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.