=\s^{\perp}+\g^{\perp}$. From , we deduce $|Du|(S)=0$ which implies $|\mu|(S)=\sum|c_j|=0$. Hence $c_j=0$ for $j=0,\dots,N$ and $\s$ writes in the form $U(M_\s,\t_\s,\xi_\s)$.

Let us now use the local results of Section \[sec:LocRes\] to prove the lower bound.

Let $(\s_{{\varepsilon}},\phi_{{\varepsilon}})$ as in the statement of the theorem. Without loss of generality, we can suppose that ${\mathcal{F}}_{{\varepsilon}}(\s_{{\varepsilon}},\phi_{{\varepsilon}})<+{\infty}$. Theorem \[teo:sigma\_equicoercive\] then ensures the existence of a rectifiable measure $\s=U(M_\s,\t_\s,\xi_\s)$ with $\operatorname{supp}(\s)\subset\overline{\Omega}$ such that $\s_{{\varepsilon}}\rwstar\s$.\
Let $\hat{\Omega}$ be as in the previous proof and let us define $\mu=\Gamma-\liminf_{{\varepsilon}}{\mathcal{F}}_{{\varepsilon}}(\s_{{\varepsilon}},\phi_{{\varepsilon}})$ and $\lambda= \a|\s|+\H^{1}{ \!\!\mbox{{ \Large$\llcorner$}} }M_\s$. Consider the countable family of sets $\{O_{i}\}\subset\A_{S}(\hat{\Omega})$ made of the open rectangles $O_i\subset \hat{\Omega}\setminus S$ with vertices in ${\mathbf{Q}}^{2}$ and let $\psi_{i}:=1_{O_{i}}$. The local result stated in Theorem \[teo:localResult\] gives for any $i\in{\mathbf{N}}$ $$\mu(A)\geq\mu(O_{i}\cap A)\geq\lambda(O_{i}\cap A)=\int_{A}\psi_{i}\dif \lambda.$$ Therefore Lemma \[lem:braides\] gives $$\begin{aligned}
\Gamma-\liminf_{{\varepsilon}{\downarrow}0}{\mathcal{F}}_{{\varepsilon}}(\s_{{\varepsilon}},\phi_{{\varepsilon}})&=\mu(\hat{\Omega})\geq\lambda(\hat{\Omega})=\a|\s|(\overline{\Omega})+\H^{1}(M_\s) \end{aligned}$$ since $\sup_{i}\psi_{i}$ is the constant function $1$.

Upper bound {#sec:UppBound}
===========

A density result
----------------

In order to obtain the upper bound we first provide a density lemma. We show that measures which have support contained in a finite union of segments, are dense in energy.\
Without loss of generality let us assume that $\s\in \M_S({\overline}{\Omega})$ is such that ${\mathcal{E}}_\alpha(\s)<\infty$. In particular $\s=U(M_\s,\t_\s,\xi_\s)$ is a $\H^1$-rectifiable measure. Applying 