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This paper is concerned with the intrinsic geometric structures of conductivetransmission eigenfunctions. The geometric properties of interior transmissioneigenfunctions were first studied in [9]. It is shown in two scenarios that theinterior transmission eigenfunction must be locally vanishing near a corner ofthe domain with an interior angle less than $\pi$. We significantly extend andgeneralize those results in several aspects. First, we consider the conductivetransmission eigenfunctions which include the interior transmissioneigenfunctions as a special case. The geometric structures established for theconductive transmission eigenfunctions in this paper include the results in [9]as a special case. Second, the vanishing property of the conductivetransmission eigenfunctions is established for any corner as long as itsinterior angle is not $\pi$ when the conductive transmission eigenfunctionssatisfy certain Herglotz functions approximation properties. That means, aslong as the corner singularity is not degenerate, the vanishing property holdsif the underlying conductive transmission eigenfunctions can be approximated bya sequence of Herglotz functions under mild approximation rates. Third, theregularity requirements on the interior transmission eigenfunctions in [9] aresignificantly relaxed in the present study for the conductive transmissioneigenfunctions. Finally, as an interesting and practical application of theobtained geometric results, we establish a unique recovery result for theinverse problem associated with the transverse electromagnetic scattering by asingle far-field measurement in simultaneously determining a polygonalconductive obstacle and its surface conductive parameter.

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