The 2018/2019 work of Klibanov and Kolesov developed a globally convergent 'convexification' method for the 3-D coefficient inverse scattering problem (CISP): recover the spatially varying dielectric coefficient c(x)c(\mathbf{x}) of the Helmholtz equation Δu+k2c(x)u=0\Delta u+k^{2}c(\mathbf{x})u=0 in R3\mathbb{R}^{3} from the total field measured on the illuminated (backscattering) face Γ\Gamma of a bounded domain, produced by a single incident plane wave over an interval of wavenumbers. The method's global-convergence theorems explicitly assume, but do not prove, that this data determines cc uniquely — the authors state that uniqueness of this CISP 'is an open and long standing problem'. The problem remains open: all known uniqueness theorems for multi-frequency backscattering-type data require extra structure (an open set of incident directions, two opposite directions, closeness to a constant coefficient, small or time-dependent potentials, or impenetrable obstacles). The problem asks for a rigorous proof that the stated single-direction multi-frequency backscattering data determines cc uniquely within the admissible class, or an explicit counterexample.

Background

In inverse scattering theory one probes an unknown medium with waves and tries to recover its material coefficient. In the acoustic/dielectric model studied here, the total field u(x,k)u(\mathbf{x},k) satisfies the Helmholtz equation Δu+k2c(x)u=0in R3,\Delta u+k^{2}c(\mathbf{x})u=0\quad\text{in }\mathbb{R}^{3}, where k>0k>0 is the wavenumber and c(x)c(\mathbf{x}) is the spatially distributed dielectric constant (the squared refractive index; in acoustics, the squared reciprocal sound speed). The medium is illuminated from below by the plane wave ui(x,k)=eikzu_{i}(\mathbf{x},k)=e^{ikz} traveling along the positive zz-axis; the scattered field us=u−uiu_{s}=u-u_{i} satisfies the Sommerfeld radiation condition, which selects the physically outgoing solution at infinity. The coefficient equals the vacuum value outside a bounded domain, c=1+βc=1+\beta with β≥0\beta\geq 0 compactly supported in Ω‾\overline{\Omega}, so all scattering is due to the inhomogeneity inside Ω\Omega.

The measurement model of Klibanov and Kolesov (arXiv:1801.04404; Computers & Mathematics with Applications, 2019) is the monostatic radar / ground-penetrating radar geometry: only the backscattered total field is measured, on the single illuminated face Γ={z=−ξ, ∣x∣,∣y∣<b}\Gamma=\{z=-\xi,\,|x|,|y|<b\} of the box Ω={∣x∣,∣y∣<b, −ξ<z<d}\Omega=\{|x|,|y|<b,\,-\xi<z<d\}, for wavenumbers kk in an interval [k‾,k‾]⊂(0,∞)[\underline{k},\overline{k}]\subset(0,\infty). Both the incident direction and the measurement surface are fixed; there is a single measurement event. The paper built a 'convexification' numerical method for this CISP: a Tikhonov-like minimization functional weighted by a Carleman Weight Function — the exponential weight appearing in a Carleman estimate, which is an a priori estimate for a PDE in a weighted norm and is the classical tool for proving uniqueness and unique continuation — made strictly convex on a ball of a Hilbert space, so that the minimizer can be reached from any first guess (global convergence). The analytical theorems require strong regularity of the coefficient, c∈C15(R3)c\in C^{15}(\mathbb{R}^{3}), used to justify the high-frequency (k→∞k\to\infty) asymptotic behavior of the field, and a travel-time condition: with respect to the Riemannian metric dτ=c(x) ∣dx∣d\tau=\sqrt{c(\mathbf{x})}\,|d\mathbf{x}|, every point is connected to the plane {z=−a}\{z=-a\} below the domain by a unique geodesic.

The central theoretical gap is stated plainly in the source: uniqueness of the solution of this CISP is an open and long-standing problem, and uniqueness of a similar coefficient inverse problem is currently provable only when the governing equation has a right-hand side that does not vanish in Ω‾\overline{\Omega} (via the Bukhgeim–Klibanov method, the Carleman-estimate technique for coefficient inverse problems); the present equation is homogeneous, so the method does not apply and uniqueness is assumed as a hypothesis. The question sits inside a well-known open family. In the far-field formulation, the backscattered near field on Γ\Gamma determines, by angular-spectrum propagation (Fourier decomposition of the measured field into plane waves), at least the scattering amplitude for the single incident direction and all observation directions in the back hemisphere and all k>0k>0 — the data of the fixed angle inverse scattering problem, which consists in recovering a compactly supported coefficient (sound speed, potential, or contrast) from scattering data generated by probing the medium with a single plane wave; this formally determined problem is open in general, as restated by Oksanen, Rakesh and Salo (2024; published SIAM J. Math. Anal. 2026). All known positive results add structure: uniqueness holds for two opposite fixed angles (Rakesh–Salo), for almost symmetric or horizontally controlled potentials from one angle (Rakesh–Salo), locally for sound speeds close to a constant (Ma–Potenciano-Machado–Salo), for small potentials in the time-domain backscattering problem (Stefanov–Uhlmann), and for small time-dependent potentials (Nursultanov–Oksanen–Stefanov); Guo and Liu (2026), who study multi-frequency backscattering for all incident directions, prove only a local result — uniqueness when the difference of two contrasts is an affine function — and confirm the problem remains 'a long-standing open issue'. For impenetrable obstacles (shape and impedance, not a penetrable coefficient), global uniqueness from multi-frequency near-field backscattering data was proved in 2026 under convexity assumptions (Li–Liu). None of these covers the penetrable coefficient with single-direction data.

Problem Statement

Let Ω={(x,y,z):∣x∣,∣y∣<b, z∈(−ξ,d)}⊂R3\Omega=\{(x,y,z):|x|,|y|<b,\ z\in(-\xi,d)\}\subset\mathbb{R}^{3} and let Γ={(x,y,z):∣x∣,∣y∣<b, z=−ξ}\Gamma=\{(x,y,z):|x|,|y|<b,\ z=-\xi\} be the illuminated (backscattering) face of ∂Ω\partial\Omega. Consider the forward problem Δu+k2c(x)u=0 in R3,u=us+eikz,\Delta u+k^{2}c(\mathbf{x})u=0\ \text{in }\mathbb{R}^{3},\qquad u=u_{s}+e^{ikz}, where the scattered field usu_{s} satisfies the Sommerfeld radiation condition and the real coefficient satisfies c=1+βc=1+\beta with β≥0\beta\geq 0 compactly supported in Ω‾\overline{\Omega} (vacuum outside), c∈C15(R3)c\in C^{15}(\mathbb{R}^{3}), and the travel-time regularity assumption: every point of R3\mathbb{R}^{3} is connected to the plane {z=−a}\{z=-a\}, a>ξa>\xi, by a unique geodesic of the Riemannian metric dτ=c(x)∣dx∣d\tau=\sqrt{c(\mathbf{x})}|d\mathbf{x}|, meeting that plane perpendicularly. The data are the total field u(x,k)u(\mathbf{x},k) measured on Γ\Gamma for all kk in a nonempty interval [k‾,k‾]⊂(0,∞)[\underline{k},\overline{k}]\subset(0,\infty), generated by the single incident plane wave eikze^{ikz} (by the analytic dependence of the field on kk, any such interval is equivalent to all k>0k>0).

Question: does this single-direction multi-frequency backscattering data determine the coefficient uniquely? Specifically, prove that if two admissible coefficients c1,c2c_{1},c_{2} generate solutions u1,u2u_{1},u_{2} of the forward problem with u1(x,k)=u2(x,k)u_{1}(\mathbf{x},k)=u_{2}(\mathbf{x},k) for all x∈Γ\mathbf{x}\in\Gamma and all k∈[k‾,k‾]k\in[\underline{k},\overline{k}], then c1=c2c_{1}=c_{2} in Ω\Omega — with the theorem valid at least for the admissible class above (or any natural class containing it), and not relying on additional structural hypotheses such as symmetry, smallness of the contrast, closeness to a constant, or data for more than one incident direction. Alternatively, refute the claim by exhibiting two distinct admissible coefficients whose total fields coincide on Γ\Gamma for a continuum of wavenumbers.

References

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