Uniqueness in the matched-parameter regime for inverse scattering by penetrable obstacles with embedded objects without local non-degeneracy of the refractive index

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 $\lambda=1$ acoustically, $\lambda_H=\mu_0/\mu_1=1$ 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 $|n-1|$ is bounded below near the obstacle boundary, because the argument divides by $n-1$ 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 $e^{ikx\cdot d}$ at wavenumber $k>0$, 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 $u^\infty(\hat{x};d)$, a function of the observation direction $\hat{x}$ and incident direction $d$ on the unit sphere. A penetrable (transmission) obstacle is a bounded domain $D\subset\mathbb{R}^3$ with $C^2$ boundary that the wave enters rather than reflects; across $\partial D$ the total field and a weighted normal derivative are continuous, the weighting given by a transmission coefficient $\lambda>0$ describing the media on the two sides. Inside $D$ the material is characterized by a refractive index $n(x)$ (a complex-valued $L^\infty$ function, allowing absorption, equal to $1$ outside $D$), and $D$ may contain embedded obstacles $D_b$, i.e. impenetrable cores on which a boundary condition $B$ is imposed: sound-soft (Dirichlet), sound-hard (Neumann), or impedance. The case $\lambda=1$ 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 $n$ with embedded objects. In electromagnetics the analogous parameter is $\lambda_H=\mu_0/\mu_1$, the ratio of the magnetic permeabilities outside and inside, and $\lambda_H=1$ is the matched electromagnetic case, which the source paper formulates as the medium scattering problem without embedded obstacles ($D_b=\emptyset$, $n\in L^\infty(D)$).

A central question is identifiability: does the far-field pattern for all incident directions at one fixed frequency determine the obstacle? For $\lambda\neq 1$, global uniqueness of the support $D$ 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 $\lambda$ across $\partial D$ generates the required singularity. When $\lambda=1$ 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 $n$ and with index $1$ in the whole obstacle domain, coupled through Cauchy data on its boundary - breaks down whenever $k$ 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 $\Omega$ strictly inside $D$ 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 $\Omega$ the wavenumber $k$ is not a transmission eigenvalue and the ITP is well posed. However, their well-posedness construction represents the ITP solution as $U=[1/(n-1)](\Delta+k^2)w$, which requires the local non-degeneracy Assumption (A): there exist an open neighborhood $\mathcal{O}(\partial D)\Subset D\setminus\overline{D_b}$ of $\partial D$ and $\varepsilon_0>0$ with $|n(x)-1|\ge\varepsilon_0$ a.e. there. Under (A) they prove uniqueness of $D$ (acoustic Theorem 2.9(ii), where the embedded obstacles $D_b$ may be present; electromagnetic Theorem 3.10, proved for the matched medium scattering problem without embedded obstacles, which additionally requires $n\in C^1(\overline{D})$), disregarding the contents $n$ and $D_b$; for $\lambda\neq 1$ (Theorem 2.9(i), Theorem 3.8) no assumption on $n$ is needed.

Assumption (A) is a genuine restriction: it forces the contrast $n-1$ to stay bounded away from zero at the interface. Without it the question must be posed for the maximal support, the essential closure of $\{x:n(x)\neq 1\}$ together with the embedded obstacles $D_b$ (which scatter by themselves, independently of $n$), since if $n\equiv 1$ on an open collar inside $\partial D$ - or on an open neighborhood of $\overline{D_b}$ inside $D\setminus\overline{D_b}$ - the far field cannot distinguish $D$ from the smaller maximal support. For $D_b=\emptyset$ 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 $n$ in the Maxwell case without embedded obstacles, uniqueness of the full index $n$ 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 $D\subset\mathbb{R}^3$ be a bounded domain with $C^2$ boundary, let $D_b\subset D$ be unknown embedded obstacles on which one of the boundary conditions $B$ (sound-soft, sound-hard, or impedance with nonnegative imaginary part $\mathrm{Im}(\beta)\ge 0$ of the impedance coefficient $\beta$) is imposed, and let $n\in L^\infty(D\setminus\overline{D_b})$ be a complex-valued refractive index with $n=1$ outside $D$. In the matched-parameter case, acoustically $\lambda=1$ and electromagnetically $\lambda_H=\mu_0/\mu_1=1$, 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 $D_b=\tilde{D}_b=\emptyset$ and $n\in L^\infty(D)$ in the electromagnetic branch). Assume the far-field patterns $u^\infty(\hat{x};d)$ for all observation directions $\hat{x}\in S^2$ and all incident plane-wave directions $d\in S^2$ (for Maxwell, additionally all polarizations) at one fixed wavenumber $k>0$ coincide for two such configurations $(D,n,D_b)$ and $(\tilde{D},\tilde{n},\tilde{D}_b)$, 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 $1$ and the embedded obstacles $D_b$ (so that $n\equiv 1$ on an open collar inside $\partial D$, and likewise on an open neighborhood of $\overline{D_b}$ inside $D\setminus\overline{D_b}$, is excluded by convention; the second condition is vacuous when $D_b=\emptyset$). Must $D=\tilde{D}$? Prove this uniqueness without any hypothesis of the form of Assumption (A) of Yang, Zhang and Zhang - that is, without assuming $|n-1|\ge\varepsilon_0>0$ in a neighborhood of $\partial D$ - 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 $D_b=\emptyset$ 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

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