by , we have $$\begin{split}
{\delta}^2\int_{\Omega}|D\omega_{\delta}|^2=&\sum_{j=1}^m\int_{{\Omega}_j}\Biggl[\left(P_{{\delta},Z}+\omega_{\delta}-\kappa_j-\frac{2\pi
q(x)}{|\ln{\varepsilon}|}\right)_+^{p}
-\left(W_{{\delta},z_j,a_{{\delta},j}}-a_{{\delta},j}\right)_+^{p}\Biggr]\omega_{\delta}\\
&+\sum_{j=1}^m\sum_{\bar h=1}^2b_{j,\bar
h}\int_{\Omega}\left(-{\delta}^2\Delta\frac{\partial P_{{\delta},Z,j}}{\partial
z_{j,\bar h}}\right)\omega_{\delta}\\
=&p\sum_{j=1}^m\int_{{\Omega}_j}\left(W_{{\delta},z_j,a_{{\delta},j}}-a_{{\delta},j}\right)_+^{p-1}\left(\frac{s_{{\delta},j}}{|\ln{\varepsilon}|}+\omega_{\delta}\right)\omega_{\delta}+O\left(\sum_{j=1}^m\sum_{\bar h=1}^2\frac{{\varepsilon}|b_{j,\bar
h}|\|\omega_{\delta}\|_\infty}{|\ln{\varepsilon}|^{p}}\right)\\
=&O\left(\frac{{\varepsilon}^4}{|\ln{\varepsilon}|^{p-1}}\right).
\end{split}$$

So we can obtain that

$$K(Z)= I\left(\sum_{j=1}^m P_{{\delta},Z,j}
\right)+O\left(
 \frac{{\varepsilon}^3}{|\ln{\varepsilon}|^{p}}\right).$$

\[l43\]

We have

$$\frac{\partial K(Z)}{\partial z_{i,h}}=\frac{\partial }{\partial
z_{i,h}}I\left(\sum_{j=1}^m P_{{\delta},Z,j} \right)+O\Bigl(
\frac{{\varepsilon}^3}{|\ln{\varepsilon}|^{p-1}}\Bigr).$$

First, we have

$$\label{4.4}
\begin{split}
&\frac{\partial K(Z)}{\partial z_{i,h}}=\left\langle I^\prime\Bigl(
P_{{\delta},Z} +\omega_{\delta}\Bigr),\frac{\partial P_{{\delta},Z}}{\partial
z_{i,h}}+ \frac{\partial
\omega_{\delta}}{\partial z_{i,h}}\right\rangle\\
=&\frac{\partial }{\partial z_{i,h}}I\Bigl(P_{{\delta},Z} \Bigr)
+\left\langle I^\prime\big( P_{{\delta},Z}+\omega_{\delta}\big),
\frac{\partial\omega_{\delta}}{\partial z_{i,h}}\right\rangle
\\
&-\sum_{j=1}^m \int_{\Omega_j}\Biggl[
\biggl(P_{{\delta},Z}+\omega_{\delta}-\kappa_j-\frac{2\pi
q(x)}{|\ln{\varepsilon}|}\biggr)_+^{p}-\biggl(P_{{\delta},Z}-\kappa_j-\frac{2\pi
q(x)}{|\ln{\varepsilon}|}\biggr)_+^{p}\Biggr]\frac{\partial
P_{{\delta},Z}}{\partial z_{i,h}}.
\end{split}$$

Since $\omega_{\delta}\in E_{{\delta},Z}$, we have $$\int_{\Omega}\left(W_{{\delta},z_j,a_{{\delta},j}}-a_{{\delta},j}\right)_+^{p-1}\left(\frac{\partial
W_{{\delta},z_j,a_{{\delta},j}}}{\partial z_{j,h}}-\frac{\partial
a_{{\delta},j}}{\partial z_{j,h}}\right)\omega_{\delta}=0.$$ Differentiating the above relation with respect to $z_{i, h}$, we can deduce

$$\begin{split}
\Bigg\langle & I^\prime\bigl(P_{{\delta},Z} +\omega_{\delta}\bigr),
\frac{\partial \omega_{\delta}}{\partial z_{i, h}}
\Bigg\rangle=\sum_{j=1}^m\sum_{\bar h=1}^2 b_{j,\bar h}
\int_{\Omega}\left(-{\delta}^2\Delta\frac{\partial P_{{\delta},Z,j}}{\partial
z_{j,\bar h}}\right) \frac{\partial \omega_{\delta}}{\partial z_{i, h}}\\
=&\sum_{j=1}^m\sum_{\bar h=1}^2 p b_{j,\bar h}
\int_{\Omega}\left(W_{{\delta},z_j,a_{{\delta},j}}-a_{{\delta},j}\right)_+^{p-1}\left(\frac{\partial
W_{{\delta},z_j,a_{{\delta},j}}}{\partial z_{j,\bar h}}-\frac{\partial
a_{{\delta},j}}{\partial z_{j,\bar h}}\right) \frac{\partial
\omega_{\delta}}{\partial z_{i, h}}\\
=&O\left(\sum_{j=1}^m\sum_{\bar h=1}^2 \frac{{\varepsilon}|b_{j,\bar
h}|}{|\ln{\varepsilon}|^{p}}\right)=O\left(\frac{{\varepsilon}^3}{|\ln{\varepsilon}|^{p-1}}\right).
\end{split}$$

On the other hand, using (for the definition of $R_{\delta}(\omega)$, see ), we obtain

$$\begin{split}
&\sum_{j=1}^m\int_{\Omega_j}\Biggl[
\biggl(P_{{\delta},Z}+\omega_{\delta}-\kappa_j-\frac{2\pi
q(x)}{|\ln{\varepsilon}|}\biggr)_+^{p}-\biggl(P_{{\delta},Z}-\kappa_j-\frac{2\pi
q(x)}{|\ln{\varepsilon}|}\biggr)_+^{p}\Biggr]\frac{\partial
P_{{\delta},Z,i}}{\partial z_{i,h}}\\
= & \sum_{j=1}^m\int_{\Omega_j}\Biggl[
\biggl(P_{{\delta},Z}+\omega_{\delta}-\kappa_j-\frac{2\pi q(x)}{|\ln{\varepsilon}|}\biggr)_+^{p}-\biggl(P_{{\delta},Z}-\kappa_j-\frac{2\pi q(x)}{|\ln{\varepsilon}|}\biggr)_+^{p}\\
&-p\biggl(P_{{\delta},Z}-\kappa_j-\frac{2\pi
q(x)}{\kappa|\ln{\varepsilon}|}\biggr)_+^{p-1}\omega_{\delta}\Biggr]\frac{\partial
P_{{\delta},Z,i}}{\partial z_{i,h}} +\sum_{j=1}^m
p\int_{\Omega_j}\Biggl[\biggl(P_{{\delta},Z}-\kappa_j-\frac{2\pi q(x)}{|\ln{\varepsilon}|}\biggr)_+^{p-1}\\
&-\bigl(W_{{\delta},z_j,a_{{\delta},j}}-a_{{\delta},j}\bigr)_+^{p-1}\Biggr]\frac{\partial
P_{{\delta},Z,i}}{\partial z_{i,h}}\omega_{\delta}+O\left(\frac{s_{{\delta},j}^2\|\omega_{\delta}\|_\infty}{|\ln{\varepsilon}|^{p}}\right)\\
=&\int_\Omega R_{\delta}(\omega_{\delta})\frac{\partial P_{{\delta},Z,i}}{\partial
z_{i,h}} +
\sum_{j=1}^mp\int_{\Omega_j}\Biggl[\biggl(P_{{\delta},Z}-\kappa_j-\frac{2\pi
q(x)}{|\ln{\varepsilon}|}\biggr)_+^{p-1}-\bigl(W_{{\delta},z_i,a_{{\delta},i}}-a_{{\delta},i}\bigr)_+^{p-1}\Biggr]\frac{\partial
P_{{\delta},Z,i}}{\partial
z_{i,h}}\omega_{\delta}\\
&+O\left(\frac{{\varepsilon}^3}{|\ln{\varepsilon}|^{p}}\right)\\
 =& O\left(\frac{{\varepsilon}^3}{|\ln{\varepsilon}|^{p-1}}\right).
\end{split}$$ Thus, the estimate follows.

Define

$$c_{{\delta},1}=\frac{ C {\delta}^2
 }{\ln \frac{R}{{\varepsilon}}
 }-\frac{ \pi{\delta}^2\ln\frac1{\varrho}
 