In a bounded $C^{1,\alpha_0}$ domain $\Omega \subset \mathbb{R}^n$, $0<\alpha_0\leq1$, contains two simply connected and strictly convex $C^{1,\alpha_0}$ subdomains (inclusions) $D_1$ and $D_2$ that satisfy $D_1\cup D_2 \subset \subset \Omega$ and $\overline{D_1}\cap \overline{D_2}=\lbrace 0 \rbrace$, we study the following elliptic differential equation \begin{equation}\label{eq:eq2-intro-abs} \begin{split} \begin{cases} \text{div} \left(a(x)\nabla u\right)=0 \quad\text{in} \;\; \Omega,\\ \partial_{\nu}u(x)=g \qquad \qquad\text{on} \;\; \partial \Omega,\\ \int_{\partial \Omega} u=0, \end{cases} \end{split} \end{equation} where $$a(x)=1+(k-1)\chi_{(D_1\cup D_2)}(x),\quad 00$. Then we begin by separating the inclusions by a distance $\delta >0$, \emph{that is}, we set $$\Ddelta{1}=D_1-\frac{\delta}{2}\bold{e_n} ,\;\text{and}\;\; \Ddelta{2}=D_2+\frac{\delta}{2}\bold{e_n}, $$ where $\bold{e_n}=(0',1)$. Then we study the approximate differential equation corresponding to the separated inclusions which is \begin{equation}\label{eq:eq3-introduction-abs} \begin{split} \begin{cases} \text{div} \left(a_{\delta}(x)\nabla u_{\delta}\right)=0 \quad\text{in} \;\; \Omega,\\ \partial_{\nu}u_{\delta}(x)=g \qquad \qquad\text{on} \;\; \partial \Omega,\\ \int_{\partial \Omega} u_{\delta}=0,\\ \end{cases} \end{split} \end{equation} where $g\in L^2_0(\partial \Omega)$ and $$a_{\delta}(x)=1+(k-1)\chi_{(D_1^{\delta}\cup D^{\delta}_2)}(x).$$The solution of the elliptic equation (\ref{eq:eq3-introduction-abs}) has an integral representation in terms of potential functions defined on theboundary of each subdomain. From the representation formula, we derive uniform piecewise $C^{1,\alpha}$, $0<\alpha<\alpha_0$, estimates for this solution which are independent of the distance between the subdomains. That is, we find the estimate \begin{align*} \norm{\ud}_{C^{1,\alpha}\Big(\Omega_{\tilde{\epsilon}}\setminus\overline{D_1^{\delta}\cup D_2^{\delta} }\Big)}+ \norm{\ud}_{C^{1,\alpha}(\overline{D_1^{\delta} })}+\norm{\ud}_{C^{1,\alpha}(\overline{D_2^{\delta} })}\leq C \norm{g}_{L^2(\partial \Omega)}, \end{align*} where $\Omega_{\tilde{\epsilon}}=\lbrace x\in \Omega : \text{dist}(x,\partial \Omega)>\tilde{\epsilon} \rbrace$ and $C$ is independent of $\delta$. Our result extends the earlier result for dimension $n=2$ \cite{Ammari:2015}, but the analysis is much more complicated. Final estimates rely on detailed analysis near the touching point and collective compactness of some integral operators.
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