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 journalArticlepeer-review

8 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 languageEnglish
Pages (from-to)131-156
Number of pages26
JournalJournal of Mathematical Analysis and Applications
Volume421
Issue number1
DOIs
Publication statusPublished - 2015 Jan 1
Externally publishedYes

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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