A sharp regularity result of solutions of a transmission problem

In this paper we study the regularity of solutions of a transmission problem arising in studying a fiber-reinforced composite media. It is described through a divergence type equation div$(A \nabla u) = h,$ in an open set $D$, where $h$ is a bounded function and $A$ is a uniformly elliptic matrix, bounded and with piecewise Hölder continuous coefficients. The subdomains $D_i$ where $A$ is of class $C^{\alpha}$ have disjoint closure and are of class $C^{1,\alpha}$. Exploiting an idea contained in a paper by Li and Vogelius, we obtain the optimal regularity result for solutions, proving that they are of class $C^{1,\alpha}(\bar D_i)$.