can choose arbitrarily in $(0, 1)$.

Let $w_\epsilon$ be defined as previous, we have a parallel version of \[lem:key\]:

\[lem:Robin problem estimation 1/2\] Assume $u_0 \in H^2(\Omega)$, then $\forall v \in H^1(\Omega)$ $$\int_\Omega A^\epsilon \nabla w_\epsilon \cdot \nabla v + \int_{\Gamma} \alpha w_\epsilon v \leq C(\Omega, \kappa_1, \kappa_2, \alpha_1, \alpha_2) {\left\lVert u_0 \right\rVert}_{2,\Omega} {\left( \epsilon^{1/2}{\left\lVert \nabla v \right\rVert}_{0,\Omega_{2\epsilon}}+\epsilon{\left\lVert \nabla v \right\rVert}_{0,\Omega}+\epsilon {\left\lVert v \right\rVert}_{0, \Gamma} \right)} .$$

Compare with the proof of \[lem:key\], we are left to show: $$J_3 \coloneqq \int_{\Gamma} \epsilon \alpha N^\epsilon_l {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0) v \leq C \epsilon {\left\lVert u_0 \right\rVert}_{2,\Omega} {\left\lVert v \right\rVert}_{0,\Gamma}  .$$ By calculation: $${\left\lvert J_3 \right\rvert} \leq C\int_{\Gamma} {\left\lvert \epsilon N^\epsilon_l {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0)\theta_\epsilon v \right\rvert} \leq C{\left\lVert \epsilon N^\epsilon_l {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0) \right\rVert}_{0,\Gamma} {\left\lVert v \right\rVert}_{0,\Gamma} .$$ Then $$\begin{aligned}
    & \int_{\Gamma} {\left\lvert \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 \right\rvert}^2\\  
    \underset{\text{use }  \eqref{lem:boundary estimation}}{\leq}&   Ct \sum_i \int_\Omega {\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}^2+{\left\lvert N^\epsilon_l {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0)\epsilon \partial_i \theta_\epsilon \right\rvert}^2+\epsilon^2 {\left\lvert N^\epsilon_l {\mathbf{S}}_\epsilon(\partial^2_{il}{\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0)\theta_\epsilon \right\rvert}^2 \\
    &+ t^{-1}\epsilon^2 \int_{\Omega} {\left\lvert N^\epsilon_l {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0)\theta_\epsilon \right\rvert}^2\\
    \underset{\text{take $t=\epsilon$}}{\leq} &C\epsilon \sum_i \int_{\tilde{\Omega}_{2\epsilon}}{\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) \right\rvert}^2+{\left\lvert N^\epsilon_l {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0) \right\rvert}^2+\epsilon^2\int_{\Omega}{\left\lvert N^\epsilon_l {\mathbf{S}}_\epsilon(\partial^2_{il}{\mkern 