= Solution
The <De la Vallée-Poussin criterion> characterizes <uniform integrability> by existence of a nonnegative increasing <convex function> $\Psi$ with $\Psi(s)/s\to\infty$ and a uniform bound on $\int\Psi(|v_j|)$. On a finite interval this gives a <weakly convergent> <subsequence> in $L^1$ by the <Dunford-Pettis theorem>.
After discarding a finite prefix of the <minimizing sequence> if needed, put $H(t)=\Phi(t)-M\geq0$. Its derivative <sequence> has $\int H(u_j')\leq C$. The superlinear bound directly gives <uniform integrability>: for $H(t)\geq A|t|$ outside $[-R_A,R_A]$,
$$
\int_E|u_j'|\leq R_A|E|+C/A.
$$
If the convex version of the criterion is used, choose increasing $R_m\to\infty$ so $H(t)\geq m^2|t|$ for $|t|\geq R_m$, and set $\Psi(s)=\sum_{m\geq1}(s-R_m)_+$. This is locally a finite sum, convex and increasing. It is superlinear because for fixed $N$ and $s\geq2R_N$, $\Psi(s)\geq Ns/2$. If exactly $m$ terms are active, $\Psi(s)\leq ms\leq H(t)$ for $s=|t|$, so its integrals have the same bound. The criterion therefore supplies $u_{j_k}'\rightharpoonup v$ in $L^1$.
Define $u(x)=\alpha+\int_a^xv$. Interval-indicator tests give $u_{j_k}(x)\to u(x)$ for every $x$, including $b$, so its endpoint values are correct and $u\in W^{1,1}$. The common derivative $L^1$ bound makes these <functions> uniformly bounded. <Uniform integrability> also gives common equicontinuity, since an increment is bounded by the derivative integral on the intervening interval. The <Arzelà-Ascoli theorem> yields uniform convergence to the same identified limit, and in particular
$$
\boxed{u\in\mathcal A,\quad u_{j_k}'\rightharpoonup u'\text{ in }L^1,\quad u_{j_k}\to u\text{ in }L^1.}
$$
Here the printed admissible space is $W^{1,1}$, so this <compactness> argument gives precisely the needed membership.
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