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

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