On the geometric structures of transmission eigenfunctions with a conductive boundary condition and applications

Abstract

This paper is concerned with the intrinsic geometric structures of conductive transmission eigenfunctions. The geometric properties of interior transmission eigenfunctions were first studied in Blåsten, E., Liu, H. (2017). On vanishing near corners of transmission eigenfunctions. J. Funct. Anal. 273(11):3616–3632. It is shown in two scenarios that the interior transmission eigenfunction must be locally vanishing near a corner of the domain with an interior angle less than π. We significantly extend and generalize those results in several aspects. First, we consider the conductive transmission eigenfunctions which include the interior transmission eigenfunctions as a special case. The geometric structures established for the conductive transmission eigenfunctions in this paper include the results in Blåsten, E., Liu, H. (2017). On vanishing near corners of transmission eigenfunctions. J. Funct. Anal. 273(11):3616–3632 as a special case. Second, the vanishing property of the conductive transmission eigenfunctions is established for any corner as long as its interior angle is not π when the conductive transmission eigenfunctions satisfy certain Herglotz functions approximation properties. That means, as long as the corner singularity is not degenerate, the vanishing property holds if the underlying conductive transmission eigenfunctions can be approximated by a sequence of Herglotz functions under mild approximation rates. Third, the regularity requirements on the interior transmission eigenfunctions in Blåsten, E., Liu, H. (2017). On vanishing near corners of transmission eigenfunctions. J. Funct. Anal. 273(11):3616–3632 are significantly relaxed in the present study for the conductive transmission eigenfunctions. In order to establish the geometric properties for the conductive transmission eigenfunctions, we develop technically new methods and the corresponding analysis is much more complicated than that in Blåsten, E., Liu, H. (2017). On vanishing near corners of transmission eigenfunctions. J. Funct. Anal. 273(11):3616–3632. Finally, as an interesting and practical application of the obtained geometric results, we establish a unique recovery result for the inverse problem associated with the transverse electromagnetic scattering by a single far-field measurement in simultaneously determining a polygonal conductive obstacle and its surface conductive parameter.

Keywords:

• Conductive transmission eigenfunctions
• corner singularity
• geometric structures
• inverse scattering
• single far-field pattern
• uniqueness
• vanishing

2010 Mathematics Subject Classification:

• 35Q60
• 78A46 (primary)
• 35P25
• 78A05
• 81U40 (secondary)

Additional information

Funding

The work of H Diao was supported in part by the Fundamental Research Funds for the Central Universities under the grant 2412017FZ007. The work of H Liu was supported by a startup fund from City University of Hong Kong and the Hong Kong RGC grants (projects 12301218, 12302919 and 12301420). The authors would like to thank an anonymous referee for many constructive comments and suggestions, which have led to significant improvements on the results and the presentation of this paper.