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 t...

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Bibliographic Details
Published in:arXiv.org 2022-06
Main Authors: Kim, Youchan, Shin, Pilsoo
Format: Article
Language:English
Subjects:
Boundaries
Composite materials
Coordinates
Parabolic differential equations
Partial differential equations
Online Access:Get full text
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Summary: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.
ISSN:2331-8422