Appendix A. Boundary estimates for the Green function and the Robin function
In this appendix, we give boundary estimates for the Robin function \(\varphi \) and the Green function G(y, x) used in the preceding sections.
For \(x\in \Omega \) close to \(\Gamma \), let \(d(x)=\textrm{dist}(x,\Gamma )\) and denote \(\hat{x}\in \Gamma \) to be the unique point such that \(|x-\hat{x}|=d(x)\) for \(d(x)>0\) small.
Lemma A.1
The Robin function \(\varphi \) satisfies as \(d(x)\rightarrow 0\) that
$$\begin{aligned} \varphi (x)=\frac{1}{2\pi }\ln \frac{1}{2d(x)}+O(d(x)) \end{aligned}$$
(A.1)
and
$$\begin{aligned} \frac{\partial \varphi (x)}{\partial \tau }=O(1),\quad \frac{\partial \varphi (x)}{\partial \nu }=\frac{1}{2\pi d(x)}+O(1), \end{aligned}$$
(A.2)
where \(\tau \) and \(\nu \) be the unit tangent and the unit normal of \(\partial \Omega \) at \(\hat{x}\), respectively.
Proof
By translation and rotation, we may assume that \(x=(0,d)\), \(\hat{x}=0\) and there exists a \(C^2\) function \(\eta (y_1)\) such that \(\eta (0)=0\), \(\eta ^\prime (0)=0\),
$$\begin{aligned} \partial \Omega \cap B_{\delta }(0)=\big \{y\in \mathbb {R}^2:\, y_2=\eta (y_1)\big \}\cap B_{\delta }(0) \end{aligned}$$
and
$$\begin{aligned} \Omega \cap B_{\delta }(0)=\big \{y\in \mathbb {R}^2:\, y_2>\eta (y_1)\big \}\cap B_{\delta }(0), \end{aligned}$$
where \(\delta >0\) is a small constant.
Let \(\bar{x}=(0,-d)\) be the refection of x with respect to \(y_1\) axis and set
$$\begin{aligned} H_0(y,x)=\frac{1}{2\pi }\ln \frac{1}{|y-\bar{x}|}. \end{aligned}$$
Then, \(f(y):=H(y,x)-H_0(y,x)\) is harmonic in \(\Omega \). For \(y\in \partial \Omega \cap B_\delta (0) \), since \(|y_2|=|\eta (y_1)|=O(y_1^2)\), we have
$$\begin{aligned} \begin{aligned} f(y)=&\frac{1}{2\pi }\ln \frac{1}{|y-x|}-\frac{1}{2\pi }\ln \frac{1}{|y-\bar{x}|}=\frac{1}{4\pi }\ln \left( 1+\frac{4y_2d}{|y-x|^2}\right) =O\left( \frac{y_2d}{|y-x|^2}\right) \\ =&O\left( \frac{y_2d}{|y|^2+d^2}\frac{1}{1-\frac{2y_2d}{|y|^2+d^2}}\right) =O\left( \frac{y_2d}{|y|^2+d^2}\right) =O(d). \end{aligned} \end{aligned}$$
On the other hand, if \(y\in \partial \Omega \setminus B_\delta (0)\), there holds
$$\begin{aligned} f(y)=\frac{1}{4\pi }\ln \left( 1+\frac{4y_2d}{|y-x|^2}\right) =O\left( \frac{y_2d}{|y-x|^2}\right) =O(d). \end{aligned}$$
By the maximum principle,
$$\begin{aligned} |f(y)|\le \max _{z\in \partial \Omega } |f(z)|,\quad y\in \Omega , \end{aligned}$$
which implies
$$\begin{aligned} H(y,x)=H_0(y,x)+O(d). \end{aligned}$$
(A.3)
Thus, (A.1) follows.
We now prove (A.2). For \(l=1,2\), let
$$\begin{aligned} f_l(y):=\frac{\partial H}{\partial x_l}(y,x)-\frac{\partial H_0}{\partial x_l}(y,x). \end{aligned}$$
Hence, \(\Delta f_l=0\) in \(\Omega \). Using Taylor expansion of \(\eta \) at 0, we get for \(y\in \partial \Omega \cap B_\delta (0)\) that
$$\begin{aligned} \begin{aligned} f_1(y)=&\frac{1}{2\pi }\frac{y_1}{|y-x|^2}-\frac{1}{2\pi }\frac{y_1}{|y-\bar{x}|^2} =\frac{1}{2\pi }\frac{y_1}{|y|^2+d^2}\left( \frac{1}{1-\frac{2y_2d}{|y|^2+d^2}}-\frac{1}{1+\frac{2y_2d}{|y|^2+d^2}}\right) \\ =&\frac{2y_1y_2d}{\pi (|y|^2+d^2)^2}+O(d) =\frac{2\eta ^{\prime \prime }(0)y_1^3d}{\pi (y_1^2+d^2)^2}+O(d)=O(1). \end{aligned} \end{aligned}$$
If \(y\in \partial \Omega \setminus B_\delta (0)\), we can verify that
$$\begin{aligned} \begin{aligned} f_1(y)=&\frac{1}{2\pi }\frac{y_1}{|y-x|^2}-\frac{1}{2\pi }\frac{y_1}{|y-\bar{x}|^2}=O(1). \end{aligned} \end{aligned}$$
The maximum principle yields
$$\begin{aligned} \frac{\partial H}{\partial x_1}(y,x)=\frac{\partial H_0}{\partial x_1}(y,x)+O(1). \end{aligned}$$
(A.4)
Similarly,
$$\begin{aligned} \frac{\partial H}{\partial x_2}(y,x)=\frac{\partial H_0}{\partial x_2}(y,x)+O(1). \end{aligned}$$
(A.5)
Therefore, the estimates (A.2) follows from (A.4), (A.5) and the equation
$$\begin{aligned} \frac{\partial \varphi (x)}{\partial x_l}=2\frac{\partial H}{\partial x_l}(y,x)\big |_{y=x},\quad l=1,2. \end{aligned}$$
\(\square \)
Now, we estimate the Green function G(y, x) near the boundary of \(\Omega \).
For any \(d_i,d_j \in \left[ \Big (\bar{b}-\frac{L}{|\ln \varepsilon |^{\theta _0}}\Big )\frac{1}{\lambda _\varepsilon |\ln \varepsilon |},\; \Big (\bar{b}+\frac{L}{|\ln \varepsilon |^{\theta _0}}\Big )\frac{1}{\lambda _\varepsilon |\ln \varepsilon |}\right] \) and \(\hat{x}_i,\hat{x}_j\in \Gamma \), let
$$\begin{aligned} x_i=\hat{x}_i-d_i\nu (\hat{x}_i),\quad x_j=\hat{x}_j-d_j\nu (\hat{x}_j). \end{aligned}$$
Lemma A.2
Suppose that \(|\hat{x}_i-\hat{x}_j|\ge d_j\), we have
$$\begin{aligned} G(x_i,x_j)=\frac{d_j^2}{\pi |\hat{x}_i-\hat{x}_j|^2}+O\left( \frac{d_j^4}{|\hat{x}_i-\hat{x}_j|^4}+\frac{1}{|\ln \varepsilon |^{\theta _0}}\right) \end{aligned}$$
(A.6)
and
$$\begin{aligned} \frac{\partial G(x_i,x_j)}{\partial x_{i,l}}=O\left( \frac{d_j}{|\hat{x}_i-\hat{x}_j|^2}\right) ,\quad l=1,2. \end{aligned}$$
(A.7)
Proof
By (A.3), for \(y\in \Omega \),
$$\begin{aligned} H(y,x_j)=\frac{1}{2\pi }\ln \frac{1}{|y-\bar{x}_j|}+O(d_j), \end{aligned}$$
where \(\bar{x}_j=\hat{x}_j+d_j\nu (\hat{x}_j)\). So we have
$$\begin{aligned} \begin{aligned} G(x_i,x_j)=\frac{1}{2\pi }\ln \frac{1}{|x_i-x_j|}-\frac{1}{2\pi }\ln \frac{1}{|x_i-\bar{x}_j|}+O(d_j). \end{aligned} \end{aligned}$$
Apparently,
$$\begin{aligned} |x_i-\bar{x}_j|^2=|x_i-x_j|^2+|x_j-\bar{x}_j|^2+2\langle x_i-x_j,x_j-\bar{x}_j \rangle \end{aligned}$$
and
$$\begin{aligned} \begin{aligned} \langle x_i-x_j,x_j-\bar{x}_j \rangle =&-2d_j\langle x_i-x_j,\nu (\hat{x}_j)\rangle \\ =&-2d_j\langle \hat{x}_i-\hat{x}_j,\nu (\hat{x}_j)\rangle +2d_j \langle d_i\nu (\hat{x}_i)-d_j\textbf{n}(\hat{x}_j),\nu (\hat{x}_j)\rangle \\ =&-2d_j\langle \hat{x}_i-\hat{x}_j,\nu (\hat{x}_j)\rangle +2d_j d_i\langle \nu (\hat{x}_i)-\nu (\hat{x}_j),\nu (\hat{x}_j)\rangle +2d_j(d_i-d_j) \\ =&O\left( d_j| \hat{x}_i-\hat{x}_j|^2+d_id_j| \hat{x}_i-\hat{x}_j|+d_j|d_i-d_j|\right) . \end{aligned} \end{aligned}$$
On the other hand,
$$\begin{aligned} |x_i-x_j|=|\hat{x}_i-\hat{x}_j|+O(|d_i\nu (\hat{x}_i)-d_j\nu (\hat{x}_j)|)=|\hat{x}_i-\hat{x}_j|+O(|d_i-d_j|+d_j|\hat{x}_i-\hat{x}_j|). \end{aligned}$$
Therefore,
$$\begin{aligned} \begin{aligned} G(x_i,x_j)=&\frac{1}{4\pi }\ln \left( 1+\frac{|x_i-\bar{x}_j|^2-|x_i-x_j|^2 }{|x_i-x_j|^2}\right) +O(d_j) \\ =&\frac{1}{4\pi }\ln \left( 1+\frac{4d_j^2+ O\big (d_j| \hat{x}_i-\hat{x}_j|^2+d_id_j| \hat{x}_i-\hat{x}_j|+d_j|d_i-d_j|\big )}{|x_i-x_j|^2}\right) +O(d_j) \\ =&\frac{1}{\pi }\frac{d_j^2}{|\hat{x}_i-\hat{x}_j|^2}+O\left( d_j +\frac{d_jd_j}{| \hat{x}_i-\hat{x}_j|}+ \frac{d_j|d_i-d_j|}{| \hat{x}_i-\hat{x}_j|^2}+\frac{d_j^4}{| \hat{x}_i-\hat{x}_j|^4}\right) \\ =&\frac{1}{\pi }\frac{d_j^2}{|\hat{x}_i-\hat{x}_j|^2}+O\left( \frac{d_j^4}{| \hat{x}_i-\hat{x}_j|^4}+\frac{1}{|\ln \varepsilon |^{\theta _0}}\right) . \end{aligned} \end{aligned}$$
By (A.4) and (A.5), for \(y\in \Omega \),
$$\begin{aligned} \frac{\partial H(y,x_j)}{\partial y_l}=\frac{\partial }{\partial y_l}\left( \frac{1}{2\pi }\ln \frac{1}{|y-\bar{x}_j|}\right) +O(1), \end{aligned}$$
and then
$$\begin{aligned} \begin{aligned} \frac{\partial G(x_i,x_j)}{\partial x_{i,l}}&=\frac{\partial }{\partial x_{i,l}}\left( \frac{1}{2\pi }\ln \frac{1}{|x_i-x_j|}-\frac{1}{2\pi }\ln \frac{1}{|x_i-\bar{x}_j|}\right) +O(1) \\&=-\frac{1}{2\pi }\left( \frac{x_{i,l}-x_{j,l}}{|x_i-x_j|^2}-\frac{x_{i,l}-\bar{x}_{j,l}}{|x_i-\bar{x}_j|^2}\right) +O(1) \\&=-\frac{x_{i,l}-x_{j,l}}{2\pi }\cdot \frac{|x_i-\bar{x}_j|^2-|x_i-x_j|^2}{|x_i-x_j|^2|x_i-\bar{x}_j|^2}+O\left( \frac{d_j}{|\hat{x}_i-\hat{x}_j|^2}\right) \\&=O\left( \frac{d_j}{|\hat{x}_i-\hat{x}_j|^2}\right) . \end{aligned} \end{aligned}$$
The proof is complete. \(\square \)
We conclude from Lemma A.2 that
Lemma A.3
For any \(\textbf{x}\in \Omega _k\), there holds
$$\begin{aligned} G(x_i,x_j)\le \frac{C_1}{|\ln \varepsilon |^{\theta _0}}. \end{aligned}$$
Moreover, if \(|\hat{x}_i-\hat{x}_j|=\frac{1}{\lambda _\varepsilon |\ln \varepsilon |^{1-\theta _0/2}}\), then
$$\begin{aligned} G(x_i,x_j)\ge \frac{C_2}{|\ln \varepsilon |^{\theta _0}}, \end{aligned}$$
where \(C_1, C_2\) are two positive constants.
Appendix B. The estimates for the plasma set
In this appendix, we give estimates for the radius of plasma set. The similar results can be found in Lemma A.1 in [4]. For the sake of completeness, we give the proof.
Lemma B.1
Suppose that \(\omega \) is a function satisfying
$$\begin{aligned} \Vert \omega \Vert _{L^\infty (\Omega )}=O\left( \varepsilon \right) . \end{aligned}$$
Then for each constant \( 0<\sigma <1/k \), there is a constant \(\varepsilon _{\sigma ,k}>0\) such that for any \(0<\varepsilon < \varepsilon _{\sigma ,k}\),
$$\begin{aligned} \sum \limits _{j=1}^kPU_{\varepsilon ,x_j,a_{\varepsilon ,j}}(y)+\omega (y)-\kappa +\lambda _\varepsilon \eta (y)>0,\quad \, y\in B_{s\varepsilon (1-\varepsilon ^{\sigma })}(x_i),\;\; i=1,\cdots ,k, \end{aligned}$$
while
$$\begin{aligned} \sum \limits _{j=1}^kPU_{\varepsilon ,x_j,a_{\varepsilon ,j}}(y)+\omega (y)-\kappa +\lambda _\varepsilon \eta (y)<0,\quad \, y\in \Omega \setminus \cup _{i=1}^k B_{s\varepsilon (1+\varepsilon ^{\sigma })}(x_i). \end{aligned}$$
Proof
If \(y\in B_{s\varepsilon (1-\varepsilon ^{\sigma })}(x_i)\), by (2.13), \(\varphi _1(s)=0\) and \(\varphi ^\prime _1(t)<0\) for \(0< t \le s \), we have
$$\begin{aligned} \begin{aligned}&\sum \limits _{j=1}^kPU_{\varepsilon ,x_j,a_{\varepsilon ,j}}(y)+\omega (y)-\kappa +\lambda _\varepsilon \eta (y) =U_{\varepsilon ,x_i,a_{\varepsilon ,i}}(y)-a_{\varepsilon ,i}+O(\varepsilon ) \\&=\frac{a_{\varepsilon ,i}}{s\varphi _1^\prime (s)\ln \frac{s\varepsilon }{R}}\varphi _1\left( \frac{|y-x_i|}{\varepsilon }\right) +O(\varepsilon ) \\&>\frac{a_{\varepsilon ,i}}{s\varphi _1^\prime (s)\ln \frac{s\varepsilon }{R}}\varphi _1\left( s(1-\varepsilon ^{\sigma })\right) +O(\varepsilon ) \\&=-\frac{a_{\varepsilon ,i}}{\ln \frac{s\varepsilon }{R}}\varepsilon ^{\sigma }+O\left( \varepsilon +\frac{\varepsilon ^{2\sigma }}{|\ln \varepsilon |}\right) >0. \end{aligned} \end{aligned}$$
For the case that \(y\in \Omega {\setminus } \cup _{i=1}^k B_{\varepsilon ^{\sigma _1}}(x_i)\), where \(\sigma<\sigma _1<1/k\) is a fixed constant, using the fact that
$$\begin{aligned} G(y,x_j)=\frac{1}{2\pi }\ln \frac{1}{|y-x_j|}-\frac{1}{2\pi }\ln \frac{1}{|y-\bar{x}_j|}+O(d_j), \end{aligned}$$
where \(\bar{x}_j\) is the reflection point of \(x_j\) with respect the boundary \(\Gamma \) of \(\Omega \), we find
$$\begin{aligned} \begin{aligned}&\sum \limits _{j=1}^kPU_{\varepsilon ,x_j,a_{\varepsilon ,j}}(y)+\omega (y)-\kappa +\lambda _\varepsilon \eta (y) =2\pi \sum \limits _{j=1}^k\frac{a_{\varepsilon ,j}}{\ln \frac{R}{s\varepsilon }}G(y,x_j)-\kappa +O(\lambda _\varepsilon ) \\&=\sum \limits _{j=1}^k\frac{a_{\varepsilon ,j}}{\ln \frac{R}{s\varepsilon }}\ln \frac{|y-\bar{x}_j|}{|y-x_j|}-\kappa +O(\lambda _\varepsilon ) \\&<\sum \limits _{j=1}^k\frac{a_{\varepsilon ,j}}{\ln \frac{R}{s\varepsilon }}\ln \left( 1+\frac{2d_j}{\varepsilon ^{\sigma _1}}\right) -\kappa +O(\lambda _\varepsilon ) \\&=\kappa (k\sigma _1-1)+O(\lambda _\varepsilon )<0. \end{aligned} \end{aligned}$$
Finally, if \(y\in B_{\varepsilon ^{\sigma _1}}(x_i){\setminus } B_{s\varepsilon (1+\varepsilon ^{\sigma _1}|\ln \varepsilon |)}(x_i)\) for some i, then
$$\begin{aligned} \begin{aligned}&\sum \limits _{j=1}^kPU_{\varepsilon ,x_j,a_{\varepsilon ,j}}(y)+\omega (y)-\kappa +\lambda _\varepsilon \eta (y) =U_{\varepsilon ,x_i,a_{\varepsilon ,i}}(y)-a_{\varepsilon ,i}+O(\lambda _\varepsilon \varepsilon ^{\sigma _1}) \\&=a_{\varepsilon ,i}\frac{\ln \frac{|y-x_i|}{s\varepsilon }}{\ln \frac{s\varepsilon }{R}}+O(\lambda _\varepsilon \varepsilon ^{\sigma _1}) \\&<a_{\varepsilon ,i}\frac{\ln (1+\varepsilon ^{\sigma _1}|\ln \varepsilon |)}{\ln \frac{s\varepsilon }{R}}+O(\lambda _\varepsilon \varepsilon ^{\sigma _1}) \\&=\frac{a_{\varepsilon ,i}}{\ln \frac{s\varepsilon }{R}}\varepsilon ^{\sigma _1}|\ln \varepsilon |+O\left( \lambda _\varepsilon \varepsilon ^{\sigma _1}+\varepsilon ^{2\sigma _1}|\ln \varepsilon |\right) <0. \end{aligned} \end{aligned}$$
Since \(0<\sigma <\sigma _1\), we have \(B_{s\varepsilon (1+\varepsilon ^{\sigma _1}|\ln \varepsilon |)}(x_i)\subset B_{s\varepsilon (1+\varepsilon ^{\sigma })}(x_i)\) for \(\varepsilon >0\) small. We therefore complete the proof. \(\square \)
Appendix C. Energy expansion
In this appendix, we will estimate
$$\begin{aligned} \mathcal {E}\left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}+\omega _{\varepsilon ,\textbf{x}}\right) \end{aligned}$$
and its derivatives with respect to \(x_{i,l}\) for \(i=1,\cdots , k\), \(l=1,2\), where \(\mathcal {E}\) is defined by (1.14).
Lemma C.1
There holds
$$\begin{aligned} \mathcal {E}\left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}+ \omega _{\varepsilon ,\textbf{x}}\right) =\mathcal {E}\left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}\right) +O\left( \frac{\lambda _\varepsilon \varepsilon ^3}{|\ln \varepsilon |}\right) . \end{aligned}$$
Proof
We have
$$\begin{aligned} \begin{aligned}&\mathcal {E}\left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}+ \omega _{\varepsilon ,\textbf{x}}\right) \\&=\mathcal {E}\left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}\right) +\varepsilon ^2\sum \limits _{j=1}^k \int _\Omega \;\nabla PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}\cdot \nabla \omega _{\varepsilon ,\textbf{x}}+\frac{\varepsilon ^2}{2}\int _\Omega \; |\nabla \omega _{\varepsilon ,\textbf{x}}|^2 \\&\quad -\frac{1}{2}\int _\Omega \bigg [\left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}+ \omega _{\varepsilon ,\textbf{x}}-\kappa +\lambda _\varepsilon \eta \right) _+^2- \left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}-\kappa +\lambda _\varepsilon \eta \right) _+^2\bigg ]. \end{aligned} \end{aligned}$$
By Proposition 2.5 and Lemma B.1,
$$\begin{aligned} \begin{aligned}&\int _\Omega \bigg [\left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}+ \omega _{\varepsilon ,\textbf{x}}-\kappa +\lambda _\varepsilon \eta \right) _+^2- \left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}-\kappa +\lambda _\varepsilon \eta \right) _+^2\bigg ] \\&= \int _{\cup _{i=1}^k B_{2s\varepsilon }(x_i)} \bigg [\left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}+ \omega _{\varepsilon ,\textbf{x}}-\kappa +\lambda _\varepsilon \eta \right) _+^2- \left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}-\kappa +\lambda _\varepsilon \eta \right) _+^2\bigg ] \\&=O\left( \frac{\varepsilon ^2}{|\ln \varepsilon |}\Vert \omega _{\varepsilon ,\textbf{x}}\Vert _{L^\infty (\Omega )} \right) = O\left( \frac{\lambda _\varepsilon \varepsilon ^3}{|\ln \varepsilon |}\right) . \end{aligned} \end{aligned}$$
On the other hand, for \(j=1,\cdots ,k\),
$$\begin{aligned} \begin{aligned} \varepsilon ^2\int _\Omega \;\nabla PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}\cdot \nabla \omega _{\varepsilon ,\textbf{x}}=&-\varepsilon ^2\int _\Omega \;\Delta (PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}) \omega _{\varepsilon ,\textbf{x}}=\int _\Omega \; (U_{\varepsilon ,x_j,a_{\varepsilon ,j}}-a_{\varepsilon ,j})_+ \omega _{\varepsilon ,\textbf{x}} \\ =&\int _{B_{s\varepsilon }(x_j)}\; (U_{\varepsilon ,x_j,a_{\varepsilon ,j}}-a_{\varepsilon ,j})_+ \omega _{\varepsilon ,\textbf{x}}=O\left( \frac{\lambda _\varepsilon \varepsilon ^3}{|\ln \varepsilon |}\right) .\end{aligned} \end{aligned}$$
Finally, we estimate the term \(\frac{\varepsilon ^2}{2}\int _\Omega \; |\nabla \omega _{\varepsilon ,\textbf{x}}|^2\). To do this, we first need to estimate the constants \(b_{jh}\), \(j=1,\cdots ,k\), \(h=1,2\) in (2.26), which satisfies
$$\begin{aligned} \begin{aligned}&\int _\Omega \; \big [-\varepsilon ^2\Delta u_{\varepsilon ,\textbf{x}}-(u_{\varepsilon ,\textbf{x}}-\kappa +\lambda _\varepsilon \eta )_+\big ]\frac{\partial PU_{\varepsilon ,x_i,a_{\varepsilon ,i}}}{\partial x_{i,l}} \\ =&\sum _{j=1}^k\sum _{h=1}^2b_{jh} \int _\Omega \; \xi \left( \frac{|y-x_j|}{s\varepsilon } \right) \frac{\partial U_{\varepsilon ,x_j,a_{\varepsilon ,j}}}{\partial x_{j,h}} \frac{\partial PU_{\varepsilon ,x_i,a_{\varepsilon ,i}}}{\partial x_{i,l}}. \end{aligned} \end{aligned}$$
(C.1)
By (2.11) and (2.12),
$$\begin{aligned} \begin{aligned}&\int _\Omega \; \xi \left( \frac{|y-x_j|}{s\varepsilon } \right) \frac{\partial U_{\varepsilon ,x_j,a_{\varepsilon ,j}}}{\partial x_{j,h}} \frac{\partial PU_{\varepsilon ,x_i,a_{\varepsilon ,i}}}{\partial x_{i,l}} \\ =&\int _{B_{2s\varepsilon }(x_j)}\; \xi \left( \frac{|y-x_j|}{s\varepsilon } \right) \frac{\partial U_{\varepsilon ,x_j,a_{\varepsilon ,j}}}{\partial x_{j,h}} \frac{\partial U_{\varepsilon ,x_i,a_{\varepsilon ,i}}}{\partial x_{i,l}}+O\left( \frac{\lambda _\varepsilon \varepsilon }{|\ln \varepsilon |}\right) \\ =&(\delta _{ij}\delta _{hl}c^\prime +o(1))\frac{1}{|\ln \varepsilon |^2}, \end{aligned} \end{aligned}$$
where \(c^\prime >0\) is a constant, \(\delta _{ij}=1\) if \(i=j\), otherwise, \(\delta _{ij}=0\). On the other hand,
$$\begin{aligned} \begin{aligned} \text {LHS of }{(C.1)}=&\int _\Omega \; \Big [\sum \limits _{j=1}^k(U_{\varepsilon ,x_j,a_{\varepsilon ,j}}-a_{\varepsilon ,j})_+-(u_{\varepsilon ,\textbf{x}}-\kappa +\lambda _\varepsilon \eta )_+\Big ]\frac{\partial PU_{\varepsilon ,x_i,a_{\varepsilon ,i}}}{\partial x_{i,l}}\\&-\varepsilon ^2\int _\Omega \; (\Delta \omega _{\varepsilon ,\textbf{x}})\frac{\partial PU_{\varepsilon ,x_i,a_{\varepsilon ,i}}}{\partial x_{i,l}}. \end{aligned} \end{aligned}$$
We derive from
$$\begin{aligned} \begin{aligned}&\int _\Omega \; \Big [\sum \limits _{j=1}^k(U_{\varepsilon ,x_j,a_{\varepsilon ,j}}-a_{\varepsilon ,j})_+-(u_{\varepsilon ,\textbf{x}}-\kappa +\lambda _\varepsilon \eta )_+\Big ]\frac{\partial PU_{\varepsilon ,x_i,a_{\varepsilon ,i}}}{\partial x_{i,l}} \\ =&\sum \limits _{j=1}^k\int _{B_{2s\varepsilon }(x_j)}\; \Big [(U_{\varepsilon ,x_j,a_{\varepsilon ,j}}-a_{\varepsilon ,j})_+-(U_{\varepsilon ,x_j,a_{\varepsilon ,j}}-a_{\varepsilon ,j}+O(\lambda _\varepsilon \varepsilon ))_+\Big ]\frac{\partial PU_{\varepsilon ,x_i,a_{\varepsilon ,i}}}{\partial x_{i,l}} \\ =&\left( \frac{\lambda _\varepsilon \varepsilon ^2}{|\ln \varepsilon |}\right) , \end{aligned} \end{aligned}$$
and
$$\begin{aligned} \begin{aligned} \varepsilon ^2\int _\Omega \; (\Delta \omega _{\varepsilon ,\textbf{x}})\frac{\partial PU_{\varepsilon ,x_i,a_{\varepsilon ,i}}}{\partial x_{i,l}}=&\varepsilon ^2\int _\Omega \; \omega _{\varepsilon ,\textbf{x}}\Delta \left( \frac{\partial PU_{\varepsilon ,x_i,a_{\varepsilon ,i}}}{\partial x_{i,l}}\right) \\ =&-\int _{B_{s\varepsilon }(x_i)} \left( \frac{\partial U_{\varepsilon ,x_i,a_{\varepsilon ,i}}}{\partial x_{i,l}}-\frac{\partial a_{\varepsilon ,i}}{\partial x_{i,l}}\right) \omega _{\varepsilon ,\textbf{x}} \\ =&\left( \frac{\lambda _\varepsilon \varepsilon ^2}{|\ln \varepsilon |}\right) \end{aligned} \end{aligned}$$
that
$$\begin{aligned} b_{jh}=O(\lambda _\varepsilon |\ln \varepsilon |\varepsilon ^2). \end{aligned}$$
(C.2)
Testing (2.26) by \(\omega _{\varepsilon ,\textbf{x}}\) and integrating on \(\Omega \), we obtain
$$\begin{aligned} \begin{aligned} \varepsilon ^2\int _\Omega \; |\omega _{\varepsilon ,\textbf{x}}|^2=&\int _\Omega \; \Big ((u_{\varepsilon ,\textbf{x}}-\kappa +\lambda _\varepsilon \eta )_+-\sum \limits _{j=1}^k(U_{\varepsilon ,x_j,a_{\varepsilon ,j}}-a_{\varepsilon ,j})_+\Big )\omega _{\varepsilon ,\textbf{x}} \\&+\sum _{j=1}^k\sum _{h=1}^2b_{jh}\int _\Omega \; \xi \left( \frac{|y-x_j|}{s\varepsilon } \right) \frac{\partial U_{\varepsilon ,x_j,a_{\varepsilon ,j}}}{\partial x_{j,h}}\omega _{\varepsilon ,\textbf{x}} \\ =&O\left( \lambda _\varepsilon \varepsilon ^3\Vert \omega _{\varepsilon ,\textbf{x}}\Vert _{L^\infty (\Omega )}+\sum _{j=1}^k\sum _{h=1}^2|b_{jh}|\frac{\varepsilon }{|\ln \varepsilon |}\Vert \omega _{\varepsilon ,\textbf{x}}\Vert _{L^\infty (\Omega )}\right) \\ =&O(\lambda _\varepsilon ^2\varepsilon ^4). \end{aligned} \end{aligned}$$
The conclusion follows. \(\square \)
Lemma C.2
We have
$$\begin{aligned} \begin{aligned} \mathcal {E}\left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}\right) =&\frac{k\pi \varepsilon ^2}{\ln \frac{R}{s\varepsilon }}(\kappa -\lambda _\varepsilon )\left( \kappa -\lambda _\varepsilon +\frac{\kappa }{\ln \frac{R}{s\varepsilon }}\ln R\right) \\&+\frac{\kappa \pi \varepsilon ^2}{|\ln \varepsilon |}\sum \limits _{i=1}^k \Bigg [ 2\lambda _\varepsilon \frac{\eta (\hat{x}_i)}{\partial \nu }d_i+\frac{\kappa }{|\ln \varepsilon |}\ln \frac{1}{2d_i} -2\pi \frac{\kappa }{|\ln \varepsilon |}\sum \limits _{j\ne i}^kG(x_i,x_j) \Bigg ] \\&+O\left( \frac{\varepsilon ^2}{\lambda _\varepsilon |\ln \varepsilon |^3}+\frac{\lambda _\varepsilon \varepsilon ^2\ln |\ln \varepsilon |}{|\ln \varepsilon |^2}\right) . \end{aligned} \end{aligned}$$
Proof
Straightforwardly, we have
$$\begin{aligned} \begin{aligned}&\varepsilon ^2\int _\Omega \,\Big |\nabla \Big (\sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}\Big )\Big |^2 \\ =&\sum \limits _{i=1}^k \int _\Omega \,\big (-\varepsilon ^2\Delta PU_{\varepsilon ,x_i,a_{\varepsilon ,i}}\big )PU_{\varepsilon ,x_i,a_{\varepsilon ,i}}+\sum \limits _{ i\ne j}\int _\Omega \,\big (-\varepsilon ^2\Delta PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}\big )PU_{\varepsilon ,x_i,a_{\varepsilon ,i}}. \end{aligned} \end{aligned}$$
Note that \(\int _{B_s(0)}\varphi _1=-2\pi s\varphi _1^\prime (s)\) and \(\int _{B_s(0)}\varphi _1^2=\pi (s\varphi _1^\prime (s))^2\). The definition of \(U_{\varepsilon ,x_i,a_{\varepsilon ,i}}\) in (1.8) enables us to show that
$$\begin{aligned} \begin{aligned} \int _\Omega \, \big (U_{\varepsilon ,x_i,a_{\varepsilon ,i}}-a_{\varepsilon ,i}\big )_+=&\frac{a_{\varepsilon ,i}}{s\varphi _1^\prime (s)\ln \frac{s\varepsilon }{R}}\int _{B_{s\varepsilon }(x_i)}\; \varphi _1\left( \frac{|y-x_i|}{\varepsilon }\right) \\ =&\frac{a_{\varepsilon ,i}}{s\varphi _1^\prime (s)\ln \frac{s\varepsilon }{R}}\varepsilon ^2\int _{B_{s}(0)}\; \varphi _1(z)dz \\ =&2\pi \varepsilon ^2 \frac{a_{\varepsilon ,i}}{\ln \frac{R}{s\varepsilon }} \end{aligned} \end{aligned}$$
and
$$\begin{aligned} \begin{aligned} \int _\Omega \, \big (U_{\varepsilon ,x_i,a_{\varepsilon ,i}}-a_{\varepsilon ,i}\big )_+^2=&\left( \frac{a_{\varepsilon ,i}}{s\varphi _1^\prime (s)\ln \frac{s\varepsilon }{R}}\right) ^2\int _{B_{s\varepsilon }(x_i)}\; \varphi _1^2\left( \frac{|y-x_i|}{\varepsilon }\right) \\ =&\left( \frac{a_{\varepsilon ,i}}{s\varphi _1^\prime (s)\ln \frac{s\varepsilon }{R}}\right) ^2\varepsilon ^2\int _{B_{s}(0)}\; \varphi ^2_1(z)dz \\ =&\pi \varepsilon ^2\left( \frac{a_{\varepsilon ,i}}{\ln \frac{R}{s\varepsilon }}\right) ^2. \end{aligned} \end{aligned}$$
By Lemma A.1, we have
$$\begin{aligned} \begin{aligned}&\int _\Omega \,\big (-\varepsilon ^2\Delta PU_{\varepsilon ,x_i,a_{\varepsilon ,i}}\big )PU_{\varepsilon ,x_i,a_{\varepsilon ,i}}=\int _\Omega \,\big ( U_{\varepsilon ,x_i,a_{\varepsilon ,i}}-a_{\varepsilon ,i}\big )_+PU_{\varepsilon ,x_i,a_{\varepsilon ,i}} \\&=\int _\Omega \, \big (U_{\varepsilon ,x_i,a_{\varepsilon ,i}}-a_{\varepsilon ,i}\big )_+^{2} +a_{\varepsilon ,i}\int _\Omega \, \big (U_{\varepsilon ,x_i,a_{\varepsilon ,i}}-a_{\varepsilon ,i}\big )_+ -\frac{a_{\varepsilon ,i}}{\ln \frac{R}{s\varepsilon }}\int _\Omega \, g(y,x_i)\big (U_{\varepsilon ,x_i,a_{\varepsilon ,j}}-a_{\varepsilon ,i}\big )_+ \\&=\pi \varepsilon ^2\left( \frac{a_{\varepsilon ,i}}{\ln \frac{R}{s\varepsilon }}\right) ^2+2\pi \varepsilon ^2 \frac{a_{\varepsilon ,i}}{\ln \frac{R}{s\varepsilon }}\left( a_{\varepsilon ,i} - \frac{a_{\varepsilon ,i}}{\ln \frac{R}{s\varepsilon }}g(x_i,x_i)\right) +O\left( \frac{\lambda _\varepsilon \varepsilon ^3}{|\ln \varepsilon |}\right) , \end{aligned} \end{aligned}$$
and for \(j\ne i\),
$$\begin{aligned} \begin{aligned}&\int _\Omega \,\big (-\varepsilon ^2\Delta PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}\big )PU_{\varepsilon ,x_i,a_{\varepsilon ,i}}=\int _\Omega \,\big ( U_{\varepsilon ,x_j,a_{\varepsilon ,j}}-a_{\varepsilon ,j}\big )_+PU_{\varepsilon ,x_i,a_{\varepsilon ,i}} \\&=2\pi \frac{a_{\varepsilon ,i}}{\ln \frac{R}{s\varepsilon }}\int _\Omega \,G(y,x_i)\big ( U_{\varepsilon ,x_j,a_{\varepsilon ,j}}-a_{\varepsilon ,j}\big )_+ \\&=4\pi ^2 \varepsilon ^2\frac{a_{\varepsilon ,i}}{\ln \frac{R}{s\varepsilon }} \frac{a_{\varepsilon ,j}}{\ln \frac{R}{s\varepsilon }} G(x_j,x_i)+O\left( \frac{\lambda _\varepsilon \varepsilon ^3}{|\ln \varepsilon |}\right) . \end{aligned} \end{aligned}$$
On the other hand, we deduce from (2.13) that
$$\begin{aligned} \begin{aligned} \int _\Omega \Big (\sum \limits _{i=1}^k PU_{\varepsilon ,x_i,a_{\varepsilon ,i}}-\kappa +\lambda _\varepsilon \eta \Big )_+^2 =&\sum \limits _{i=1}^k\int _{B_{2s\varepsilon }(x_i)}\, \Big (U_{\varepsilon ,x_i,a_{\varepsilon ,i}}-a_{\varepsilon ,i}+O(\lambda _\varepsilon \varepsilon )\Big )_+^2 \\ =&\sum \limits _{i=1}^k\int _{B_{s \varepsilon }(x_i)}\, \big (U_{\varepsilon ,x_i,a_{\varepsilon ,i}}-a_{\varepsilon ,i}\big )_+^{2}+O\left( \frac{\lambda _\varepsilon \varepsilon ^3}{|\ln \varepsilon |}\right) \\ =&\pi \varepsilon ^2\sum \limits _{i=1}^k\left( \frac{a_{\varepsilon ,i}}{\ln \frac{R}{s\varepsilon }}\right) ^2 +O\left( \frac{\lambda _\varepsilon \varepsilon ^3}{|\ln \varepsilon |}\right) . \end{aligned} \end{aligned}$$
Equations (2.7) and (2.10) then allow us to find
$$\begin{aligned} \mathcal {E}{} & {} \left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}\right) \\= & {} \pi \varepsilon ^2 \sum \limits _{i=1}^k\frac{a_{\varepsilon ,i}}{\ln \frac{R}{s\varepsilon }}\left( a_{\varepsilon ,i}- \frac{a_{\varepsilon ,i}}{\ln \frac{R}{s\varepsilon }}g(x_i,x_i) +2\pi \sum \limits _{j\ne i}\frac{a_{\varepsilon ,j}}{\ln \frac{R}{s\varepsilon }}G(x_j,x_i)\right) +O\left( \frac{\lambda _\varepsilon \varepsilon ^3}{|\ln \varepsilon |}\right) \\= & {} \pi \varepsilon ^2 \sum \limits _{i=1}^k\frac{a_{\varepsilon ,i}}{\ln \frac{R}{s\varepsilon }}\big (\kappa -\lambda _\varepsilon \eta (x_i)\big ) +O\left( \frac{\lambda _\varepsilon \varepsilon ^3}{|\ln \varepsilon |}\right) \\= & {} \pi \varepsilon ^2 \sum \limits _{i=1}^k\frac{1}{\ln \frac{R}{s\varepsilon }} \left( \kappa -\lambda _\varepsilon +\lambda _\varepsilon \frac{\partial \eta (\hat{x}_i)}{\partial \nu }d_i+O(\lambda _\varepsilon d_i^2) \right) \Bigg [\left( \kappa -\lambda _\varepsilon +\frac{\kappa }{\ln \frac{R}{s\varepsilon }}\ln R\right) +\lambda _\varepsilon \frac{\eta (\hat{x}_i)}{\partial \nu }d_i \\{} & {} +\frac{\kappa }{\ln \frac{R}{s\varepsilon }}\ln \frac{1}{2d_i}-2\pi \frac{\kappa }{\ln \frac{R}{s\varepsilon }}\sum \limits _{j\ne i}^kG(x_i,x_j) +O\left( \frac{1}{\lambda _\varepsilon |\ln \varepsilon |^2}+\frac{\lambda _\varepsilon \ln |\ln \varepsilon |}{|\ln \varepsilon |}\right) \Bigg ]+O\left( \frac{\lambda _\varepsilon \varepsilon ^3}{|\ln \varepsilon |}\right) \\= & {} \pi \varepsilon ^2 \sum \limits _{i=1}^k\frac{1}{\ln \frac{R}{s\varepsilon }} \left( \kappa -\lambda _\varepsilon +\lambda _\varepsilon \frac{\partial \eta (\hat{x}_i)}{\partial \nu }d_i\right) \Bigg [\left( \kappa -\lambda _\varepsilon +\frac{\kappa }{\ln \frac{R}{s\varepsilon }}\ln R\right) +\lambda _\varepsilon \frac{\eta (\hat{x}_i)}{\partial \nu }d_i\\{} & {} +\frac{\kappa }{\ln \frac{R}{s\varepsilon }}\ln \frac{1}{2d_i}-2\pi \frac{\kappa }{\ln \frac{R}{s\varepsilon }}\sum \limits _{j\ne i}^kG(x_i,x_j)\Bigg ] +O\left( \frac{\varepsilon ^2}{\lambda _\varepsilon |\ln \varepsilon |^3}+\frac{\lambda _\varepsilon \varepsilon ^2\ln |\ln \varepsilon |}{|\ln \varepsilon |^2}\right) \\= & {} \frac{k\pi \varepsilon ^2}{\ln \frac{R}{s\varepsilon }}(\kappa -\lambda _\varepsilon )\left( \kappa -\lambda _\varepsilon +\frac{\kappa }{\ln \frac{R}{s\varepsilon }}\ln R\right) \\{} & {} +\frac{\kappa \pi \varepsilon ^2}{\ln \frac{R}{s\varepsilon }}\sum \limits _{i=1}^k \Bigg [2\lambda _\varepsilon \frac{\eta (\hat{x}_i)}{\partial \nu }d_i+\frac{\kappa }{\ln \frac{R}{s\varepsilon }}\ln \frac{1}{2d_i}-2\pi \frac{\kappa }{\ln \frac{R}{s\varepsilon }}\sum \limits _{j\ne i}^kG(x_i,x_j)\Bigg ]\\{} & {} +O\left( \frac{\varepsilon ^2}{\lambda _\varepsilon |\ln \varepsilon |^3}+\frac{\lambda _\varepsilon \varepsilon ^2\ln |\ln \varepsilon |}{|\ln \varepsilon |^2}\right) . \end{aligned}$$
The conclusion follows from the estimate \(\frac{1}{\ln \frac{R}{s\varepsilon }}=\frac{1}{|\ln \varepsilon |}+O\left( \frac{1}{|\ln \varepsilon |^2}\right) \). \(\square \)
Lemma C.3
We have
$$\begin{aligned} \frac{\partial }{\partial x_{i,l}}\mathcal {E}\left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}+ \omega _{\varepsilon ,\textbf{x}}\right) =\frac{\partial }{\partial x_{i,l}}\mathcal {E}\left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}\right) +O\left( \frac{\lambda _\varepsilon \varepsilon ^{2+\sigma }}{|\ln \varepsilon |}\right) , \end{aligned}$$
where \(\sigma >0\) is a small constant.
Proof
It is readily to verify that
$$\begin{aligned} \begin{aligned} \frac{\partial }{\partial x_{i,l}}\mathcal {E}\left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}+ \omega _{\varepsilon ,\textbf{x}}\right)&= \left\langle \mathcal {E}^\prime \left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}+ \omega _{\varepsilon ,\textbf{x}}\right) , \sum \limits _{j=1}^k \frac{\partial PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}}{\partial x_{i,l}}\right\rangle \\&\quad +\left\langle \mathcal {E}^\prime \left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}+ \omega _{\varepsilon ,\textbf{x}}\right) , \frac{\partial \omega _{\varepsilon ,\textbf{x}} }{\partial x_{i,l}}\right\rangle \\&= I +II. \end{aligned} \end{aligned}$$
We estimate I and II separately. It can be derived from Proposition 2.5 and (C.2) that
$$\begin{aligned} \begin{aligned} II=&\sum _{j=1}^k\sum _{h=1}^2b_{jh} \int _\Omega \; \xi \left( \frac{|y-x_j|}{s\varepsilon } \right) \frac{\partial U_{\varepsilon ,x_j,a_{\varepsilon ,j}}}{\partial x_{j,h}} \frac{\partial \omega _{\varepsilon ,\textbf{x}} }{\partial x_{i,l}} \\ =&O\left( \sum _{j=1}^k\sum _{h=1}^2|b_{jh}| \left\| \frac{\partial \omega _{\varepsilon ,\textbf{x}} }{\partial x_{i,l}} \right\| _{L^\infty (\Omega )} \frac{\varepsilon }{|\ln \varepsilon |}\right) \\ =&O\left( \frac{\lambda _\varepsilon \varepsilon ^{2+\sigma }}{|\ln \varepsilon |}\right) . \end{aligned} \end{aligned}$$
While for the estimation of I, since \(\omega _{\varepsilon ,\textbf{x}}\in E_{\varepsilon ,\textbf{x}}\), it follows from (2.12) and Lemma B.1 that
$$\begin{aligned}{} & {} \left\langle \mathcal {E}^\prime \left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}+ \omega _{\varepsilon ,\textbf{x}}\right) , \sum \limits _{j=1}^k \frac{\partial PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}}{\partial x_{i,l}}\right\rangle \\{} & {} -\left\langle \mathcal {E}^\prime \left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}\right) , \sum \limits _{j=1}^k \frac{\partial PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}}{\partial x_{i,l}}\right\rangle \\= & {} \int _\Omega \; \left[ \left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}-\kappa +\lambda _\varepsilon \eta \right) _+-\left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}+ \omega _{\varepsilon ,\textbf{x}}-\kappa +\lambda _\varepsilon \eta \right) _+\right] \\{} & {} \times \sum \limits _{j=1}^k \frac{\partial PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}}{\partial x_{i,l}}-\varepsilon ^2 \sum \limits _{j=1}^k \int _\Omega \; \Delta \left( \frac{\partial PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}}{\partial x_{i,l}}\right) \omega _{\varepsilon ,\textbf{x}} \\= & {} \int _{B_{s\varepsilon (1+\varepsilon ^{\sigma })}(x_i)}\; \left[ \left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}-\kappa +\lambda _\varepsilon \eta \right) _+ \right. \\ {}{} & {} \left. -\left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}+ \omega _{\varepsilon ,\textbf{x}}-\kappa +\lambda _\varepsilon \eta \right) _+\right] \frac{\partial U_{\varepsilon ,x_i,a_{\varepsilon ,i}}}{\partial x_{i,l}} \\{} & {} +O\left( \lambda _\varepsilon \varepsilon ^2 \Vert \omega _{\varepsilon ,\textbf{x}}\Vert _{L^\infty (\Omega )} \right) \\= & {} -\int _{B_{s\varepsilon (1-\varepsilon ^{\sigma })}(x_i)}\; \omega _{\varepsilon ,\textbf{x}} \frac{\partial U_{\varepsilon ,x_i,a_{\varepsilon ,i}}}{\partial x_{i,l}} +O\left( \lambda _\varepsilon \varepsilon ^2 \Vert \omega _{\varepsilon ,\textbf{x}}\Vert _{L^\infty (\Omega )}+ \frac{\varepsilon ^{1+\sigma }}{|\ln \varepsilon |}\Vert \omega _{\varepsilon ,\textbf{x}}\Vert _{L^\infty (\Omega )}\right) \\= & {} \varepsilon ^2\int _{\Omega }\; \omega _{\varepsilon ,\textbf{x}} \Delta \left( \frac{\partial PU_{\varepsilon ,x_i,a_{\varepsilon ,i}}}{\partial x_{i,l}}\right) +O\left( \lambda _\varepsilon \varepsilon ^2 \Vert \omega _{\varepsilon ,\textbf{x}}\Vert _{L^\infty (\Omega )}+ \frac{\varepsilon ^{1+\sigma }}{|\ln \varepsilon |}\Vert \omega _{\varepsilon ,\textbf{x}}\Vert _{L^\infty (\Omega )}\right) \\= & {} O\left( \frac{\lambda _\varepsilon \varepsilon ^{2+\sigma }}{|\ln \varepsilon |}\right) . \end{aligned}$$
We complete the proof. \(\square \)
Lemma C.4
We have
$$\begin{aligned} \begin{aligned}&\frac{\partial }{\partial x_{i,l}}\mathcal {E}\left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}\right) \\ =&-\frac{2\pi \kappa \varepsilon ^2}{|\ln \varepsilon |} \left( \lambda _\varepsilon \frac{\partial \eta (x_i)}{\partial x_{i,l}}-\pi \frac{\kappa }{|\ln \varepsilon |} \frac{\partial \varphi (x_i)}{\partial x_{i,l}}+2\pi \sum \limits _{j\ne i} ^k\frac{\kappa }{|\ln \varepsilon |}\frac{\partial G(x_i,x_j)}{\partial x_{i,l}}+O(\lambda _\varepsilon ^2)\right) . \end{aligned} \end{aligned}$$
In particular, it follows from (A.2) and (A.7) that
$$\begin{aligned} \begin{aligned} \frac{\partial }{\partial \nu _i}\mathcal {E}\left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}\right) =-\frac{\pi \kappa \varepsilon ^2}{|\ln \varepsilon |} \left( 2\lambda _\varepsilon \frac{\partial \eta (x_i)}{\partial \nu _i}-\frac{\kappa }{|\ln \varepsilon |} \frac{1}{d_i}+O\Big (\lambda _\varepsilon ^2+\frac{1}{|\ln \varepsilon |}+\frac{\lambda _\varepsilon }{|\ln \varepsilon |^{\theta _0}}\Big )\right) , \end{aligned} \end{aligned}$$
where \(\nu _i=\nu (\hat{x}_i)\) is the unit outward normal of \(\partial \Omega \) at \(\hat{x}_i\).
Proof
It can be shown that
$$\begin{aligned} \begin{aligned}&\frac{\partial }{\partial x_{i,l}}\mathcal {E}\left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}\right) =\left\langle \mathcal {E}^\prime \left( \sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}\right) , \sum \limits _{j=1}^k \frac{\partial PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}}{\partial x_{i,l}}\right\rangle \\ =&\int _\Omega \bigg ( -\varepsilon ^2\Delta \Big (\sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}\Big )-\Big (\sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}-\kappa +\lambda _\varepsilon \eta \Big )_+\bigg ) \sum \limits _{j=1}^k \frac{\partial PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}}{\partial x_{i,l}} \\ =&\int _\Omega \bigg (\sum \limits _{j=1}^k \Big (U_{\varepsilon ,x_j,a_{\varepsilon ,j}}-a_{\varepsilon ,j}\Big )_+-\Big (\sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}-\kappa +\lambda _\varepsilon \eta \Big )_+\bigg ) \sum \limits _{j=1}^k \frac{\partial PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}}{\partial x_{i,l}}. \end{aligned} \end{aligned}$$
By (2.13) and Lemma B.1, we have
$$\begin{aligned} \begin{aligned}&\int _\Omega \bigg (\sum \limits _{j=1}^k \Big (U_{\varepsilon ,x_j,a_{\varepsilon ,j}}-a_{\varepsilon ,j}\Big )_+-\Big (\sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}-\kappa +\lambda _\varepsilon \eta \Big )_+\bigg ) \sum \limits _{j\ne i}^k \frac{\partial PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}}{\partial x_{i,l}} \\ =&\sum \limits _{j=1}^k \int _{B_{2s\varepsilon }(x_j)}\bigg (\Big (U_{\varepsilon ,x_j,a_{\varepsilon ,j}}-a_{\varepsilon ,j}\Big )_+-\Big (U_{\varepsilon ,x_j,a_{\varepsilon ,j}}-a_{\varepsilon ,j}+O(\lambda _\varepsilon \varepsilon )\Big )_+\bigg ) \sum \limits _{j\ne i}^k \frac{\partial PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}}{\partial x_{i,l}} \\ =&O\left( \lambda _\varepsilon \sum \limits _{j=1}^k \int _{B_{2s\varepsilon }(x_j)}\bigg |\Big (U_{\varepsilon ,x_j,a_{\varepsilon ,j}}-a_{\varepsilon ,j}\Big )_+-\Big (U_{\varepsilon ,x_j,a_{\varepsilon ,j}}-a_{\varepsilon ,j}+O(\lambda _\varepsilon \varepsilon )\Big )_+\bigg | \right) \\ =&O(\lambda _\varepsilon ^2 \varepsilon ^3). \end{aligned} \end{aligned}$$
On the other hand, by (2.12), (2.13) and Lemma B.1 we have
$$\begin{aligned} \begin{aligned}&\int _\Omega \bigg (\sum \limits _{j=1}^k \Big (U_{\varepsilon ,x_j,a_{\varepsilon ,j}}-a_{\varepsilon ,j}\Big )_+-\Big (\sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}-\kappa +\lambda _\varepsilon \eta \Big )_+\bigg ) \frac{\partial PU_{\varepsilon ,x_i,a_{\varepsilon ,i}}}{\partial x_{i,l}} \\ =&\int _{B_{2s\varepsilon }(x_i)}\bigg (\Big (U_{\varepsilon ,x_i,a_{\varepsilon ,i}}-a_{\varepsilon ,i}\Big )_+-\Big (\sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}-\kappa +\lambda _\varepsilon \eta \Big )_+\bigg ) \frac{\partial PU_{\varepsilon ,x_i,a_{\varepsilon ,i}}}{\partial x_{i,l}} \\&+\sum \limits _{j\ne i}^k \int _{B_{2s\varepsilon }(x_j)}\bigg (\Big (U_{\varepsilon ,x_j,a_{\varepsilon ,j}}-a_{\varepsilon ,j}\Big )_+-\Big (\sum \limits _{j=1}^k PU_{\varepsilon ,x_j,a_{\varepsilon ,j}}-\kappa +\lambda _\varepsilon \eta \Big )_+\bigg ) \frac{\partial PU_{\varepsilon ,x_i,a_{\varepsilon ,i}}}{\partial x_{i,l}} \\ =&\left( \int _{B_{s\varepsilon (1-\varepsilon ^{\sigma })}(x_i)} + \int _{B_{s\varepsilon (1+\varepsilon ^{\sigma })}(x_i)\setminus B_{s\varepsilon (1-\varepsilon ^{\sigma })}(x_i)}\right) \bigg (\Big (U_{\varepsilon ,x_i,a_{\varepsilon ,i}}-a_{\varepsilon ,i}\Big )_+ \\&-\Big (U_{\varepsilon ,x_i,a_{\varepsilon ,i}}-a_{\varepsilon ,i}+h_{\varepsilon ,i}+O(\varepsilon ^{1+\sigma })\Big )_+\bigg ) \frac{\partial PU_{\varepsilon ,x_i,a_{\varepsilon ,i}}}{\partial x_{i,l}}+O(\lambda _\varepsilon ^2\varepsilon ^3) \\ =&-\int _{B_{s\varepsilon (1-\varepsilon ^{\sigma })}(x_i)}h_{\varepsilon ,i} \frac{\partial PU_{\varepsilon ,x_i,a_{\varepsilon ,i}}}{\partial x_{i,l}}+O\left( \frac{\varepsilon ^{2+\sigma }}{|\ln \varepsilon |}\right) \\ =&-\int _{B_{s\varepsilon }(x_i)}h_{\varepsilon ,i} \frac{\partial U_{\varepsilon ,x_i,a_{\varepsilon ,i}}}{\partial x_{i,l}}+O\left( \frac{\varepsilon ^{2+\sigma }}{|\ln \varepsilon |}\right) , \end{aligned} \end{aligned}$$
where
$$\begin{aligned} h_{\varepsilon ,i}(y)=\lambda _\varepsilon \langle \nabla \eta (x_i),y-x_i \rangle -\frac{a_{\varepsilon ,i}}{\ln \frac{R}{s\varepsilon }}\langle \nabla g(x_i,x_i),y-x_i\rangle +2\pi \sum \limits _{j\ne i} ^k\frac{a_{\varepsilon ,j}}{\ln \frac{R}{s\varepsilon }}\langle \nabla G(x_i,x_j),y-x_i\rangle . \end{aligned}$$
Since \(\int _{B_s(0)} \varphi ^\prime _1(|z|)\frac{z_l^2}{|z|}=2\pi s \varphi _1^\prime (s),\,\, a_{\varepsilon ,i}=\kappa +O(\lambda _\varepsilon )\,\,\textrm{and }\,\,\frac{1}{\ln \frac{R}{s\varepsilon }}=\frac{1}{|\ln \varepsilon |}+O(\frac{1}{|\ln \varepsilon |^2})\), by (2.12), we deduce
$$\begin{aligned} \begin{aligned}&\int _{B_{s\varepsilon }(x_i)}h_{\varepsilon ,i} \frac{\partial U_{\varepsilon ,x_i,a_{\varepsilon ,i}}}{\partial x_{i,l}}=\int _{B_{s\varepsilon }(x_i)}h_{\varepsilon ,i}(y) \frac{a_{\varepsilon ,i}}{s\varepsilon \varphi _1^\prime (s)\ln \frac{s\varepsilon }{R}} \varphi _1^\prime \left( \frac{|y-x_i|}{\varepsilon }\right) \frac{x_{i,l}-y_l}{|y-x_i|}+O(\lambda _\varepsilon ^2\varepsilon ^3)\\ =&-\frac{a_{\varepsilon ,i}\varepsilon }{s\varphi _1^\prime (s)\ln \frac{s\varepsilon }{R}}\int _{B_s(0)}h_{\varepsilon ,i}(\varepsilon z+x_i)\varphi ^\prime _1(|z|)\frac{z_l}{|z|}+O(\lambda _\varepsilon ^2\varepsilon ^3)\\ =&-\frac{a_{\varepsilon ,i}\varepsilon ^2}{s\varphi _1^\prime (s)\ln \frac{s\varepsilon }{R}}\left( \lambda _\varepsilon \frac{\partial \eta (x_i)}{\partial x_{i,l}}-\frac{a_{\varepsilon ,i}}{\ln \frac{R}{s\varepsilon }} \frac{\partial g(x_i,x_i)}{\partial x_{i,l}}+2\pi \sum \limits _{j\ne i} ^k\frac{a_{\varepsilon ,j}}{\ln \frac{R}{s\varepsilon }}\frac{\partial G(x_i,x_j)}{\partial x_{i,l}}\right) \\&\int _{B_s(0)}\varphi ^\prime _1(|z|)\frac{z_l^2}{|z|}+O(\lambda _\varepsilon ^2\varepsilon ^3)\\ =&\frac{2\pi a_{\varepsilon ,i}\varepsilon ^2}{\ln \frac{R}{s\varepsilon }}\left( \lambda _\varepsilon \frac{\partial \eta (x_i)}{\partial x_{i,l}}-\pi \frac{a_{\varepsilon ,i}}{\ln \frac{R}{s\varepsilon }} \frac{\partial \varphi (x_i)}{\partial x_{i,l}}+2\pi \sum \limits _{j\ne i} ^k\frac{a_{\varepsilon ,j}}{\ln \frac{R}{s\varepsilon }}\frac{\partial G(x_i,x_j)}{\partial x_{i,l}}\right) +O(\lambda _\varepsilon ^2\varepsilon ^3) \\ =&\frac{2\pi \kappa \varepsilon ^2}{|\ln \varepsilon |} \left( \lambda _\varepsilon \frac{\partial \eta (x_i)}{\partial x_{i,l}}-\pi \frac{\kappa }{|\ln \varepsilon |} \frac{\partial \varphi (x_i)}{\partial x_{i,l}}+2\pi \sum \limits _{j\ne i} ^k\frac{\kappa }{|\ln \varepsilon |}\frac{\partial G(x_i,x_j)}{\partial x_{i,l}}+O(\lambda _\varepsilon ^2)\right) , \end{aligned} \end{aligned}$$
which leads to the conclusion. \(\square \)
