An H-convergence-based implicit function theorem for homogenization of nonlinear non-smooth elliptic systems

Abstract

Abstract We consider homogenization of semilinear elliptic PDE systems of the type $$ \partial _{x_i}\Big (a^{\alpha \beta }_{ij}(\varepsilon ,x) \partial _{x_j}u^\beta (x) +f_i^\alpha (x,u(x))\Big )=0 \text{ in } \Omega , \; \alpha =1,\ldots ,n, $$ ∂ x i ( a ij α β ( ε , x ) ∂ x j u β ( x ) + f i α ( x , u ( x ) ) ) = 0 in Ω , α = 1 , … , n , with homogeneous Dirichlet boundary conditions. Here $$\varepsilon >0$$ ε > 0 is the small homogenization parameter, $$\Omega \subset \mathbb {R}^N$$ Ω ⊂ R N is a bounded Lipschitz domain, $$a^{\alpha \beta }_{ij}(\varepsilon ,\cdot )\in L^\infty (\Omega )$$ a ij α β ( ε , · ) ∈ L ∞ ( Ω ) satisfy the Legendre ellipticity condition, and the maps $$u \in C(\overline{\Omega };\mathbb {R}^n) \mapsto f_i^\alpha (\cdot ,u(\cdot )) \in L^{p_0}(\Omega )$$ u ∈ C ( Ω ¯ ; R n ) ↦ f i α ( · , u ( · ) ) ∈ L p