}{|\ln\frac{R}{{\varepsilon}}|^2
 },\quad
c_{{\delta},2}=\frac{(C+\eta) {\delta}^2
 }{\ln\frac{R}{{\varepsilon}}
 },$$ where $\eta>0$ is a small constant and $\varrho>0$ is a fixed small constant. Let

$$D=\bigl\{ Z=(z_1,\cdots,z_m): \; z_i\in\Omega_\varrho,
i=1,\cdots,m,\; |z_i-z_j|\ge \varrho^{\bar L}, i\ne j\bigr\},$$ where $\Omega_\varrho=\bigl\{y: y\in\Omega,\; d(y,\partial\Omega)\ge
\varrho\bigr\}$, and $\bar L>0$ is a large constant.

Denote $K^c=\bigl\{Z: Z\in D,\; K(Z)\le c\bigr\}$. Consider

$$\left\{
 \begin{array}{ll}
\frac{d Z(t)}{dt}=- DK(Z(t)), \; t\ge 0,&\\ \vspace{0.05cm}\\
Z(0)\in K^{c_{{\delta},2}}.&
\end{array}
\right.$$

\[l44\] $Z(t)$ does not leave $D$ before it reaches $K^{c_{{\delta},1}}$.

Note that $$\label{4.5}
h(x,z)=\frac{1}{2\pi}\ln\frac{1}{|x-\bar{z}|}+o(1),\quad
\frac{\partial h(x,z)}{\partial
n}=-\frac{1}{2\pi|x-\bar{z}|}\left\langle\frac{x-\bar{z}}{|x-\bar{z}|},n\right\rangle+o(1),$$ if $z$ is close to $\partial\Omega$, where $n$ is the outward normal unit vector of $\partial\{x: x\in\Omega, d(x,
\partial\Omega)\leq d(z, \partial\Omega)\}$ and $\bar{z}$ is the reflection point of $z$ with respect to $\partial\Omega$.

Suppose that there is $t_0>0$, such that $Z(t_0)=:(z_1,\cdots,z_m)\in
\partial D$.

1.  Suppose that there are $i, j\in\{1,2,\cdots,m\}$, such that $i\neq j$ and $|z_i-z_j|=\varrho^{\bar
    L}$.

    Since $d(\bar{z},\partial\Omega)\geq \varrho$ and $\bar{z}\notin\Omega$, using , we get $|h(z_j,z_j)|\le
    C^\prime\ln\frac1{\varrho}$ for any $i, j$, where $C^\prime>0$. Thus, we have

    $$\bar G(z_i,z_j)\ge
    \ln\frac1{|z_i-z_j|}-C^\prime\ln\frac1{\varrho}\ge \bar
    L\ln\frac1{\varrho}-C^\prime\ln\frac1{\varrho}.$$ Then, by Lemma \[l42\] and Proposition \[ap1\], we have $$K(Z)\le \frac{C {\delta}^2
     }{\ln \frac{R}{{\varepsilon}}
     }+ \frac{ mC^\prime {\delta}^2\ln\frac1\varrho
     }{(\ln\frac{R}{{\varepsilon}})^2}-\frac{ \bar L {\delta}^2\ln\frac1\varrho\
     }{(\ln\frac{R}{{\varepsilon}})^2}<c_{{\delta},1},$$ if $\bar L>0$ is large.\

2.  Suppose that there is $i$, such that $z_i\in\partial\Omega_\varrho$.

    Let $n$ 