For time-harmonic Maxwell scattering in R3\mathbb{R}^{3} by a perfectly conducting planar screen — a thin, perfectly conducting plate modeled as a compact surface piece of zero thickness lying in an affine plane — Ola, Päivärinta and Sadique (2023) proved that the far-field pattern produced by a single incident plane wave uniquely determines both the supporting hyperplane and the screen, for C2C^{2} orientable screens and polarizations not parallel to the propagation direction. The proof, however, presumes unique solvability of the electric-field integral equation for the screen in the trace space TH˙Div−1/2(S)\dot{TH}^{-1/2}_{\mathrm{Div}}(S), a step that needs verification because the available theory of Buffa and Christiansen uses a different, variational scale of spaces and does not assume orientability. The open problem is to complete and generalize this uniqueness program: establish (or refute) the solvability of the screen integral equation in the natural trace space, extend single-measurement uniqueness to screens of lower regularity, and extend it from perfectly conducting to penetrable (impedance or material-contrast) planar screens.

Background

In time-harmonic electromagnetic inverse scattering one illuminates an unknown object with an incident wave and measures the scattered field far away. For Maxwell's equations ∇×E=iωμH\nabla\times E=i\omega\mu H, ∇×H=−iωεE\nabla\times H=-i\omega\varepsilon E outside the object, an outgoing scattered field has an asymptotic form Esc(x)≈eik∣x∣x^×(x^×E∞(x^))/(4π∣x∣)E_{\rm sc}(x)\approx e^{ik|x|}\hat{x}\times(\hat{x}\times E^{\infty}(\hat{x}))/(4\pi|x|) as ∣x∣→∞|x|\to\infty (with kk the wavenumber, μ,ε\mu,\varepsilon the permeability and permittivity, and the outgoing character encoded by the Silver–Müller radiation condition); the vector function E∞E^{\infty} on the unit sphere is the far-field pattern. The classical uniqueness question — does E∞E^{\infty} determine the scatterer? — is known as Schiffer's problem. For general bounded obstacles, uniqueness is proved only with far fields for infinitely many incident plane waves (Schiffer; see Colton–Kress), so identifying scatterer classes that a single measurement pins down is a central theme: single far-field uniqueness holds for polyhedral scatterers (Alessandrini–Rondi; Elschner–Yamamoto; Liu–Zou), for balls (Liu Changmei), and related classes. A screen is an idealization of a thin plate or crack: a compact CkC^{k} submanifold with boundary of an affine plane LL, the supporting hyperplane, of zero thickness. A perfectly conducting (PC, also called supraconductive in the source paper; the total tangential electric field vanishes on the screen) planar screen is the electromagnetic analogue of a sound-soft acoustic screen. For acoustic screens, single-measurement shape determination was settled by Blåsten–Päivärinta–Sadique (2020): given the supporting plane, the far field of any single incident wave determines the screen shape, apart from an antisymmetry caveat. For electromagnetic screens the earlier literature gave only reconstruction heuristics without uniqueness: Cakoni–Colton–Darrigrand (2003) and Cakoni–Darrigrand (2005) developed linear-sampling-type methods for screen problems. Ola–Päivärinta–Sadique (2023) proved the first single-measurement uniqueness theorem for the electromagnetic problem: if SS is a C2C^{2} planar PC screen and the incident plane wave has electric and magnetic polarizations p,qp,q neither of which is parallel to the propagation direction θ\theta, then the far-field pattern of one scattered field determines both LL and SS. The mechanism: the scattered field is represented by a magnetic-dipole (vector single-/double-layer) potential whose density is the jump ν×[Hsc]\nu\times[H_{\rm sc}] of the tangential magnetic field across the screen (ν\nu a chosen unit normal), so the screen is exactly the support of that jump; injectivity of the density-to-far-field map on a plane (a Fourier-transform argument), a reflection/parity argument showing the density has full support, and a Sobolev-regularity obstruction for densities supported on codimension-2 sets together force coincidence of any two candidate screens. This representation presumes the direct problem: solvability of the electric-field integral equation (EFIE) −ν×E0=iN(ν×[Hsc])/ωε-\nu\times E_{0}=iN(\nu\times[H_{\rm sc}])/\omega\varepsilon for the unknown jump density in the space TH˙Div−1/2(S)\dot{TH}^{-1/2}_{\mathrm{Div}}(S) — tangential vector fields on SS whose surface divergence lies in H−1/2(S)H^{-1/2}(S), the dot marking closure of smooth compactly supported tangential fields (the natural trace space for Maxwell fields near an open surface). The only available solvability theory, Buffa–Christiansen (2003), is set in a different scale of variational (multitrace) spaces on Lipschitz screens and assumes no orientability; the 2023 preprint states explicitly in a footnote that the transfer 'needs to be checked', and that prior references 'do not assume orientability of the screen, i.e for example a Möbius–strip would be ok' (the published version drops this caveat and instead cites [4] for this solvability). Their uniqueness statement also requires, in the hyperplane-determination step, that the measured far fields be non-vanishing (not identically zero) — a property their own full-support argument in fact derives once the integral equation is known to be solvable, which is why the solvability gap is load-bearing. Penetrable screens (thin sheets with impedance or dielectric contrast, rather than perfectly conducting) are not treated at all in the 2023 work.

Problem Statement

Complete the single-measurement uniqueness theory for electromagnetic scattering by a planar screen in R3\mathbb{R}^{3}, begun by Ola–Päivärinta–Sadique (2023). The overarching objective is that the far-field pattern of one incident plane wave (measured in all directions) should be provable — with full rigor and in natural generality — to determine a planar screen and its supporting hyperplane. The linked open components are: (1) Direct problem (load-bearing): prove or disprove unique solvability, in the trace space TH˙Div−1/2(S)\dot{TH}^{-1/2}_{\mathrm{Div}}(S), of the electric-field integral equation −ν×E0=iN(ν×[Hsc])/ωε-\nu\times E_{0}=iN(\nu\times[H_{\rm sc}])/\omega\varepsilon for a perfectly conducting C2C^{2} orientable planar screen SS, or identify the correct space in which the screen scattering problem is well posed; the 2023 uniqueness proof presumes this solvability (the preprint flags that the existing theory of Buffa–Christiansen uses a variational scale of spaces and assumes no orientability, so the transfer 'needs to be checked'), and the non-vanishing of the measured far field assumed in the hyperplane-determination step is likewise derived from it, so a resolution of (1) completes the existing proof. (2) Geometry: assuming the direct problem, extend the single-measurement uniqueness theorem to screens of lower regularity (Lipschitz or C1C^{1} submanifolds of a plane). (3) Material model: extend single-plane-wave shape uniqueness from perfectly conducting to penetrable planar screens — thin sheets with impedance or dielectric-contrast transmission conditions — where the far field no longer has a representation by a single tangential jump density. The standing of the components is: (1) is the load-bearing core, and settling it — establishing or refuting solvability in TH˙Div−1/2(S)\dot{TH}^{-1/2}_{\mathrm{Div}}(S), or, if that space is not the right one, identifying the correct well-posed space and transferring the proof to it — completes the verification of the 2023 proof; (2) presupposes the direct problem of (1); (3) is independent of (1) and (2).

References

  1. Ola, Petri; Päivärinta, Lassi; Sadique, Sadia (2023). Unique Determination of a Planar Screen in Electromagnetic Inverse Scattering. Mathematics 11(22):4655. DOI 10.3390/math11224655, https://doi.org/10.3390/math11224655. Preprint arXiv:2307.10100, DOI 10.48550/arXiv.2307.10100, https://arxiv.org/abs/2307.10100. Note: the arXiv v1 version misspells the title ("Electromagentic") and lists Sadique as first author; the MDPI version has the correct title and author order Ola, Päivärinta, Sadique.
  2. Blåsten, Emilia; Ola, Petri; Päivärinta, Lassi (2024). Inverse Problems for Screens. arXiv:2409.02591, DOI 10.48550/arXiv.2409.02591, https://arxiv.org/abs/2409.02591
  3. Cakoni, Fioralba; Monk, Peter (2025). Target Signatures for Anisotropic Screens in Electromagnetic Scattering. SIAM Journal on Applied Mathematics 85(6):2758–2783. DOI 10.1137/24m1713120, https://doi.org/10.1137/24m1713120. Preprint arXiv:2504.02078, https://arxiv.org/abs/2504.02078
  4. Buffa, A.; Christiansen, S. H. (2003). The electric field integral equation on Lipschitz screens: definitions and numerical approximation. Numerische Mathematik 94:229–267. DOI 10.1007/s00211-002-0422-0, https://doi.org/10.1007/s00211-002-0422-0
  5. Blåsten, Emilia; Päivärinta, Lassi; Sadique, Sadia (2020). Unique Determination of the Shape of a Scattering Screen from a Passive Measurement. Mathematics 8(7):1156. DOI 10.3390/math8071156, https://doi.org/10.3390/math8071156
  6. Cakoni, F.; Colton, D.; Darrigrand, E. (2003). The inverse electromagnetic scattering problem for screens. Inverse Problems 19(3):627–642. DOI 10.1088/0266-5611/19/3/310, https://doi.org/10.1088/0266-5611/19/3/310
  7. Cakoni, F.; Darrigrand, E. (2005). The inverse electromagnetic scattering problem for a mixed boundary value problem for screens. Journal of Computational and Applied Mathematics 174(2):251–269. DOI 10.1016/j.cam.2004.04.012, https://doi.org/10.1016/j.cam.2004.04.012
  8. Blåsten, Emilia; Päivärinta, Lassi; Sadique, Sadia (2023). The Fourier, Hilbert, and Mellin Transforms on a Half-Line. SIAM Journal on Mathematical Analysis 55(6):7529–7548. DOI 10.1137/23m1560628, https://doi.org/10.1137/23m1560628