For inverse scattering by a bounded inhomogeneous penetrable obstacle containing unknown embedded obstacles, Yang, Zhang and Zhang (J. Differential Equations 265 (2018) 6352-6383) proved that a fixed-frequency far-field pattern for all incident plane waves uniquely determines the obstacle support, disregarding its contents, in both the acoustic and electromagnetic settings. In the matched-parameter case (transmission coefficient λ=1\lambda=1 acoustically, λH=μ0/μ1=1\lambda_H=\mu_0/\mu_1=1 electromagnetically), where the transmission problem reduces to the inhomogeneous-medium scattering problem (with embedded objects acoustically, and without embedded obstacles in the source paper's electromagnetic formulation), their proof works only under the local non-degeneracy Assumption (A) that the contrast ∣n−1∣|n-1| is bounded below near the obstacle boundary, because the argument divides by n−1n-1 inside a small-domain interior transmission problem. Whether the (maximal) support of the inhomogeneity is uniquely determined at a single fixed frequency without this hypothesis remains open; the question is a fixed-energy support-identifiability problem for inhomogeneous media with embedded obstacles, and it may in principle be answered either by a uniqueness proof or by an explicit nonuniqueness (transparency) construction.

Background

In time-harmonic inverse scattering, an unknown object in a homogeneous background is probed with incident plane waves eikx⋅de^{ikx\cdot d} at wavenumber k>0k>0, and the scattered field is observed far away, where it has the asymptotic form of an outgoing spherical wave multiplied by the far-field pattern u∞(x^;d)u^\infty(\hat{x};d), a function of the observation direction x^\hat{x} and incident direction dd on the unit sphere. A penetrable (transmission) obstacle is a bounded domain D⊂R3D\subset\mathbb{R}^3 with C2C^2 boundary that the wave enters rather than reflects; across ∂D\partial D the total field and a weighted normal derivative are continuous, the weighting given by a transmission coefficient λ>0\lambda>0 describing the media on the two sides. Inside DD the material is characterized by a refractive index n(x)n(x) (a complex-valued L∞L^\infty function, allowing absorption, equal to 11 outside DD), and DD may contain embedded obstacles DbD_b, i.e. impenetrable cores on which a boundary condition BB is imposed: sound-soft (Dirichlet), sound-hard (Neumann), or impedance. The case λ=1\lambda=1 is called matched: the transmission conditions reduce to continuity of the field and its normal derivative, and the scattering problem becomes exactly the inhomogeneous-medium (medium) scattering problem for nn with embedded objects. In electromagnetics the analogous parameter is λH=μ0/μ1\lambda_H=\mu_0/\mu_1, the ratio of the magnetic permeabilities outside and inside, and λH=1\lambda_H=1 is the matched electromagnetic case, which the source paper formulates as the medium scattering problem without embedded obstacles (Db=∅D_b=\emptyset, n∈L∞(D)n\in L^\infty(D)).

A central question is identifiability: does the far-field pattern for all incident directions at one fixed frequency determine the obstacle? For λ≠1\lambda\neq 1, global uniqueness of the support DD has been known since 1990 through variational and integral-equation methods that construct singular point-source solutions whose blow-up at a common boundary point forces the two supports to coincide; the jump of λ\lambda across ∂D\partial D generates the required singularity. When λ=1\lambda=1 all of these arguments fail, and the classical approach through the interior transmission problem (ITP) - the problem of finding a pair of fields satisfying the Helmholtz equation with index nn and with index 11 in the whole obstacle domain, coupled through Cauchy data on its boundary - breaks down whenever kk is a transmission eigenvalue (a wavenumber at which this homogeneous coupled problem has a nontrivial solution). Yang, Zhang and Zhang circumvented this by posing the ITP only in a small auxiliary domain Ω\Omega strictly inside DD near a boundary point: by a result of Cakoni, Gintides and Haddar the lowest transmission eigenvalue of a domain tends to infinity as the domain shrinks, so for small Ω\Omega the wavenumber kk is not a transmission eigenvalue and the ITP is well posed. However, their well-posedness construction represents the ITP solution as U=[1/(n−1)](Δ+k2)wU=[1/(n-1)](\Delta+k^2)w, which requires the local non-degeneracy Assumption (A): there exist an open neighborhood O(∂D)⋐D∖Db‾\mathcal{O}(\partial D)\Subset D\setminus\overline{D_b} of ∂D\partial D and ε0>0\varepsilon_0>0 with ∣n(x)−1∣≥ε0|n(x)-1|\ge\varepsilon_0 a.e. there. Under (A) they prove uniqueness of DD (acoustic Theorem 2.9(ii), where the embedded obstacles DbD_b may be present; electromagnetic Theorem 3.10, proved for the matched medium scattering problem without embedded obstacles, which additionally requires n∈C1(D‾)n\in C^1(\overline{D})), disregarding the contents nn and DbD_b; for λ≠1\lambda\neq 1 (Theorem 2.9(i), Theorem 3.8) no assumption on nn is needed.

Assumption (A) is a genuine restriction: it forces the contrast n−1n-1 to stay bounded away from zero at the interface. Without it the question must be posed for the maximal support, the essential closure of {x:n(x)≠1}\{x:n(x)\neq 1\} together with the embedded obstacles DbD_b (which scatter by themselves, independently of nn), since if n≡1n\equiv 1 on an open collar inside ∂D\partial D - or on an open neighborhood of Db‾\overline{D_b} inside D∖Db‾D\setminus\overline{D_b} - the far field cannot distinguish DD from the smaller maximal support. For Db=∅D_b=\emptyset the resulting question - whether fixed-frequency far-field data determines the support of a general inhomogeneous medium in three dimensions without sign-type conditions - is a recognized open problem of inverse scattering theory, closely tied to the fixed-energy uniqueness problem for the three-dimensional Schrödinger equation. What is known without sign conditions concerns only special structure: interfaces of piecewise constant media can be reconstructed by complex geometrical optics (CGO) solutions, that is, special high-frequency-type solutions with oscillating complex phases used to probe interior discontinuities (Nagayasu, Uhlmann and Wang); convex polyhedral supports and corner or edge singularities are identifiable from the absence of non-scattering (Elschner and Hu; Hu, Salo and Vesalainen, who recover a convex polygon from a single far-field pattern); and for smooth compactly supported nn in the Maxwell case without embedded obstacles, uniqueness of the full index nn is classical (Colton and Päivärinta). None of these covers a general inhomogeneity whose contrast degenerates at the interface, least of all with unknown embedded obstacles present.

Problem Statement

Let D⊂R3D\subset\mathbb{R}^3 be a bounded domain with C2C^2 boundary, let Db⊂DD_b\subset D be unknown embedded obstacles on which one of the boundary conditions BB (sound-soft, sound-hard, or impedance with nonnegative imaginary part Im(β)≥0\mathrm{Im}(\beta)\ge 0 of the impedance coefficient β\beta) is imposed, and let n∈L∞(D∖Db‾)n\in L^\infty(D\setminus\overline{D_b}) be a complex-valued refractive index with n=1n=1 outside DD. In the matched-parameter case, acoustically λ=1\lambda=1 and electromagnetically λH=μ0/μ1=1\lambda_H=\mu_0/\mu_1=1, the transmission scattering problem coincides with the corresponding inhomogeneous-medium scattering problem: with embedded objects in the acoustic case, and, as formulated in the source paper, without embedded obstacles in the electromagnetic case (so Db=D~b=∅D_b=\tilde{D}_b=\emptyset and n∈L∞(D)n\in L^\infty(D) in the electromagnetic branch). Assume the far-field patterns u∞(x^;d)u^\infty(\hat{x};d) for all observation directions x^∈S2\hat{x}\in S^2 and all incident plane-wave directions d∈S2d\in S^2 (for Maxwell, additionally all polarizations) at one fixed wavenumber k>0k>0 coincide for two such configurations (D,n,Db)(D,n,D_b) and (D~,n~,D~b)(\tilde{D},\tilde{n},\tilde{D}_b), both taken to be maximal in the sense that each support is the essential closure of the union of the set where the refractive index differs from 11 and the embedded obstacles DbD_b (so that n≡1n\equiv 1 on an open collar inside ∂D\partial D, and likewise on an open neighborhood of Db‾\overline{D_b} inside D∖Db‾D\setminus\overline{D_b}, is excluded by convention; the second condition is vacuous when Db=∅D_b=\emptyset). Must D=D~D=\tilde{D}? Prove this uniqueness without any hypothesis of the form of Assumption (A) of Yang, Zhang and Zhang - that is, without assuming ∣n−1∣≥ε0>0|n-1|\ge\varepsilon_0>0 in a neighborhood of ∂D\partial D - allowing the contrast to degenerate or vanish on approach to the interface, for the acoustic case, the electromagnetic case, or both. Alternatively, disprove it by an explicit construction or rigorous existence proof of two distinct maximal-support configurations (with possibly different embedded obstacles and contents) whose fixed-frequency far-field patterns coincide for all incident plane waves. The special case Db=∅D_b=\emptyset is the classical open question whether fixed-frequency far-field data in three dimensions determines the support of a general inhomogeneous medium without sign-type conditions; the embedded-obstacle setting of the source paper's matched acoustic theorem (Theorem 2.9(ii)) is the full target of the acoustic branch, while the electromagnetic branch is the matched medium scattering problem of Theorem 3.10, which carries no embedded obstacles.

References

  1. Jiaqing Yang, Bo Zhang, Haiwen Zhang, Uniqueness in inverse acoustic and electromagnetic scattering by penetrable obstacles with embedded objects, Journal of Differential Equations 265 (2018) 6352-6383. DOI: 10.1016/j.jde.2018.07.033, https://doi.org/10.1016/j.jde.2018.07.033 (preprint arXiv:1305.0917, https://arxiv.org/abs/1305.0917)
  2. Fioralba Cakoni, Drossos Gintides, Houssem Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM Journal on Mathematical Analysis 42 (2010) 237-255. DOI: 10.1137/090769338, https://doi.org/10.1137/090769338
  3. David Colton, Peter Monk, Jiguang Sun, Analytical and computational methods for transmission eigenvalues, Inverse Problems 26 (2010) 045011. DOI: 10.1088/0266-5611/26/4/045011, https://doi.org/10.1088/0266-5611/26/4/045011
  4. David Colton, Lassi Päivärinta, The uniqueness of a solution to an inverse scattering problem for electromagnetic waves, Archive for Rational Mechanics and Analysis 119 (1992) 59-70. DOI: 10.1007/BF00376010, https://doi.org/10.1007/BF00376010
  5. Johannes Elschner, Guanghui Hu, Corners and edges always scatter, Inverse Problems 31 (2015) 015003. DOI: 10.1088/0266-5611/31/1/015003, https://doi.org/10.1088/0266-5611/31/1/015003
  6. Guanghui Hu, Mikko Salo, Esa V. Vesalainen, Shape Identification in Inverse Medium Scattering Problems with a Single Far-Field Pattern, SIAM Journal on Mathematical Analysis 48 (2016) 152-165. DOI: 10.1137/15M1032958, https://doi.org/10.1137/15M1032958
  7. Sei Nagayasu, Gunther Uhlmann, Jenn-Nan Wang, Reconstruction of Penetrable Obstacles in Acoustic Scattering, SIAM Journal on Mathematical Analysis 43 (2011) 189-211. DOI: 10.1137/09076218X, https://doi.org/10.1137/09076218X
  8. Jiaqing Yang, Meng Ding, Keji Liu, On the uniqueness of inverse problems for the reduced wave equation with unknown embedded obstacles, Inverse Problems 37 (2021) 125009. DOI: 10.1088/1361-6420/ac3711, https://doi.org/10.1088/1361-6420/ac3711
  9. Yu Jia, Jiaqing Yang, Determination of embedded obstacle and its surrounding medium in electrodynamics, Inverse Problems 41 (2025) 065008. DOI: 10.1088/1361-6420/add75e, https://doi.org/10.1088/1361-6420/add75e
  10. Chengyu Wu, Jiaqing Yang, On uniqueness of inverse conductive scattering problem with unknown embedded obstacles, Journal of Differential Equations 459 (2026) 114101. DOI: 10.1016/j.jde.2026.114101, https://doi.org/10.1016/j.jde.2026.114101 (preprint arXiv:2412.12452, https://arxiv.org/abs/2412.12452)
  11. Jiaqing Yang, Jianliang Li, Bo Zhang, Simultaneous recovery of a locally rough interface and the embedded obstacle with its surrounding medium, Inverse Problems 38 (2022) 045011. DOI: 10.1088/1361-6420/ac572a, https://doi.org/10.1088/1361-6420/ac572a
  12. Chun Liu, Jiaqing Yang, Bo Zhang, Elastic scattering problems by penetrable obstacles with embedded objects, preprint (2025). arXiv:2512.03624, https://arxiv.org/abs/2512.03624
  13. Yulong Lu, Bo Zhang, Direct and inverse scattering problem by an unbounded rough interface with buried obstacles, preprint (2016). arXiv:1610.03515, https://arxiv.org/abs/1610.03515
  14. Fenglong Qu, Jiaqing Yang, Bo Zhang, An approximate factorization method for inverse medium scattering with unknown buried objects, Inverse Problems 33 (2017) 035007. DOI: 10.1088/1361-6420/aa58d8, https://doi.org/10.1088/1361-6420/aa58d8
  15. Fenglong Qu, Yubo Wang, Yanli Cui, Xiliang Li, Simultaneous identification of a two-layered medium and the embedded obstacle with Maxwell's coefficients, Journal of Differential Equations 469 (2026) 114406. DOI: 10.1016/j.jde.2026.114406, https://doi.org/10.1016/j.jde.2026.114406