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Geometric series expansion of the Neumann–Poincaré operator: Application to composite materials

Part of: Partial differential equations Two-dimensional theory Geometric function theory Eigenvalue problems

Published online by Cambridge University Press:  11 May 2021

ELENA CHERKAEV ,
MINWOO KIM  and
MIKYOUNG LIM Open the ORCID record for MIKYOUNG LIM [Opens in a new window]
ELENA CHERKAEV
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA email: elena@math.utah.edu
MINWOO KIM
Affiliation:
School of Electrical Engineering, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea email: epsilon4b@kaist.ac.kr
MIKYOUNG LIM
Affiliation:
Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea email: mklim@kaist.ac.kr
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Abstract

The Neumann–Poincaré (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the NP operator was developed in two dimensions based on geometric function theory [34]. In this paper, we investigate geometric properties of composite materials using this series expansion. In particular, we obtain explicit formulas for the polarisation tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the NP operator has a monotonic behaviour with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain using the properties of the polarisation tensor corresponding to the domain.

Keywords

Homogenisationintegral equations methods in two dimensionseigenvalues of operatorsconformal mappings

MSC classification

Primary: 35B27: Homogenization; equations in media with periodic structure 31A10: Integral representations, integral operators, integral equations methods 45C05: Eigenvalue problems
Secondary: 30C35: General theory of conformal mappings
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 3 , June 2022 , pp. 560 - 585
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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