Lemma \[lemma:completediv\] we obtain a $\H^1$-rectifiable measure $\g=U(M_\g,\t_\g,\xi_\g)$ and a partition of ${\Omega}$ made of polyhedrons $\{{\Omega}_i\}$ such that $M_\g\subset \cup_i\de{\Omega}_i$, $\H^1(M_\s \cap\cup_i\de{\Omega}_i)=0$ and $\s+\g$ is divergence free.\
From the above properties, we can write $$\s^\perp+\g^\perp=Du$$ for some $u\in\P({\Omega})$. Our strategy is the following, using existing results [@Bel_Cham_Gold], we build an approximating sequence for $u$ on each ${\Omega}_j$ whose gradient is supported on a finite union of segments. We then glue these approximations together to obtain a sequence $(w_j)$ approximating $u$ in $\hat{\Omega}$. The main difficulty is to establish that $Dw_j { \!\!\mbox{{ \Large$\llcorner$}} }[\cup_i\de{\Omega}_i]$ is close to $Du{ \!\!\mbox{{ \Large$\llcorner$}} }[\cup_i\de{\Omega}_i]=\g^\perp$.

\[lem:uapproximation\] There exists a sequence $(w_j)\subset\P(\hat{\Omega})$ with the following properties:

1.  $w_j\to u$ weakly in $BV(\hat{\Omega})$,

2.  $\operatorname{supp}w_j\subset {\overline}{\Omega}$,

3.  $\limsup_{j{\rightarrow}{\infty}}{\mathcal{E}}_\a(w_j,1)\leq{\mathcal{E}}_\a(u,1)$,

4.  $J_{w_j}$ is contained in a finite union of segments for any $j\in {\mathbf{N}}$,

5.  $|Dw_j-Du|(\cup \de{\Omega}_i)\to0$.

*Step 1.* In order to apply the results of [@Bel_Cham_Gold], we first need to modify $u$ and the energy. Let us note the energy density function $f(t)=1+\a t$ and for $k\geq0$ and $t\geq 0$ let us introduce the approximation $$\label{eq:definizionefk}
 f_k(t):=\min\{(2^{k/2}+\a 2^{-k/2})\sqrt{t},f(t)\}.$$

![Graph of f and two of its approximations $f_{k_1} $ and $f_{k_2}$ with $k_1<k_2$.[]{data-label="fig:approssimazionef"}](approssimazionef){width=".3\textwidth"}

We have $0\leq f_k\leq 