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

TL;DR

This paper establishes the existence, uniqueness, and convergence of solutions for nonlinear homogenization problems with localized defects in boundary value problems for semilinear ODE systems, using an implicit function theorem approach.

Contribution

It provides new results on the existence, local uniqueness, and asymptotic behavior of solutions in nonlinear homogenization with localized defects, extending classical linear theory.

Findings

Existence of weak solutions for small ε

Convergence of solutions to homogenized problem as ε→0

Solution difference is O(ε) under certain regularity conditions

Abstract

We consider periodic homogenization with localized defects of boundary value problems for semilinear ODE systems of the type $$ \Big((A(x/\varepsilon)+B(x/\varepsilon))u'(x)+c(x,u(x))\Big)'= d(x,u(x)) \mbox{ for } x \in (0,1),\; u(0)=u(1)=0. $$ 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 $u=u_0$ is a given solution to the homogenized problem $$ \Big(A_0u'+c(x,u(x))\Big)'= d(x,u(x)) \mbox{ for } x \in (0,1),\; u(0)=u(1)=0,\; A_0:=\left(\int_0^1A(y)^{-1}dy\right)^{-1} $$ such that the linearized problem $$ \Big(A_0u'+\partial_uc(x,u_0(x))u(x)\Big)'= \partial_ud(x,u_0(x))u(x) \mbox{ for } x \in (0,1),\; u(0)=u(1)=0 $$ does not have weak solutions $u\not=0$. Further, we prove that $\|u_\varepsilon-u_0\|_\infty\to 0$ and, if $c(\cdot,u)\in W^{1,\infty}((0,1);\mathbb{R}^n)$, that $\|u_\varepsilon-u_0\|_\infty=O(\varepsilon)$ for $\varepsilon \to 0$. Moreover, all these statements are true, roughly speaking, uniformly with respect to the localized defects $B$. We assume that $A \in L^\infty(\mathbb{R};\mathbb{M}_n)$ 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 $y$, $c(x,\cdot),d(x,\cdot)\in C^1(\mathbb{R}^n;\mathbb{R}^n)$ and $c(\cdot,u),d(\cdot,u) \in L^\infty((0,1);\mathbb{R}^n)$. The main tool of the proofs is an abstract result of implicit function theorem type which has been tailored for applications to nonlinear singular perturbation and homogenization problems.