\nabla \theta_\epsilon \right\rVert}_{L^\infty({\mathbb{R}}^d)} \leq C(\Omega)/\epsilon$. Then split $J_1$ as $$J_1 = \int_\Omega (\partial_j E_{ijl})^\epsilon \theta_\epsilon {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0) \partial_i v +\int_\Omega (\partial_j E_{ijl})^\epsilon (1-\theta_\epsilon) {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0) \partial_i v  \coloneqq J_{11}+J_{12} .$$ Use \[lem:3.4\], we have: $$\begin{aligned}
    {\left\lvert J_{11} \right\rvert} \leq &{\left\lVert \nabla v \right\rVert}_{0,\Omega_{2\epsilon}} {\left[ \sum_i\int_{\Omega_{2\epsilon}} {\left\lvert (\partial_j E_{ijl})^\epsilon \right\rvert}^2 {\left\lvert {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0) \right\rvert}^2 \theta_\epsilon^2\right]}^{1/2} \\
    \leq & {\left\lVert \nabla v \right\rVert}_{0,\Omega_{2\epsilon}} {\left[ \sum_i\int_{\tilde{\Omega}_{2\epsilon}} {\left\lvert (\partial_j E_{ijl})^\epsilon \right\rvert}^2 {\left\lvert {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0) \right\rvert}^2\right]}^{1/2} \\
    \leq  & C \epsilon^{1/2} {\left\lVert u_0 \right\rVert}_{2,\Omega}{\left\lVert \nabla v \right\rVert}_{0,\Omega_{2\epsilon}}
    \end{aligned}.$$ Due to $1-\theta_\epsilon \equiv 0$ on $\partial \Omega$, $(1-\theta_\epsilon) {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0) \partial_i v \in C^\infty_0(\Omega)$, and $(\partial_j E_{ijl})^\epsilon=\epsilon \partial_j E_{ijl}^\epsilon$. According to integration by parts: $$\begin{aligned}
    J_{12}=&\int_{\Omega} \epsilon \partial_j E_{ijl}^\epsilon (1-\theta_\epsilon) {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0) \partial_i v 
    = -\int_{\Omega} \epsilon E_{ijl}^\epsilon \partial_j{\left[ (1-\theta_\epsilon){\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0)\partial_i v\right]} \\
    = & \int_{\Omega} E_{ijl}^\epsilon \epsilon \partial_j \theta_\epsilon {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0) \partial_i v \\
    & - \int_{\Omega} \epsilon E_{ijl}^\epsilon (1-\theta_\epsilon) {\mathbf{S}}_\epsilon(\partial^2_{jl}{\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0) \partial_i v \\
    & - \int_{\Omega} \epsilon E_{ijl}^\epsilon (1-\theta_\epsilon) {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0) \partial^2_{ij} v   
    \end{aligned},$$ $E_{ijl}=-E_{jil}$ implies $\int_{\Omega} \epsilon E_{ijl}^\epsilon (1-\theta_\epsilon) {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0) \partial^2_{ij} v =0$, and $$\begin{aligned}
    \int_{\Omega} {\left\lvert E_{ijl}^\epsilon \epsilon \partial_j 