Nonlinear non-periodic homogenization: Existence, local uniqueness and estimates

We consider periodic homogenization with localized defects of boundary value problems for semilinear ODE systems of the type $$ \Big((A(x/\varepsilon)+B(x/\varepsilon))u'(x)+c(x,u(x))\Big)'= d(x,u(x)) \mbox{ for } x \in (0,1),\; u(0)=u(1)=0. $$ Our assumptions are, roughly speaking, as follows: $A \in L^\infty(\mathbb{R};\mathbb{M}_n)$ is 1-periodic, $B \in L^\infty(\mathbb{R};\mathbb{M}_n))\cap L^1(\mathbb{R};\mathbb{M}_n))$, $A(y)$ and $A(y)+B(y)$ are positive definite uniformly with respect to $y$, $c(x,\cdot),d(x,\cdot)\in C^1(\mathbb{R}^n;\mathbb{R}^n))$, $c(\cdot,u) \in C([0,1];\mathbb{R}^n)$ and $d(\cdot,u) \in L^\infty((0,1);\mathbb{R}^n)$. For small $\varepsilon>0$ we show 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 solution to the homogenized problem, and we prove that $\|u_\varepsilon-u_0\|_\infty\to 0$ and, if $c(\cdot,u)$ is $C^1$-smooth, that $\|u_\varepsilon-u_0\|_\infty=O(\varepsilon)$ for $\varepsilon \to 0$. The main tool of the proofs is an abstract result of implicit function theorem type which in the past has been applied to singular perturbation as well as to periodic homogenization of nonlinear ODEs and PDEs and, hence, which permits a common approach to existence and local uniqueness results for singularly perturbed problems and for homogenization problems.