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

TL;DR

This paper proves that composite materials with components having $C^{1,eta}$-boundaries allow for coordinate systems that ensure gradient boundedness and H"older continuity in elliptic PDEs, extending classical results.

Contribution

It establishes the existence of coordinate systems for composite materials with $C^{1,eta}$ boundaries, enabling gradient regularity results in elliptic systems.

Findings

Gradient boundedness for elliptic systems in composite materials.

Piecewise gradient H"older continuity established.

Extension of classical boundary regularity assumptions.

Abstract

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 H\"{o}lder continuity results for linear elliptic systems related to composite materials.