Asymptotics of the solution to the conductivity equation in the presence of adjacent circular inclusions with finite conductivities

Mikyoung Lim; Sanghyeon Yu

doi:10.1016/j.jmaa.2014.07.002

Asymptotics of the solution to the conductivity equation in the presence of adjacent circular inclusions with finite conductivities

Mikyoung Lim, Sanghyeon Yu

Research output: Contribution to journal › Article › peer-review

10 Citations (Scopus)

Abstract

We consider the conductivity problem in the presence of adjacent circular inclusions with constant conductivities. When two inclusions get closer and their conductivities degenerate to zero or infinity, the gradient of the solution can be arbitrary large. In this paper we derive an asymptotic formula of the solution, which characterizes the gradient blow-up of the solution in terms of conductivities of inclusions as well as the distance between inclusions. The asymptotic formula is expressed in bipolar coordinates in terms of the Lerch transcendent function, and it is valid for inclusions with arbitrary constant conductivities. We illustrate our results with numerical calculations.

Original language	English
Pages (from-to)	131-156
Number of pages	26
Journal	Journal of Mathematical Analysis and Applications
Volume	421
Issue number	1
DOIs	https://doi.org/10.1016/j.jmaa.2014.07.002
Publication status	Published - 2015 Jan 1
Externally published	Yes

Keywords

Anti-plane elasticity
Bipolar coordinates
Conductivity equation
Gradient blow-up
Lerch transcendent function

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.jmaa.2014.07.002

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title = "Asymptotics of the solution to the conductivity equation in the presence of adjacent circular inclusions with finite conductivities",

abstract = "We consider the conductivity problem in the presence of adjacent circular inclusions with constant conductivities. When two inclusions get closer and their conductivities degenerate to zero or infinity, the gradient of the solution can be arbitrary large. In this paper we derive an asymptotic formula of the solution, which characterizes the gradient blow-up of the solution in terms of conductivities of inclusions as well as the distance between inclusions. The asymptotic formula is expressed in bipolar coordinates in terms of the Lerch transcendent function, and it is valid for inclusions with arbitrary constant conductivities. We illustrate our results with numerical calculations.",

keywords = "Anti-plane elasticity, Bipolar coordinates, Conductivity equation, Gradient blow-up, Lerch transcendent function",

author = "Mikyoung Lim and Sanghyeon Yu",

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AU - Lim, Mikyoung

AU - Yu, Sanghyeon

N1 - Funding Information: This research was supported by the Korean Ministry of Science, ICT and Future Planning through NRF grant Nos. 2013R1A1A3012931 and 2013003192 . Publisher Copyright: © 2014 Elsevier Inc.

PY - 2015/1/1

Y1 - 2015/1/1

N2 - We consider the conductivity problem in the presence of adjacent circular inclusions with constant conductivities. When two inclusions get closer and their conductivities degenerate to zero or infinity, the gradient of the solution can be arbitrary large. In this paper we derive an asymptotic formula of the solution, which characterizes the gradient blow-up of the solution in terms of conductivities of inclusions as well as the distance between inclusions. The asymptotic formula is expressed in bipolar coordinates in terms of the Lerch transcendent function, and it is valid for inclusions with arbitrary constant conductivities. We illustrate our results with numerical calculations.

AB - We consider the conductivity problem in the presence of adjacent circular inclusions with constant conductivities. When two inclusions get closer and their conductivities degenerate to zero or infinity, the gradient of the solution can be arbitrary large. In this paper we derive an asymptotic formula of the solution, which characterizes the gradient blow-up of the solution in terms of conductivities of inclusions as well as the distance between inclusions. The asymptotic formula is expressed in bipolar coordinates in terms of the Lerch transcendent function, and it is valid for inclusions with arbitrary constant conductivities. We illustrate our results with numerical calculations.

KW - Anti-plane elasticity

KW - Bipolar coordinates

KW - Conductivity equation

KW - Gradient blow-up

KW - Lerch transcendent function

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U2 - 10.1016/j.jmaa.2014.07.002

DO - 10.1016/j.jmaa.2014.07.002

M3 - Article

AN - SCOPUS:84925712303

SN - 0022-247X

VL - 421

SP - 131

EP - 156

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

ER -

Asymptotics of the solution to the conductivity equation in the presence of adjacent circular inclusions with finite conductivities

Abstract

Keywords

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