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

TL;DR

This paper studies nonlinear periodic homogenization with localized defects in semilinear elliptic equations, proving existence, uniqueness, and convergence of solutions as the scale parameter tends to zero, using an implicit function theorem approach.

Contribution

It introduces a unified method for nonlinear singular perturbation and homogenization, establishing solution existence, local uniqueness, and convergence rates for equations with localized defects.

Findings

Existence of weak solutions for small ε

Convergence of solutions to homogenized limit

Rate estimates for solution convergence

Abstract

We consider periodic homogenization with localized defects for semilinear elliptic equations and systems of the type $$ \nabla\cdot\Big(\Big(A(x/\varepsilon)+B(x/\varepsilon)\Big)\nabla u(x)+c(x,u(x)\Big)=d(x,u(x)) \mbox{ in } \Omega $$ with Dirichlet boundary conditions. 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 weak solution to the homogenized problem. Moreover, we prove that $\|u_\varepsilon-u_0\|_\infty\to 0$ for $\varepsilon \to 0$, and we estimate the corresponding rate of convergence. Our assumptions are, roughly speaking, as follows: $\Omega$ 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.