Quantitative Homogenization and Convergence of Moving Averages

We study homogenization it its most basic form $$-{(a\left(\frac{x}{\epsilon }\right)u_{\epsilon }^{\mathrm{\prime }}(x))}^{\mathrm{\prime }}=f(x)forx\in (0,1),$$ where $a(\cdot )$ is a positive $1-$periodic continuous function, $f$ is smooth and $u_{\epsilon }$ is subjected to Dirichlet boundary conditions. Classically, there is a homogenized equation with $a(\cdot )$ replaced by a constant coefficient $\bar{a}>0$ whose solution $u$ satisfies $|u-u_{\varepsilon}|_{L^{\infty}} \lesssim \varepsilon$. We show that local averages can result in faster convergence: for example, if $a(x)=a(1-x)$, then for $x\in (\epsilon ,1-\epsilon )$ $$\mid \frac{1}{\epsilon }{\int }_{x-\epsilon \mathrm{/}2}^{x+\epsilon \mathrm{/}2}{u}_{\epsilon }(y)dy-u(x)\mid {\lesssim }_{a,f}{\epsilon }^2\mathrm{.}$$ If the condition on $a(\cdot )$ is not satisfied, then subtracting an explicitly given linear function (depending on $a(\cdot ),f$) results in the same bound. We also describe another approach to quantitative homogenization problems and illustrate it on the same example.