Consider the functional $\begin{align*} \mathcal{F} (u) = \int_0^1 \frac{1}{2} |u(x)'|^2 - f(x)u(x)dx \end{align*}$ where $f \in L^2(0,1)$ is fixed. I want to show that there is a unique minimizer in the class $\mathcal{A}= \left \{ u \in W^{1,2}_0(0,1) : |u'| \leq 1 \text{ a.e. on } (0,1) \right \}$.
Consider a minimizing sequence $(a_k)_k \subset \mathcal{A}$. Can I conclude that there is a subsequence $(u_{n_k})$ and a function $u$ s.th. $u_{n_k}$ convergence strongly to $u$ (in $L^2$) ? If so, why is $u \in \mathcal{A}$? Any help is appreciated.