A decomposition theorem of surface vector fields and spectral structure of the Neumann-Poincaré operator in elasticity
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- by Shota Fukushima, Yong-Gwan Ji and Hyeonbae Kang;
- Trans. Amer. Math. Soc. 377 (2024), 2065-2123
- DOI: https://doi.org/10.1090/tran/9078
- Published electronically: December 11, 2023
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Abstract:
We prove that the space of vector fields on the boundary of a bounded domain with the Lipschitz boundary in three dimensions is decomposed into three subspaces: elements of the first one extend to inside the domain as divergence-free and rotation-free vector fields, the second one to the outside as divergence-free and rotation-free vector fields, and the third one to both the inside and the outside as divergence-free harmonic vector fields. We then show that each subspace in the decomposition is infinite-dimensional. We also prove under a mild regularity assumption on the boundary that the decomposition is almost direct in the sense that any intersection of two subspaces is finite-dimensional. We actually prove that the dimension of intersection is bounded by the first Betti number of the boundary. In particular, if the boundary is simply connected, then the decomposition is direct. We apply this decomposition theorem to investigate spectral properties of the Neumann-Poincaré operator in elasticity, whose cubic polynomial is known to be compact. We prove that each linear factor of the cubic polynomial is compact on each subspace of decomposition separately and those subspaces characterize eigenspaces of the Neumann-Poincaré operator. We then prove all the results for three dimensions, decomposition of surface vector fields and spectral structure, are extended to higher dimensions. We also prove analogous but different results in two dimensions.References
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Bibliographic Information
- Shota Fukushima
- Affiliation: Department of Mathematics and Institute of Applied Mathematics, Inha University, Incheon 22212, S. Korea
- MR Author ID: 1571649
- Email: shota.fukushima.math@gmail.com
- Yong-Gwan Ji
- Affiliation: School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, S. Korea
- MR Author ID: 1170886
- Email: ygji@kias.re.kr
- Hyeonbae Kang
- Affiliation: Department of Mathematics and Institute of Applied Mathematics, Inha University, Incheon 22212, S. Korea
- MR Author ID: 268781
- Email: hbkang@inha.ac.kr
- Received by editor(s): December 6, 2022
- Received by editor(s) in revised form: July 31, 2023, and September 7, 2023
- Published electronically: December 11, 2023
- Additional Notes: This work was supported by NRF (of S. Korea) grants 2019R1A2B5B01069967 and 2022R1A2B5B01001445, and a KIAS Individual Grant (MG089001) at Korea Institute for Advanced Study.
- © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 377 (2024), 2065-2123
- MSC (2020): Primary 47A10; Secondary 31A10, 31B10, 35Q74
- DOI: https://doi.org/10.1090/tran/9078
- MathSciNet review: 4744751