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 [9]. 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 [9] 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. That means, as long as the corner singularity is not degenerate, the vanishing property holds. Third, the regularity requirements on the interior transmission eigenfunctions in [9] 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 [9]. 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.

On an electromagnetic problem in a corner and its applications

“…That is, the assertion of non-invisibility mainly comes from the "strong" radiating nature of the corner which is independent of the other parts of the scatterer. This is also in consistence with the corresponding studies in the literature for the acoustic case [4,9,10,19,44]. However in Maxwell scattering one does not have H 2 -or C α -smoothness a-priori.…”

“…where The study of the geometric structures of transmission eigenfunctions was initiated in [6] and then further developed in [2,8,19]. However, in all of the aforementioned literature, the transmission eigenvalue problems are associated to the Helmholtz system that arises from the time-harmonic acoustic scattering.…”

On local and global structures of transmission eigenfunctions and beyond

“…Hence, the regularity requirement (4.12) is fulfilled or not critically depends on the singular parts in the decomposition (4.13). On the other hand, the singular parts indeed belong to C α (S h ) if u, v are sufficiently regular on Γ h = ∂S h ∩ ∂Ω; see [35] for the relevant discussion. However, in the transmission conditions on Γ h in (4.10), the values of u| Γ h , v| Γ h are not a-priori specified.…”

“…Hence, the vanishing property of u/v can serve as an indicator of the regularity of u, v on Γ h . The regularity point discussed above was first explored in [35]. As shown in [35], the regularity requirement in (4.12) is a physical condition since when applying the vanishing property to inverse problems or invisibility problems associated with the physical scattering system (2.2), such a regularity requirement can always be fulfilled.…”

“…The regularity point discussed above was first explored in [35]. As shown in [35], the regularity requirement in (4.12) is a physical condition since when applying the vanishing property to inverse problems or invisibility problems associated with the physical scattering system (2.2), such a regularity requirement can always be fulfilled. On the other hand, it is numerically shown in [5] that generically, the transmission eigenfunctions possess the vanishing property near a corner.…”