exactly same as ([@Shen2016] Lem. 3.5). However, we will still elaborate its details for the coherence and self-containing of the proof. $$\begin{aligned}
    &\int_{\Omega} {\left\{ \hat{A}_{ij} \partial_j u_0-A^\epsilon_{ij}\partial_j u_0 - A^\epsilon_{ij}{\left( \partial_j N_l \right)}^\epsilon {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0)\right\}}\partial_i v-\int_{\Omega}\epsilon A^\epsilon_{ij}N^\epsilon_l {\mathbf{S}}_\epsilon(\partial^2_{lj}{\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0)\partial_i v \\
    \coloneqq & \int_\Omega B^\epsilon_{il} {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0) \partial_i v+\int_\Omega T_\epsilon \cdot \nabla v \\
    \coloneqq & J_1+J_2
    \end{aligned}.$$ Here $B^\epsilon={\left[ B_{il}(\bm{x}/\epsilon)\right]}_{1\leq i,l \leq n} \coloneqq \hat{A}-A^\epsilon-{\left( A_{ij}\partial_j N_l \right)}^\epsilon$, and $$T_\epsilon \coloneqq {\left( \hat{A}\nabla u_0 - \hat{A}{\mathbf{S}}_\epsilon(\nabla {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0) \right)}+{\left( A^\epsilon\nabla u_0-A^\epsilon {\mathbf{S}}_\epsilon(\nabla {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0) \right)}-\epsilon A^\epsilon_{ij} N^\epsilon_l {\mathbf{S}}_\epsilon(\partial^2_{lj}{\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0),$$ \[lem:3.1\] together with \[lem:3.2\] implies ${\left\lVert T_\epsilon \right\rVert}_{0,\Omega} \leq \epsilon {\left\lVert u_0 \right\rVert}_{2,\Omega}$, then: $${\left\lvert J_2 \right\rvert} \leq C \epsilon {\left\lVert u_0 \right\rVert}_{2,\Omega} {\left\lVert \nabla v \right\rVert}_{0,\Omega} .$$ Because $\partial_i B_{il}=\bm{0}$ and $\fint_Q B_{il} = \bm{0}$, we can construct $E_{ijl}(\bm{y}) \in H^1_\sharp(Q)$, such that $B_{il}=\partial_j E_{ijl}$, $E_{ijl}=-E_{jil}$, and ${\left\lVert E_{ijl} \right\rVert}_{1,Q} \leq C {\left\lVert B_{il} \right\rVert}_{0,Q} \leq C(\kappa_1, \kappa_2)$, see ([@Jikov1994] p. 6-7). Take $\theta_\epsilon \in C^\infty({\mathbb{R}}^d)$, let $\theta_\epsilon(\bm{x}) \equiv 1$ when $\text{dist}(\bm{x}, \partial \Omega) \leq \epsilon$; $\theta_\epsilon(\bm{x}) \equiv 0$ when $\text{dist}(\bm{x}, \partial \Omega) \geq 2\epsilon$; $0 \leq \theta_\epsilon(\bm{x}) \leq 1$ when $\epsilon \leq \text{dist}(\bm{x}, \partial \Omega) \leq 2\epsilon$. We have ${\left\lVert 