Spectral properties of the Neumann-Poincaré operator and uniformity of estimates for the conductivity equation with complex coefficients

Hyeonbae Kang, Kyoungsun Kim, Hyundae Lee, Jaemin Shin, Sanghyeon Yu

Research output: Contribution to journalArticlepeer-review

21 Citations (Scopus)

Abstract

We consider well-posedness of the boundary value problem in the presence of an inclusion with complex conductivity k. We first consider the transmission problem in {R}d and characterize solvability of the problem in terms of the spectrum of the Neumann-Poincaré (NP) operator. We then deal with the boundary value problem and show that the solution is bounded in its H-1-norm uniformly in k as long as k is at some distance from a closed interval in the negative real axis. We then show with an estimate that the solution depends on k in its H-1-norm Lipschitz continuously. We finally show that the boundary perturbation formula in the presence of a diametrically small inclusion is valid uniformly in k away from the closed interval mentioned before. The results for the single inclusion case are extended to the case when there are multiple inclusions with different complex conductivities: We first obtain a complete characterization of solvability when inclusions consist of two disjoint disks, and then prove solvability and uniform estimates when imaginary parts of conductivities have the same signs. The results are obtained using the spectral property of the associated NP operator and the spectral resolution.

Original languageEnglish
Pages (from-to)519-545
Number of pages27
JournalJournal of the London Mathematical Society
Volume93
Issue number2
DOIs
Publication statusPublished - 2016 Apr 1
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)

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