A geometric result for composite materials with $C^{1,\gamma }$-boundaries

In this paper, we obtain a geometric result for composite materials related to elliptic and parabolic partial differential equations. In the classical papers Li and Vogelius (2000), and Li and Nirenberg (2003), they assumed that for any scale and for any point there exists a coordinate system such that the boundaries of the individual components of a composite material locally become $C^{1,\gamma }$-graphs. We prove that if the individual components of a composite material are composed of $C^{1,\gamma }$-boundaries then such a coordinate system in Li and Vogelius (2000), and Li and Nirenberg (2003) exists, and therefore obtaining the gradient boundedness and the piecewise gradient Hölder continuity results for linear elliptic systems related to composite materials.