A Common Approach to Singular Perturbation and Homogenization II: Semilinear Elliptic Systems

We consider periodic homogenization of boundary value problems for second-order semilinear elliptic systems in 2D of the type $$\partial_{x_i}\left(a_{ij}^{\alpha \beta}(x/\varepsilon)\partial_{x_j}u(x)+b_i^\alpha(x,u(x))\right)=b^\alpha(x,u(x)) \mbox{ for } x \in \Omega.$$ For small $\varepsilon>0$ we prove existence of weak solutions $u=u_\varepsilon$ as well as their local uniqueness for $\|u-u_0\|_\infty \approx 0$, where $u_0$ is a given non-degenerate weak solution to the homogenized boundary value problem, and we estimate the rate of convergence to zero of $\|u_\varepsilon-u_0\|_\infty$ for $\varepsilon \to 0$. Our assumptions are, roughly speaking, as follows: The functions $a_{ij}^{\alpha \beta}$ are bounded, measurable and $\mathbb{Z}^2$-periodic, the functions $b_i^\alpha(\cdot,u)$ and $b^\alpha(\cdot,u)$ are bounded and measurable, the functions $b_i^\alpha(x,\cdot)$ and $b^\alpha(x,\cdot)$ are $C^1$-smooth, and $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^2$. Neither global solution uniqueness is supposed nor growth restrictions of $b_i^\alpha(x,\cdot)$ or $b^\alpha(x,\cdot)$ nor higher regularity of $u_0$, 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 and for homogenization problems.