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A decay estimate for the eigenvalues of the Neumann-Poincaré operator using the Grunsky coefficients


Authors: YoungHoon Jung and Mikyoung Lim
Journal: Proc. Amer. Math. Soc. 148 (2020), 591-600
MSC (2010): Primary 35J05, 30C35, 35P15
DOI: https://doi.org/10.1090/proc/14785
Published electronically: October 18, 2019
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Abstract: We consider the decay property of the eigenvalues of the Neumann-Poincaré operator in two dimensions. As is well known, this operator admits only a sequence of eigenvalues that accumulates to zero as its spectrum for a bounded domain having $ C^{1,\alpha }$ boundary with $ \alpha \in (0,1)$. We show that the eigenvalues $ \lambda _k$ of the Neumann-Poincaré operator ordered by size satisfy that $ \vert\lambda _k\vert = O(k^{-p-\alpha +1/2})$ for an arbitrary simply connected domain having $ C^{1+p,\alpha }$ boundary with $ p\geq 0,~ \alpha \in (0,1)$, and $ p+\alpha >\frac {1}{2}$.


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Additional Information

YoungHoon Jung
Affiliation: Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea
Email: hapy1010@kaist.ac.kr

Mikyoung Lim
Affiliation: Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea
Email: mklim@kaist.ac.kr

DOI: https://doi.org/10.1090/proc/14785
Received by editor(s): February 13, 2019
Published electronically: October 18, 2019
Additional Notes: This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIT) (No. 2016R1A2B4014530 and No. 2019R1F1A1062782)
The second author is the corresponding author
Communicated by: Ariel Barton
Article copyright: © Copyright 2019 American Mathematical Society