Spectral structure of the Neumann–Poincaré operator on tori
Ando, Kazunori1 ; Ji, Yong-Gwan2 ; Kang, Hyeonbae2 ; Kawagoe, Daisuke3 ; Miyanishi, Yoshihisa4
1 Department of Electrical and Electronic Engineering and Computer Science, Ehime University, Ehime 790-8577, Japan
2 Department of Mathematics and Institute of Applied Mathematics, Inha University, Incheon 22212, South Korea
3 Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
4 Center for Mathematical Modeling and Data Science, Osaka University, Osaka 560-8531, Japan
Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 7, pp. 1817-1828.
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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.

DOI : 10.1016/j.anihpc.2019.05.002
Classification : 47A45, 31B25
Mots-clés : Neumann-Poincaré operator, Negative eigenvalues, Tori, Stationary phase method
Ando, Kazunori 1 ; Ji, Yong-Gwan 2 ; Kang, Hyeonbae 2 ; Kawagoe, Daisuke 3 ; Miyanishi, Yoshihisa 4

1 Department of Electrical and Electronic Engineering and Computer Science, Ehime University, Ehime 790-8577, Japan
2 Department of Mathematics and Institute of Applied Mathematics, Inha University, Incheon 22212, South Korea
3 Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
4 Center for Mathematical Modeling and Data Science, Osaka University, Osaka 560-8531, Japan 
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     title = {Spectral structure of the {Neumann{\textendash}Poincar\'e} operator on tori},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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Ando, Kazunori; Ji, Yong-Gwan; Kang, Hyeonbae; Kawagoe, Daisuke; Miyanishi, Yoshihisa. Spectral structure of the Neumann–Poincaré operator on tori. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 7, pp. 1817-1828. doi : 10.1016/j.anihpc.2019.05.002. http://www.numdam.org/articles/10.1016/j.anihpc.2019.05.002/

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