construction $(a)$ holds true. Moreover, up to a small displacement of $p$ we may assume that $[p,x_i]\cap[p,x_j]=\{p\}$ for $i\neq j$ so that $(b)$ holds.\
Next, let $D_j$ be the straight line supporting $[p,x_j]$. We define the sets $({\Omega}_i)$ as the connected components of ${\Omega}\setminus \left(D_0 \cup\cdots\cup D_N\right)$. We see that $(c,d,e,f)$ hold true.\
For the last statement, we observe that by the coarea formula, we have $\H^1(M\cap \bigcup_i \de{\Omega}_i)=0$ for a.e. choice of $p$.

![Example of the construction of the $\H^1$-rectifiable measure $\g$ (red) and of the partition $\{{\Omega}_i\}$ (gray) in the case $\s$ (green) is being an $\H^1$-rectifiable vector measure.[]{data-label="fig:partizione"}](partizione){width=".4\textwidth"}

Local Result {#sec:LocRes}
============

In this section we introduce a localization of the family of functionals $({\mathcal{F}}_{\varepsilon})$ (see ). We establish a lower bound and a compactness property for these local energies.

\[procedure\] Let $O\in A_S({\Omega})$, for $u{\varepsilon}\in H^1(O)$ and $\phi_{\varepsilon}\in H^1(O)$, we define $$\LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon};O)\ :=\ {\mathcal{F}}_{\varepsilon}({\nabla}u_{\varepsilon},\phi_{\varepsilon};O).$$ Notice that for ${\varepsilon}<d(O,S)$, we have ${\nabla}\cdot \s_{\varepsilon}\equiv 0$ in $O$ for any $\s_{\varepsilon}\in V_{\varepsilon}({\Omega})$. By the Stokes theorem we have $Du_{\varepsilon}=\s_{\varepsilon}^\perp$ for some $u\in H^1(O)$ and we have $${\mathcal{F}}_{\varepsilon}(\s_{\varepsilon},\phi_{\varepsilon};O)=\LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon};O).$$

The remaining of the section is devoted to the proof of

\[teo:localResult\] Let $(u_{\varepsilon})_{{\varepsilon}\in(0,1]}\subset H^1(O)$ be a family of functions with zero 