such that $\s\e\rwstar\s$, $\phi\e{\rightarrow}\phi$ in $L^1({\Omega})$ and $$\label{eq:limsup}
\limsup_{{\varepsilon}{\downarrow}0}{\mathcal{F}}\e(\s\e,\phi\e)\leq{\mathcal{E}}_\a(\s,\phi).$$ When ${\mathcal{E}}_\a(\s,\phi)=+\infty$ the inequality is valid for any sequence therefore by definition  we can assume $\s=U(M,\t,\xi)$ and $\phi=1$. Furthermore by density Lemma \[lem:density\] is sufficient to consider measures of the form $$\label{eq:strutturadense}
\s=\sum_{i=1}^n U(M_i,\t_i,\xi_i),$$ where $M_i$ is a segment, $\t_i\in{\mathbf{R}}_+$ is $\H^1$-a.e. constant and $\xi_i$ is an orientation of $M_i$ for each $i$. Without loss of generality we can suppose that for each couple of segments $M_i$, $M_j$, for $i \neq j$, the intersection $M_i\cap M_j$ is at most a point (called branching point) not belonging to the relative interior of $M_i$ and $M_j$. We firstly produce the estimate  for $\s$ composed by a single segment thus let us assume $\s=\t e_1\cdot \H^1{ \!\!\mbox{{ \Large$\llcorner$}} }(0,l)\times\{0\}$.

*Notation:* Let us fix the values $$\label{eq:infinitesimi}
a_{\varepsilon}:=\begin{dcases}
         \frac{\t\a\,{\varepsilon}}{2} &\mbox{ if } \a>0\\
         \quad{\varepsilon}& \mbox{ if } \a=0
        \end{dcases},
\qquad\qquad
b_{\varepsilon}:={\varepsilon}\ln{\left}(\frac{1-\eta}{{\varepsilon}}{\right})
\qquad\mbox{ and }\qquad
r\e=\max\{{\varepsilon},a\e\}.$$ Let $d_\infty(x,S)$ be the distance function from $x$ to the set $S\subset {\Omega}$ relative to the infinity norm on ${\mathbf{R}}^2$ and $Q_r(P)=\{x\in{\mathbf{R}}^2: d_\infty(x,P)\leq r\}$ the square centered in $P$ of size $2r$ and sides parallel to the axes. Introduce the sets $$\begin{aligned}
\label{def:rectangular}
I_{a\e}&:=\{x\in{\mathbf{R}}^2: d_\infty(x,[0,l]\times\{0\})\leq a\e\}\cup Q_{r\e}(0,0)\cup Q_{r\e}(l,0),\\
I_{b\e}&:=\{x\in{\mathbf{R}}^2: d_\infty(x,I_{a\e})\leq 