\int_{J_u\cap O}|\nu_u\cdot \xi|{\left}[1+\a|[u]|{\right}]\dH^{d-1}.$$ In order to conclude, we use the following localization method stated by Braides in [@Bra1 Prop. 1.16].

\[lem:braides\] Let $\mu:\A(X){\rightarrow}[0,+{\infty})$ be a superadditive set function and let $\lambda$ be a positive measure on $X$. For any $i\in{\mathbf{N}}$ let $\psi_i$ be a Borel function on $X$ such that $\mu(A)\geq\int_A \psi_i\dif \lambda$ for all $A\in\A(X)$. Then $$\mu(A)\geq\int_A\psi\dif\lambda$$ where $\psi:=\sup_i\psi_i$.

We introduce the superadditive increasing set function $\mu$ defined on $\A(O)$ by $$\mu(A):=\Gamma-\liminf_{{\varepsilon}{\rightarrow}0} \LF_{\varepsilon}(u;)),\quad\text{for any }A\in\A(O)$$ and we let $\lambda$ be a Radon measure defined as $$\lambda:=[1+\a|u(x^+)-u(x^-)|]\H^{d-1}{ \!\!\mbox{{ \Large$\llcorner$}} }J_u.$$ Fix a sequence $(\xi_i)_{i\in{\mathbf{N}}}$ dense in $\S^{d-1}$. By  we have $$\mu(O)\geq \int_O \psi_i\dif \lambda, \quad i\in {\mathbf{N}},$$ where $$\psi_i(x):=
\begin{dcases}
   |\langle\nu_u(x),\xi_i\rangle|&\text{if } x\in J_u,\\
   0&\text{if } x\in O\setminus J_u.
  \end{dcases}$$ Hence by Lemma \[lem:braides\] we finally obtain $$\liminf_{{\varepsilon}{\rightarrow}0}\LF_{\varepsilon}(u_{\varepsilon},\phi_{\varepsilon};O)\geq \int_O \sup_i\psi_i(x)\dif \mu =\int_{J_u\cap O}{\left}[1+\a|[u]|{\right}]\dH^{d-1}.$$

Equicoercivity and $\Gamma$-liminf {#sec:Comp}
==================================

We first prove the compactness property stated in the introduction. Let us consider a sequence $(\s_{\varepsilon},\phi_{\varepsilon})\in\M({\Omega},{\mathbf{R}}^2)$ uniformly bounded in energy by $c_0<+{\infty}$, $$\label{eq:unifbound}
0\leq{\mathcal{F}}_{\varepsilon}(\s_{\varepsilon},\phi_{\varepsilon})\leq c_0\qquad\mbox{for } {\varepsilon}\in(0,1].$$

First observe that by definition  and equation , we have $\s_{\varepsilon}\in V_{\varepsilon}({\Omega})$ and $\phi_{\varepsilon}\in W_{\varepsilon}({\Omega})$.

Next, substituting $|\s_{\varepsilon}|$ for $|{\nabla}u_{\varepsilon}|$ in the argument of *Step 1.* of the proof of Theorem \[teo:localResult\], inequality  reads $$ 