  Li and Vogelius, and Li and Nirenberg obtained piecewise
$C^{1,\gamma}$-regularity for linear elliptic problems with piecewise
$C^{\gamma}$-coefficients which come from composite materials. In this paper,
we obtain piecewise $C^{m+1,\gamma}$-regularity for linear elliptic systems
with piecewise $C^{m,\gamma}$-coefficients which also come from composite
materials. This answers the open problem suggested by Li and Vogelius, and Li
and Nirenberg.
