Uniqueness in inverse acoustic and electromagnetic scattering by penetrable obstacles with embedded objects

Abstract

This paper considers the inverse problem of scattering of time-harmonic acoustic and electromagnetic plane waves by a bounded, inhomogeneous, penetrable obstacle with embedded objects inside. A new method is proposed to prove that the inhomogeneous penetrable obstacle can be uniquely determined from the far-field pattern at a fixed frequency, disregarding its contents. Our method is based on constructing a well-posed interior transmission problem in a small domain associated with the Helmholtz or modified Helmholtz equation and the Maxwell or modified Maxwell equations. A key role is played by the smallness of the domain which ensures that the lowest transmission eigenvalue is large so that a given wave number k is not an eigenvalue of the interior transmission problem. Another ingredient in our proofs is a priori estimates of solutions to the transmission scattering problems with data in L p (1 < p < 2), which are established in this paper by using the integral equation method. A main feature of the new method is that it can deal with the acoustic and electromagnetic cases in a unified way and can be easily applied to deal with inverse scattering by unbounded rough interfaces.

A novel method in determining a layered periodic structure

“…In this paper we intend to develop a novel method, which differs from the approach used in [34], to prove the uniqueness on the identification of the penetrable periodic structure in the three-dimensional space from the measured data only above the structure with respect to a countably infinite number of quasiperiodic incident plane waves. The technique developed in this paper can date back to the work [27,36] on the inverse scattering problems of determining the support of penetrable electromagnetic obstacles or to [28] for the fluid-solid interaction problem of identifying the bounded solid obstacle, [29] for the cavity scattering case.…”

Locating a complex inhomogeneous medium with an approximate factorization method

“…In the current paper, we are interested in the inverse problem of reconstructing the shape and location of the inhomogeneous medium D from a knowledge of the far-field pattern u ∞ for incident plane waves. The uniqueness of this inverse problem has been established in [25] for the case when n is an unknown constant, in [23] for the case when n is an unknown piecewise constant, and in [11,16,24] for other related inverse medium scattering problems.…”

The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation

“…The first uniqueness result had been shown by Schiffer in acoustic waves with Dirichlet boundary condition ( [11], [30]). And then, there are many modified methods toward this direction for acoustic scattering of impenetrable obstacles with different types of boundary conditions ( [12], [16], [26], [32], [40], [41], [43]) and also the penetrable transmission problems ( [10], [15], [25], [33], [34], [35], [44], [45]). Due to the complexity of the problem, there are not many results about the uniqueness of elastic wave scattering problems.…”

“…the impenetrable scattering, then for our main problem, i.e. inverse transmission elastic scattering problem (see [39] and [44]).…”