Determine a smooth sound-soft obstacle from one far-field pattern
Fix a wave number $k > 0$ and a direction $\theta \in S^2$. For a bounded domain $D \subset \mathbb{R}^3$ with smooth boundary and connected exterior, let $u = e^{ik x \cdot \theta} + u^s$ solve $$\Delta u + k^2 u = 0 \ \text{ in } \mathbb{R}^3 \setminus \overline D, \qquad u = 0 \ \text{ on } \partial D,$$ with $u^s$ outgoing, and define the far-field amplitude $a_D$ by $$u^s(r\omega) = r^{-1} e^{ikr} a_D(\omega; \theta, k) + O(r^{-2}), \qquad r \to \infty, \ \omega \in S^2.$$ Schiffer proved that the far-field amplitudes for all incident directions $\theta \in S^2$ at one wave number determine a sound-soft obstacle. For one incident wave, uniqueness holds for obstacles contained in a ball of radius $R$ with $kR < \pi$ (Colton and Sleeman), improved by Gintides to $kR$ below the smallest positive zero $4.49\ldots$ of the spherical Bessel function $j_1$; for balls (Liu); and for polyhedra (Alessandrini and Rondi; Liu and Zou).
Problem. Let $D_1, D_2$ be two such domains. Does $a_{D_1}(\omega; \theta, k) = a_{D_2}(\omega; \theta, k)$ for every observation direction $\omega \in S^2$ imply $D_1 = D_2$, for every $k > 0$ and every $\theta \in S^2$? The wave number and the incident direction are fixed, and no bound on the size of the obstacles relative to the wavelength is assumed.
Source: Classical, after Schiffer's uniqueness theorem for all incident directions (around 1960, published in Lax and Phillips (1967)); recorded as a challenging open problem by Colton and Kress (2018), §2. Stated in this form, for smooth obstacles in ℝ³, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- SP
- Keywords
- AIM Open Applied Problems, far-field patterns, Helmholtz equation, inverse obstacle scattering
References
@article{colton2018,
author = {Colton, David and Kress, Rainer},
title = {Looking back on inverse scattering theory},
journal = {SIAM Rev.},
volume = {60},
number = {4},
year = {2018},
pages = {779--807},
doi = {10.1137/17M1144763},
}
@book{lax1967,
author = {Lax, Peter D. and Phillips, Ralph S.},
title = {Scattering Theory},
series = {Pure and Applied Mathematics},
volume = {26},
publisher = {Academic Press},
address = {New York},
year = {1967},
}
@article{colton1983,
author = {Colton, David and Sleeman, B. D.},
title = {Uniqueness theorems for the inverse problem of acoustic scattering},
journal = {IMA J. Appl. Math.},
volume = {31},
number = {3},
year = {1983},
pages = {253--259},
doi = {10.1093/imamat/31.3.253},
}
@article{gintides2005,
author = {Gintides, Drossos},
title = {Local uniqueness for the inverse scattering problem in acoustics via the {F}aber--{K}rahn inequality},
journal = {Inverse Problems},
volume = {21},
number = {4},
year = {2005},
pages = {1195--1205},
doi = {10.1088/0266-5611/21/4/001},
}
@article{liu1997,
author = {Liu, Changmei},
title = {Inverse obstacle problem: local uniqueness for rougher obstacles and the identification of a ball},
journal = {Inverse Problems},
volume = {13},
number = {4},
year = {1997},
pages = {1063--1069},
doi = {10.1088/0266-5611/13/4/011},
}
@article{alessandrini2005,
author = {Alessandrini, Giovanni and Rondi, Luca},
title = {Determining a sound-soft polyhedral scatterer by a single far-field measurement},
journal = {Proc. Amer. Math. Soc.},
volume = {133},
number = {6},
year = {2005},
pages = {1685--1691},
doi = {10.1090/S0002-9939-05-07810-X},
eprint = {math/0307004},
archivePrefix = {arXiv},
}
@article{liu2006,
author = {Liu, Hongyu and Zou, Jun},
title = {Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers},
journal = {Inverse Problems},
volume = {22},
number = {2},
year = {2006},
pages = {515--524},
doi = {10.1088/0266-5611/22/2/008},
}
@misc{aim-single-wave-obstacle,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Determine a smooth sound-soft obstacle from one far-field pattern}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/054-single-wave-obstacle.md},
note = {Entry 054 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00559, "Determine a smooth sound-soft obstacle from one far-field pattern", clopen.solutions.