A common approach to singular perturbation and homogenization III: Nonlinear periodic homogenization with localized defects

We consider periodic homogenization with localized defects for semilinear elliptic equations and systems of the type $$\mathrm{\nabla }\cdot \left(\right(A(x\mathrm{/}\epsilon )+B(x\mathrm{/}\epsilon ))\mathrm{\nabla }u(x)+c(x,u(x))=d(x,u(x))in\mathrm{\Omega }$$ with Dirichlet boundary conditions. For small $\epsilon >0$ we show existence of weak solutions $u=u_{\epsilon }$ as well as their local uniqueness for $\mathrm{\parallel }u-u_0{\mathrm{\parallel }}_{\mathrm{\infty }}\approx 0$, where $u_0$ is a given non-degenerate weak solution to the homogenized problem. Moreover, we prove that $\mathrm{\parallel }{u}_{\epsilon }-u_0{\mathrm{\parallel }}_{\mathrm{\infty }}\to 0$ for $\epsilon \to 0$, and we estimate the corresponding rate of convergence. Our assumptions are, roughly speaking, as follows: $\mathrm{\Omega }$ is a bounded Lipschitz domain, $A$, $B$, $c(\cdot ,u)$ and $d(\cdot ,u)$ are bounded and measurable, $c(x,\cdot )$ and $d(x,\cdot )$ are $C^1$-smooth, $A$ is periodic, and $B$ is a localized defect. Neither global uniqueness is supposed nor growth restriction for $c(x,\cdot )$ or $d(x,\cdot )$. The main tool of the proofs is an abstract result of implicit function theorem type which permits a common approach to nonlinear singular perturbation and homogenization.