Relaxation of convex variational problems with linear growth defined on classes of vector-valued functions

For a bounded Lipschitz domain $\Omega\subset\mathbb R^n$ and a function $u_0\in W{}_1^1(\Omega;\mathbb R^N)$, the following minimization problem is considered:
$$ (\mathcal P)\colon\int_\Omega f(\nabla u)\,dx\to\min\quad\text{in}\quad u_0+\overset\circ W{}_1^1(\Omega;\mathbb R^N), $$
where $f\colon\mathbb R^{nN}\to[0,\infty)$ is a strictly convex integrand. Let $\mathcal M$ denote the set of all $L^1$-cluster points of minimizing sequences of problem $(\mathcal P)$. It is shown that the geometric relaxation of problem $(\mathcal P)$ coincides with the relaxation based on the notion of the extended Lagrangian; moreover, it is proved that the elements $u$ of $\mathcal M$ are in one-to-one correspondence with the solutions of the relaxed problems.