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On vanishing near corners of conductive transmission eigenfunctions

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Abstract

This paper is concerned with the geometric structure of the transmission eigenvalue problem associated with a general conductive transmission condition. We prove that under a mild regularity condition in terms of the Herglotz approximations of one of the pair of the transmission eigenfunctions, the eigenfunctions must be vanishing around a corner on the boundary. The Herglotz approximation is the Fourier extension of the transmission eigenfunction, and the growth rate of the density function can be used to characterize the regularity of the underlying wave function. The geometric structures derived in this paper include the related results in Diao et al. (Commun Partial Differ Equ 46(4):630–679, 2021) and Blåsten and Liu (J Funct Anal 273:3616–3632, 2017) as special cases and verify that the vanishing around corners is a generic local geometric property of the transmission eigenfunctions.

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Acknowledgements

The work of Y. Deng was supported by NSF grant of China No. 11971487 and NSF grant of Hunan No. 2020JJ2038. The work of H Liu was supported by the Hong Kong RGC General Research Fund (Projects 12301420, 12302919, 11300821) and NSFC/RGC Joint Research Grant, N_CityU101/21.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, Hunan, China

    Youjun Deng & Chaohua Duan

  2. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, China

    Hongyu Liu

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  1. Youjun Deng
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  2. Chaohua Duan
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  3. Hongyu Liu
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Correspondence to Hongyu Liu.

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Deng, Y., Duan, C. & Liu, H. On vanishing near corners of conductive transmission eigenfunctions. Res Math Sci 9, 2 (2022). https://doi.org/10.1007/s40687-021-00299-8

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