Abstract
We consider the spectral structure of the Neumann–Poincaré operators defined on the boundaries of thin domains of rectangular shape in two dimensions. We prove that as the aspect ratio of the domains tends to ∞, or equivalently, as the domains get thinner, the spectra of the Neumann–Poincaré operators are densely distributed in [−1/2, 1/2], the interval which contains the spectrum of Neumann–Poincaré operators.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. V. Ahlfors, Remarks on the Neumann–Poincaré integral equation, Pacific J. Math. 3 (1952), 271–280.
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin–Heidelberg, 2001.
J. Helsing, H. Kang and M. Lim, Classification of spectra of the Neumann–Poincaré operator on planar domains with corners by resonance, Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), 991–1011.
D. Khavinson, M. Putinar and H. S. Shapiro, Poincaré’s variational problem in potential theory, Arch. Ration. Mech. An. 185 (2007), 143–184.
K. M. Perfekt and M. Putinar, The essential spectrum of the Neumann–Poincaré operator on a domain with corners, Arch. Ration. Mech. An. 223 (2017), 1019–1033.
K. Yosida, Functional Analysis, Springer, Berlin, 1974.
Acknowledgement
We thank the anonymous referee for helpful comments on this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by NRF (of S. Korea) grants No. 2019R1A2B5B01069967 and by JSPS (of Japan) KAKENHI Grant Number JP19K14553, 20K03655 and 21K13805.
Rights and permissions
About this article
Cite this article
Ando, K., Kang, H. & Miyanishi, Y. Spectral structure of the Neumann–Poincaré operator on thin domains in two dimensions. JAMA 146, 791–800 (2022). https://doi.org/10.1007/s11854-022-0206-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-022-0206-7