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Decay Rate of the Eigenvalues of the Neumann-Poincaré Operator

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Abstract

If the boundary of a domain in three dimensions is smooth enough, then the decay rate of the eigenvalues of the Neumann-Poincaré operator is known and it is optimal. In this paper, we deal with domains with less regular boundaries and derive quantitative estimates for the decay rates of the Neumann-Poincaré eigenvalues in terms of the Hölder exponent of the boundary. Estimates in particular show that the less the regularity of the boundary is, the slower is the decay of the eigenvalues. We also prove that the similar estimates in two dimensions. The estimates are not only for less regular boundaries for which the decay rate was unknown, but also for regular ones for which the result of this paper makes a significant improvement over known results.

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Acknowledgements

The authors deeply appreciate valuable comments of anonymous referee(s).

Funding

H. Kang is supported by a grant from National Research Foundation of Korea, No. 2022R1A2B5B01001445. Y. Miyanishi is supported by KAKENHI grant from Japan Society for the Promotion of Science, No. 21K13805 and No. 20K03655.

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Authors and Affiliations

  1. Department of Mathematics and Institute of Applied Mathematics, Inha University, 100 Inha-ro, Incheon, 22212, South Korea

    Shota Fukushima & Hyeonbae Kang

  2. Department of Mathematical Sciences, Faculty of Science, Shinshu University, Asahi 3-1-1, Matsumoto, 390-8621, Japan

    Yoshihisa Miyanishi

Authors
  1. Shota Fukushima
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  2. Hyeonbae Kang
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  3. Yoshihisa Miyanishi
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Contributions

S. Fukushima and H. Kang wrote the main manuscript text. Y. Miyanishi proposed the problem and provided the idea of considering the iteration of integral operators, which is crucial in our paper. All authors reviewed the manuscript.

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Correspondence to Shota Fukushima.

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Fukushima, S., Kang, H. & Miyanishi, Y. Decay Rate of the Eigenvalues of the Neumann-Poincaré Operator. Potential Anal (2023). https://doi.org/10.1007/s11118-023-10120-6

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