Inverse source problems in an inhomogeneous medium with a single far-field pattern

Abstract

This paper concerns time-harmonic inverse source problems with a single far-field pattern in two dimensions, where the source term is compactly supported in an a priori given inhomogeneous background medium. For convex-polygonal source terms, we prove that the source support together with the zeroth and first order derivatives of the source function at corner points can be uniquely determined. Further, we prove that an admissible set of source functions (including harmonic functions) having a convex-polygonal support can be uniquely identified by a single far-field pattern. A class of radiating sources is characterized and the extension of the radiated field across a corner point is proven impossible. Our arguments are motivated by the uniqueness proof to inverse medium scattering from convex-polygonal penetrable scatterers [Elschner & Hu, Archive for Rational Mechanics and Analysis, 228 (2018): 653-690] which depends on the singularity analysis of solutions in a planer corner domain.

Uniqueness for time-dependent inverse problems with single dynamical data

“…Noting f (O) = 0 at O, we know that the lowest order expansion of hf is harmonic. By using Lemma 2.3 in [22], we get h(x)f (O) = 0 for x ∈ Σ, which implies h ≡ 0, hence completes the proof of Lemma 4.1. Lemma 4.2.…”

“…By the assumption of c j , we know h ∈ S(A, b). Recalling the result in [22], we know the lowest order expansion of h near O is harmonic. Furthermore, by Lemma 4.2 we know the existence of s 0 > 0 such that u 2 (O, s 0 ) = 0.…”

“…In the time-harmonic regime, the condition (iii) has already been used in [23] to prove uniqueness in recovering the support of the contrast function. For linear inverse source problems, the condition (ii) guarantees the unique identification of a source term having a convex-polygonal support (see [22]). The above Theorem 2.2 verifies that both the support and the refractive index can be uniquely identified under the conditions (i)-(iii).…”

“…by means of the data along all the time, we prove in the theory that the Laplace transform can be applied to the measurement data and the time-dependent inverse problem is thus transformed to an equivalent problem in the Fourier-Laplace domain with many parameters (frequencies). To apply the recently developed shape identification theory [13,22] to inverse coefficient problems in a corner domain, we shall show that there exists at least one parameter (frequency) for which the Laplace-transformed solution cannot vanish at the corner points (see Lemma 4.1, also [23]). We will then prove via asymptotic analysis that this parameter in two dimensions can be taken sufficiently small for wave equations and sufficiently large for the Schrödinger equation (see Lemmas 4.2 and 5.1).…”

“…We will then prove via asymptotic analysis that this parameter in two dimensions can be taken sufficiently small for wave equations and sufficiently large for the Schrödinger equation (see Lemmas 4.2 and 5.1). We refer to [5,12,13,21,20,27,16,25,24,23,22,33] for related works on target identification with a single measurement data in the stationary case.…”