Recently, I found an exercise in Hunter's Applied Analysis(last page in the link), which may be closely related to the question I raised two months ago.
Consider heat flow in a rod with rapidly varying thermal conductivity $a_n(x)=a(nx)$, where $n\in{\mathbb N}$ and $a(y)$ is a strictly positive periodic function with period one assumed continuous for simplicity. If the ends of the rod are held at an equal fixed temperature, and there is a given heat source $f(x)$ per unit length, the temperature $u_n(x)$ satisfies the boundary value problem
$$-\frac{d}{dx}\bigg(a_n(x)\frac{d}{dx}u_n\bigg)=f(x),\quad0 Here are my thoughts:
The first problem can be solve with straightforward calculation. For the second problem, one needs to estimate the absolute value of the integral
$$\bigg|\int_0^1(u_n'-u')v'dx\bigg|.$$
With Cauchy-Schwarz inequality, one may want to estimate
$$\int_0^1(u_n'-u')^2dx$$
A natural idea is that using the original equation,
$$u_n'(x)=\frac{1}{a_n(x)}g(x)$$
and $$u'(x)=\frac{1}{a^h}g(x)$$
It seems that things end up with show the convergence
$$\frac{1}{a_n(x)}\to\frac{1}{a^h}\quad \text{in}~L^2([0,1])$$
which is the part that may relate to the question whose link is given at the beginning. I get stuck here. Any idea for how to go on?