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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 systemswith piecewise $C^{m,\gamma}$-coefficients which also come from compositematerials. This answers the open problem suggested by Li and Vogelius, and Liand Nirenberg.

Piecewise smoothness for linear elliptic systems with piecewise smooth coefficients