A common approach to singular perturbation and homogenization I: Quasilinear ODE systems

Abstract

We consider periodic homogenization of boundary value problems for quasilinear second-order ODE systems in divergence form of the type $a(x,x/\varepsilon,u(x),u'(x))'= f(x,x/\varepsilon,u(x),u'(x))$ for $x \in [0,1]$. 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 solution to the homogenized boundary value problem, and we describe the rate of convergence to zero for $\varepsilon \to 0$ of the homogenization error $\|u_\varepsilon-u_0\|_\infty$. In particular, we show that this rate depends on the smoothness of the maps $a(\cdot,y,u,u')$ and $f(\cdot,y,u,u')$. Our assumptions are, roughly speaking, as follows: The maps $a,f:[0,1]\times\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}^n$ are continuous, the maps $a(x,y,\cdot,\cdot)$ and $f(x,y,\cdot,\cdot)$ are $C^1$-smooth, the maps $a(x,\cdot,u,u')$ and $f(x,\cdot,u,u')$ are 1-periodic, and the maps $a(x,y,u,\cdot)$ are strongly monotone and Lipschitz continuous uniformly with respect to $x$, $y$ and bounded $u$. No global solution uniqueness is supposed. Because $x$ is one-dimensional, no correctors and no cell problems are needed. But, because the problem is nonlinear, we have to care about commutability of homogenization and linearization. The main tool of the proofs is an abstract result of implicit function theorem type which in the past has been applied to singularly perturbed nonlinear ODEs and elliptic and parabolic PDEs and, hence, which permits a common approach to existence and local uniqueness results for singularly perturbed problems and and for homogenization problems.