\gamma_j v) \geq {\left\langle \tilde{f},v \right\rangle}
\end{aligned}
\right. .$$

Again, [@Han2017] tells us $u_0$ exists and is unique. To deduce $u_\epsilon \rightharpoonup u_0$, we first claim that $\{ u_\epsilon\}$ is uniformly bounded, then a subsequence of $\{ u_\epsilon\}$ will weakly converge in $V$. Finally, we prove the convergence point can only be $u_0$.

There is a constant $C$ independent with $\epsilon$, such that ${\left\lVert u_\epsilon \right\rVert}_V \leq C$, and $$C = 1/\Delta{\left( c_0c_j+{\left\lVert \tilde{f} \right\rVert}_{V^*} \right)} \leq 1/\Delta {\left( c_0c_j+{\left\lVert f \right\rVert}_{V^*}+c_k{\left\lVert g \right\rVert}_{0,\Gamma_N} \right)}.$$

Let $v=-u_\epsilon$ in \[eq:contact problem hemiform\], we have $$\begin{aligned}
    \kappa_1 {\left\lVert u_\epsilon \right\rVert}_V^2 &\leq \int_{\Omega} A^\epsilon \nabla u_\epsilon\cdot \nabla u_\epsilon \leq {\left\langle \tilde{f},u_\epsilon \right\rangle}+j^0(\gamma_ju_\epsilon; -\gamma_j u_\epsilon) \\
    &\leq {\left\lVert \tilde{f} \right\rVert}_{V^*} {\left\lVert u_\epsilon \right\rVert}_V+j^0(\gamma_ju_\epsilon; -\gamma_j u_\epsilon)
    \end{aligned}.$$ Recall \[ass:B2\], then $$\begin{aligned}
    j^0(\gamma_ju_\epsilon; -\gamma_j u_\epsilon) &\leq j^0(\gamma_ju_\epsilon; -\gamma_ju_\epsilon)+j^0(0;\gamma_ju_\epsilon)-j^0(0;\gamma_ju_\epsilon) \\ &\leq \alpha_j {\left\lVert \gamma_ju_\epsilon \right\rVert}_{V_j}^2-j^0(0;\gamma_ju_\epsilon) \\
    & \leq \alpha_j c_j^2 {\left\lVert u_\epsilon \right\rVert}_V^2-j^0(0;\gamma_ju_\epsilon)
    \end{aligned}.$$ By the definition of $j^0(x;h)$, we choose arbitrarily a $\xi \in \partial j(0)$ $$\begin{aligned}
    -j^0(0;\gamma_ju_\epsilon) &\leq -{\left\langle \xi,\gamma_ju_\epsilon \right\rangle} \leq {\left\lVert \xi \right\rVert}_{X_j^*} {\left\lVert \gamma_ju_\epsilon \right\rVert}_{X_j}\\
    & \leq c_0 c_j {\left\lVert u_\epsilon \right\rVert}_V
    \end{aligned}.$$ Then we establish the desired inequality.

div-curl lemma states as follows:

Let $\bm{p}_\epsilon, \bm{q}_\epsilon \in L^2(\Omega)^d$, such that: $$\bm{p}_\epsilon 