The inverse scattering problem for a conductive boundary condition and transmission eigenvalues

ABSTRACT

In this paper, we consider the inverse scattering problem associated with an inhomogeneous media with a conductive boundary. In particular, we are interested in two problems that arise from this inverse problem: the inverse conductivity problem and the corresponding interior transmission eigenvalue problem. The inverse conductivity problem is to recover the conductive boundary parameter from the measured scattering data. We prove that the measured scatted data uniquely determine the conductivity parameter as well as describe a direct algorithm to recover the conductivity. The interior transmission eigenvalue problem is an eigenvalue problem associated with the inverse scattering of such materials. We investigate the convergence of the eigenvalues as the conductivity parameter tends to zero as well as prove existence and discreteness for the case of an absorbing media. Lastly, several numerical and analytical results support the theory and we show that the inside–outside duality method can be used to reconstruct the interior conductive eigenvalues.

COMMUNICATED BY:

• Jiguang Sun

KEYWORDS:

• Inverse scattering
• conductive boundary condition
• transmission eigenvalues

AMS SUBJECT CLASSIFICATIONS:

• 35P25
• 35P30
• 78A46

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

A. Kleefeld  http://orcid.org/0000-0001-8324-821X