  In this paper, we derive an interior Schauder estimate for the divergence
form elliptic equation \begin{equation*} D_i(a(x)D_iu)=D_if_i \end{equation*}
in $\mathbb{R}^2$, where $a(x)$ and $f_i(x)$ are piecewise H\"older continuous
in a domain containing two touching balls as subdomains. When $f_i\equiv 0$ and
$a$ is piecewise constant, we prove that $u$ is piecewise smooth with bounded
derivatives. This completely answers a question raised by Li and Vogelius [7]
in dimension 2.
