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

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 $\epsilon >0$ 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 $u=u_0$ 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 $u\ne 0$. Further, we prove that $\Vert u_{\epsilon }-u_0{\Vert }_{\infty }\to 0$ and, if $c(\cdot ,u)\in W^{1,\infty }((0,1);{\mathbb{R}}^n)$, that $\Vert u_{\epsilon }-u_0{\Vert }_{\infty }=O(\epsilon )$ for $\epsilon \to 0$. 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.

arXiv admin note: substantial text overlap with arXiv:2309.15611