Piecewise regularity results for linear elliptic systems with piecewise regular coefficients

We obtain piecewise regularity results for linear elliptic systems with piecewise regular coefficients arising from composite materials. We find Hölder continuous functions related to the higher-order derivatives by compensating the possible discontinuity due to the geometry and the discontinuity of the coefficients. This leads to the piecewise regularity of the higher-order derivatives and solves an open problem introduced by Li and Vogelius, and Li and Nirenberg for general composite geometry.