H-convergence of a class of quasilinear equations in perforated domains beyond periodic setting

Abstract

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.