b\e\},\\
I_{c\e}&:=\{x\in{\mathbf{R}}^2: d_\infty(x,(I_{a\e}\cup I_{b\e}))\leq {\varepsilon}\},\\
I_{d\e}&:={\Omega}\setminus(I_{a\e}\cup I_{b\e}\cup I_{c\e}),\end{aligned}$$ and define $R\e= I_{a\e}\setminus(Q_{r\e}(0,0)\cup Q_{r\e}(l,0))$.

![Example of the neighborhoods of the segment $[0,l]\times\{0\}$. On the left the case $r\e={\varepsilon}$ on the right the case in which $r\e=a\e>{\varepsilon}$. The stripped region is $R\e$ and $I_{a\e}=R\e\cup(Q_{r\e}(0,0)\cup Q_{r\e}(l,0))$. Remark that $\operatorname{supp}(\rho\e)=B(0,{\varepsilon})$.[]{data-label="fig:intorninuovo"}](nuovointorni){width="\textwidth"}

*Costruction of $\s\e$:* We build $\s\e$ as a vector field supported on $I_{a\e}$. In particular we add together three different constructions performed respectively on $R\e$, $Q_{r\e}(0,0)$ and $ Q_{r\e}(l,0)$. Let $r=r_{\varepsilon}/{\varepsilon}$ and consider the problem

$$\begin{dcases}
\Delta u=\pm\t\delta_{x_0}*\rho &\mbox{ on }  Q_{r}(0,0),\\
\frac{\de u}{\de \nu}=\frac{\pm\t}{\H^1(\Sigma)}  &\mbox{ on }  \Sigma^\pm=\{x\in{\mathbf{R}}^2: x_1=\pm1,\;|x_2|\leq\frac{\t\a}{2}\}.
\end{dcases}$$

![image](problema00){width=".6\textwidth"}

Let $u^+$ be the solution relative to the problem in which every occurrence of $\pm$ is replaced by $+$ and let $u^-$ defined accordingly. Then set $$\label{def:se}
 \s\e=
 \begin{dcases}
       \frac{{\nabla}u^+(x/{\varepsilon})}{{\varepsilon}} &\mbox{ on } Q_{r\e}(0,0),\\
       \quad\frac{\t}{2a_{\varepsilon}}\cdot e_1 &\mbox{ on } R\e,\\
       \frac{{\nabla}u^-((x-(l,0))/{\varepsilon})}{{\varepsilon}} &\mbox{ on } Q_{r\e}(l,0).
  \end{dcases}$$ By construction we have that ${\nabla}\cdot\s\e=\t(\d_{(0,0)}-\d_{(l,0)})\ast\rho\e$ and $\s\e\rwstar\s$. Let us point out as well that there exists a constant $c(\a,\t)$ such that $$\label{eq:energiapunti}
c(\a,\t):=\int_{Q_{r\e}(l,0)}|\s\e|^2\dx=\int_{Q_{r\e}(0,0)}|\s\e|^2\dx=\int_{Q_{r}(0,0)}{\left}|{\nabla}u^+(x){\right}|^2\dx=\int_{Q_{r}(0,0)}{\left}|{\nabla}u^-(x){\right}|^2\dx.$$

*Costruction of $\phi\e$:* Most of the properties of $\phi\e$ are a consequence of the inequalities obtained in Theorem \[teo:localResult\] and the structure of $\s\e$. On one hand we need $\phi\e$ to attain 