一致等连续的，所考虑问题的均化可以分两步完成。首先，我们固定变量d，并在穿孔域中均化与\（A ^ {\ varepsilon}（\ cdot，d）\）相关的线性问题。一旦对\（（A ^ {\ varepsilon}，T ^ {\ varepsilon}）\）对中的\（H ^ {0} \）- limit \（A ^ {0}（\ cdot，d ）\）被确定，在第二步骤中，我们推断出溶液\（U ^ {\ varepsilon} \）在某种意义上，以独特的解收敛\（U ^ {0} \）在\（H ^ {1} _ { 0}（\ Omega）\）的拟线性方程\（-\ mathrm {{div}} \;（A ^ {0}（\ cdot，u ^ {0}）\ nabla u）= \ chi ^ {0} f \）（其中\（\ chi ^ { 0} \）是穿孔区域特征函数的\（L ^ {\ infty} \）弱\（^ {\ star} \）极限）。我们通过给出两种应用来完成我们的研究，一种适用于经典周期性情况，另一种适用于非周期性情况。