Linear inverse bounds for four-diamond scattering
Let $Q = (-\pi, \pi)^2$ and, for $r > 0$, let $U_r$ be the union of the open discs of radius $r$ centred at the four corners $(\pm\pi, \pm\pi)$ of $Q$. A four-diamond obstacle is a bounded set $\Omega_- = \Omega_1 \cup \Omega_2 \cup \Omega_3 \cup \Omega_4 \subset \mathbb{R}^2$, where the $\Omega_j$ are disjoint open convex sets with smooth boundary, such that for some $0 < \epsilon < \pi/2$ $$\partial Q \setminus U_\epsilon \subseteq \Gamma \subseteq \mathbb{R}^2 \setminus U_{\epsilon/2}, \qquad \Omega_- \cap Q = \varnothing,$$ where $\Gamma = \partial \Omega_-$. For a wavenumber $k > 0$ let $\Phi_k(x, y) = \frac{i}{4} H_0^{(1)}(k|x - y|)$, let $\nu$ be the unit normal pointing out of $\Omega_-$, and let $V_k$, $K_k$ and $K_k'$ be the single-layer, double-layer and adjoint double-layer operators on $L^2(\Gamma)$, $$V_k \phi(x) = \int_\Gamma \Phi_k(x, y) \phi(y) \, ds(y), \quad K_k \phi(x) = \int_\Gamma \partial_{\nu(y)} \Phi_k(x, y) \phi(y) \, ds(y), \quad K_k' \phi(x) = \int_\Gamma \partial_{\nu(x)} \Phi_k(x, y) \phi(y) \, ds(y).$$ The standard second-kind integral equations for the exterior Dirichlet problem use the invertible operators $$A_k = \tfrac12 I + K_k - ik V_k, \qquad A_k' = \tfrac12 I + K_k' - ik V_k.$$ Galkowski, Rachh and Spence proved that the Galerkin boundary element method with piecewise polynomials suffers from the pollution effect on four-diamond obstacles: for piecewise constants at frequencies outside an exceptional set off which the inverses are polynomially bounded, and for higher degrees under a bound $C k^2$ on the inverses. The bound below would remove both restrictions. For two aligned squares, where the trapping is parabolic as here, Chandler-Wilde, Spence, Gibbs and Smyshlyaev proved that $\|(A_k')^{-1}\|$ grows at most like $k^2$, and at least like $k$ along a sequence of frequencies.
Problem. Is it true that for every four-diamond obstacle and every $k_0 > 0$ there is $C < \infty$ such that $$\|A_k^{-1}\|_{L^2(\Gamma) \to L^2(\Gamma)} + \|(A_k')^{-1}\|_{L^2(\Gamma) \to L^2(\Gamma)} \le C k \qquad (k \ge k_0)?$$ The constant may depend on the obstacle and on $k_0$, and the bound must hold at every real $k \ge k_0$, with no exceptional set of frequencies.
Source: Attributed by Galkowski, Rachh and Spence (2025), Remark 2.14(1), (2.17), and Galkowski and Spence (2026), Remark 8.22(1), (8.45), to Chandler-Wilde, Spence, Gibbs and Smyshlyaev (2020); stated by AIM — Open Applied Problems with coupling parameter k.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- NA — Numerical Analysis
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, boundary integral equations, Helmholtz equation, trapping obstacles
References
@article{chandlerwilde2020,
author = {Chandler-Wilde, S. N. and Spence, E. A. and Gibbs, A. and Smyshlyaev, V. P.},
title = {High-frequency bounds for the {H}elmholtz equation under parabolic trapping and applications in numerical analysis},
journal = {SIAM J. Math. Anal.},
volume = {52},
number = {1},
year = {2020},
pages = {845--893},
doi = {10.1137/18M1234916},
eprint = {1708.08415},
archivePrefix = {arXiv},
}
@misc{galkowski2025,
author = {Galkowski, J. and Rachh, M. and Spence, E. A.},
title = {Helmholtz boundary integral methods and the pollution effect},
year = {2025},
eprint = {2507.22797},
archivePrefix = {arXiv},
note = {Version 3, March 2026},
}
@article{galkowski2026,
author = {Galkowski, Jeffrey and Spence, Euan A.},
title = {Numerical analysis of the high-frequency {H}elmholtz equation using semiclassical analysis},
journal = {Acta Numer.},
volume = {35},
year = {2026},
pages = {1--171},
doi = {10.1017/S0962492925100275},
eprint = {2511.15287},
archivePrefix = {arXiv},
}
@misc{aim-four-diamond-scattering-inverse,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Linear inverse bounds for four-diamond scattering}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/522-four-diamond-scattering-inverse.md},
note = {Entry 522 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-00344, "Linear inverse bounds for four-diamond scattering", clopen.solutions.