\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\rvert} \leq& C {\left\lVert \nabla v \right\rVert}_{0,\Omega_{2\epsilon}} {\left[ \sum_{i,j}\int_{\tilde{\Omega}_{2\epsilon}} {\left\lvert E^\epsilon_{ijl} \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 \nabla v \right\rVert}_{0,\Omega_{2\epsilon}} {\left\lVert u_0 \right\rVert}_{2,\Omega}
    \end{aligned},$$ $$\begin{aligned}
    \int_{\Omega} {\left\lvert \epsilon E_{ijl}^\epsilon (1-\theta_\epsilon){\mathbf{S}}_\epsilon(\partial_{jl}{\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0)\partial_i v \right\rvert} \leq& \epsilon {\left\lVert \nabla v \right\rVert}_{0,\Omega} {\left[ \sum_i \int_{\Omega}  {\left\lvert E_{ijl}^\epsilon \right\rvert}^2 {\left\lvert {\mathbf{S}}_\epsilon(\partial_{jl}{\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0) \right\rvert}^2\right]}^{1/2} \\ \leq& C \epsilon {\left\lVert \nabla v \right\rVert}_{0,\Omega} {\left\lVert u_0 \right\rVert}_{2,\Omega}
    \end{aligned}.$$ Finally, we derive ${\left\lvert J_1 \right\rvert}+{\left\lvert J_2 \right\rvert}\leq C {\left\lVert u_0 \right\rVert}_{2,\Omega}
    {\left( \epsilon^{1/2}{\left\lVert \nabla v \right\rVert}_{0,\Omega_{2\epsilon}}+\epsilon{\left\lVert \nabla v \right\rVert}_{0,\Omega} \right)}$.

A corollary of this lemma is following theorem:

Let the assumptions \[ass:A\]-\[ass:C\] be satisfied, and $u_0 \in H^2(\Omega)$. Then $${\left\lVert \nabla w_\epsilon \right\rVert}_{0,\Omega} \leq C\epsilon^{1/2}{\left\lVert u_0 \right\rVert}_{2,\Omega},$$ here constant $C$ depends on $\Omega, \kappa_1, \kappa_2, \Delta$.

Take $\theta_\epsilon$ defined as previous, and let $\tilde{u}_\epsilon=u_0+\epsilon N_l^\epsilon {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0)(1-\theta_\epsilon)$. Then $w_\epsilon = u_\epsilon-\tilde{u}_\epsilon-\epsilon N_l^\epsilon {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0)\theta_\epsilon \coloneqq u_\epsilon-\tilde{u}_\epsilon-r_\epsilon$. It is obvious to see $\tilde{u}_\epsilon \in V$ and $\gamma_j u_0 = \gamma_j \tilde{u_\epsilon}$. We use \[lem:3.1\]-\[lem:3.4\] to handle $r_\epsilon = \epsilon N^\epsilon_l {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0) \theta_\epsilon$: $$\begin{aligned}
    {\left\lVert \partial_i r_\epsilon \right\rVert}_{0,\Omega} \leq& {\left\lVert (\partial_i N_l)^\epsilon {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0)\theta_\epsilon \right\rVert}_{0,\Omega} + {\left\lVert 