Li and Vogelius in 2000 and Li and Nirenberg in 2003 showed piecewise $C^{1,\gamma}$ regularity for linear elliptic problems with piecewise $C^{\gamma}$ coefficients which come from composite materials.
In this talk, 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 in 2000 and Li and Nirenberg in 2003.