JournalsaihpcVol. 36, No. 7pp. 1817–1828

Spectral structure of the Neumann–Poincaré operator on tori

  • Kazunori Ando

    Department of Electrical and Electronic Engineering and Computer Science, Ehime University, Ehime 790-8577, Japan

  • Yong-Gwan Ji

    Department of Mathematics and Institute of Applied Mathematics, Inha University, Incheon 22212, South Korea

  • Hyeonbae Kang

    Department of Mathematics and Institute of Applied Mathematics, Inha University, Incheon 22212, South Korea

  • Daisuke Kawagoe

    Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan

  • Yoshihisa Miyanishi

    Center for Mathematical Modeling and Data Science, Osaka University, Osaka 560-8531, Japan
Spectral structure of the Neumann–Poincaré operator on tori cover

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Abstract

We address the question whether there is a three-dimensional bounded domain such that the Neumann–Poincaré operator defined on its boundary has infinitely many negative eigenvalues. It is proved in this paper that tori have such a property. It is done by decomposing the Neumann–Poincaré operator on tori into infinitely many self-adjoint compact operators on a Hilbert space defined on the circle using the toroidal coordinate system and the Fourier basis, and then by proving that the numerical range of infinitely many operators in the decomposition has both positive and negative values.

Cite this article

Kazunori Ando, Yong-Gwan Ji, Hyeonbae Kang, Daisuke Kawagoe, Yoshihisa Miyanishi, Spectral structure of the Neumann–Poincaré operator on tori. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 7, pp. 1817–1828

DOI 10.1016/J.ANIHPC.2019.05.002