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On vanishing near corners of conductive transmission eigenfunctions

On vanishing near corners of conductive transmission eigenfunctions 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

On vanishing near corners of conductive transmission eigenfunctions

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Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021
eISSN
2197-9847
DOI
10.1007/s40687-021-00299-8
Publisher site
See Article on Publisher Site

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.

Journal

Research in the Mathematical SciencesSpringer Journals

Published: Mar 1, 2022

Keywords: Conductive transmission eigenfunctions; Corner singularity; Geometric structures; Vanishing; Herglotz approximation; Primary 35P25; 35R30; Secondary 58J05; 78A05

References