Piecewise smoothness for linear elliptic systems with piecewise smooth coefficients

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.