$C$ only depends on $\Omega, \kappa_1, \kappa_2$. By Green formula, we have $\forall v \in V$ $$\begin{aligned}
    \int_{\Omega} A^\epsilon \nabla u_\epsilon \cdot \nabla v - \int_{\Gamma_C} \bm{n}\cdot A^\epsilon\nabla u_\epsilon v &= {\left\langle f,v \right\rangle}+\int_{\Gamma_N} g v \\
    &= \int_{\Omega} \hat{A} \nabla u_0 \cdot \nabla v - \int_{\Gamma_C}  \bm{n} \cdot \hat{A} \nabla u_0 v 
    \end{aligned}.$$ The definition of $j^0$ gives: $$\begin{aligned}
    \int_{\Omega} A^\epsilon \nabla u_\epsilon \cdot \nabla v &= \int_{\Omega} \hat{A} \nabla u_0 \cdot \nabla v + \int_{\Gamma_C}  {\left( -\bm{n} \cdot \hat{A} \nabla u_0 \right)} v + \int_{\Gamma_C} {\left( -\bm{n}\cdot A^\epsilon\nabla u_\epsilon \right)}{\left( -v \right)} \\
    & \leq \int_{\Omega} \hat{A} \nabla u_0 \cdot \nabla v+j^0(\gamma_j u_0; \gamma_j v)+ j^0(\gamma_j u_\epsilon; -\gamma_j v)
    \end{aligned}.$$ Take a direct calculation, $$\begin{aligned}
    &\int_\Omega A^\epsilon \nabla w_\epsilon \cdot \nabla v \\
    =& \int_{\Omega} A^\epsilon \nabla u_\epsilon \cdot \nabla v-\int_{\Omega} A^\epsilon \nabla u_0 \cdot \nabla v - \int_{\Omega}A^\epsilon_{ij}{\left( \partial_j N_l \right)}^\epsilon {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0) \partial_i v -\int_{\Omega}\epsilon A^\epsilon_{ij}N^\epsilon_l {\mathbf{S}}_\epsilon(\partial^2_{lj}{\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0)\partial_i v \\
    \leq& j^0(\gamma_j u_0; \gamma_j v)+ j^0(\gamma_j u_\epsilon; -\gamma_j v) \\
    &+\int_{\Omega} {\left\{ \hat{A}_{ij} \partial_j u_0-A^\epsilon_{ij}\partial_j u_0 - A^\epsilon_{ij}{\left( \partial_j N_l \right)}^\epsilon {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0)\right\}}\partial_i v\\
    &-\int_{\Omega}\epsilon A^\epsilon_{ij}N^\epsilon_l {\mathbf{S}}_\epsilon(\partial^2_{lj}{\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0)\partial_i v
    \end{aligned}.$$ The techniques for last two parts of above inequality are 