A common approach to singular perturbation and homogenization II: Semilinear elliptic systems

Abstract

We consider periodic homogenization of boundary value problems for second-order semilinear elliptic systems in 2D of the type∂xi(aijαβ(x/ε)∂xjuβ(x)+biα(x,u(x)))=bα(x,u(x)) for x∈Ω. For small ε>0 we prove existence of weak solutions u=uε as well as their local uniqueness for ‖u−u0‖∞≈0, where u0 is a given non-degenerate weak solution to the homogenized boundary value problem, and we estimate the rate of convergence to zero of ‖uε−u0‖∞ for ε→0. Our assumptions are, roughly speaking, as follows: The functions aijαβ are bounded, measurable and Z2-periodic, the functions biα(⋅,u) and bα(⋅,u) are bounded and measurable, the functions biα(x,⋅) and bα(x,⋅) are C1-smooth, and Ω is a bounded Lipschitz domain in R2. Neither global solution uniqueness is supposed nor growth restrictions of biα(x,⋅) or bα(x,⋅) nor higher regularity of u0, and cross-diffusion is allowed. The main tool of the proofs is an abstract result of implicit function theorem type which in the past has been applied to singularly perturbed nonlinear ODEs and elliptic and parabolic PDEs and, hence, which permits a common approach to existence, local uniqueness and error estimates for singularly perturbed problems and for homogenization problems.