In $L^2([0;1])$: $u(x)$ is given smooth (piecewise linear) function, s.t.$u: [0;1]\to \mathbb{R}$. Function $\rho: [0;1] \to \mathbb{R}$, s.t.
$ \rho(x) = \begin{cases} 1, & \text{if $x\in[0;{1\over2})$;} \\ 2, & \text{if $x\in[{1\over2};1]$.} \end{cases} $.
Then $$c = {1\over{\int_{0}^1{1\over{\rho(x)}}dx}} = {4\over3}.$$
We should find a sequence of functions $u_n$, $u_n$ $\Gamma$-converges to $u$, s.t. $$\lim_{n\to\inf}\int_0^1\rho(nx)(u_n^{'}(x))^2dx \to \int_0^1{4\over3}(u^{'}(x))^2dx.$$
Moreover, I know that if $u = {{b-x}\over{b-a}}$, then $u_n$ is a solution of $$\left\{ \begin{aligned} {d\over{dx}}\Big{[}\rho(nx){d\over{dx}}u_n(x)\Big{]} &= 0 \\ u(a) &= 1 \\ u(b) &= 0 \end{aligned} \right.$$ I don't know how to find any solution even a partial one for $u = {{b-x}\over{b-a}}$.
Can someone help me, please?