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

Abstract

We consider periodic homogenization with localized defects for semilinear elliptic equations and systems of the type $$ \nabla\cdot\Big(\Big(A(x/\varepsilon)+B(x/\varepsilon)\Big)\nabla u(x)+c(x,u(x)\Big)=d(x,u(x)) \mbox{ in } Ω$$ with Dirichlet boundary conditions. For small $\epsilon >0$ we show existence of weak solutions $u=u_{\epsilon }$ as well as their local uniqueness for $\Vert u-u_0{\Vert }_{\infty }\approx 0$, where $u_0$ is a given non-degenerate weak solution to the homogenized problem. Moreover, we prove that $\Vert u_{\epsilon }-u_0{\Vert }_{\infty }\to 0$ for $\epsilon \to 0$, and we estimate the corresponding rate of convergence. Our assumptions are, roughly speaking, as follows: $\hat{I}\text{c◯}$ 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.

arXiv admin note: text overlap with arXiv:2309.14108