Observability of an acoustic shell at equal bulk and surface speeds
Let $\Omega = \{x \in \mathbb{R}^2 : 1 < |x| < 2\}$, $\Gamma_1 = \{|x| = 1\}$ and $\Gamma_0 = \{|x| = 2\}$, and consider the wave equation with a dynamic (Wentzell) boundary condition on the inner circle, $$z_{tt} - \Delta z = 0 \ \text{ in } \Omega, \qquad z = 0 \ \text{ on } \Gamma_0,$$ $$z_\Gamma = z|_{\Gamma_1}, \qquad z_{\Gamma,tt} - \Delta_\Gamma z_\Gamma + \partial_\nu z = 0 \ \text{ on } \Gamma_1,$$ where $\nu$ is the outward unit normal of the annulus and $\Delta_\Gamma$ is the Laplace–Beltrami operator of the circle $\Gamma_1$. The energy $$E(t) = \int_\Omega \bigl(|z_t|^2 + |\nabla z|^2\bigr)\,dx + \int_{\Gamma_1} \bigl(|z_{\Gamma,t}|^2 + |\nabla_\Gamma z_\Gamma|^2\bigr)\,dS$$ is conserved. Baudouin, Dardé, Ervedoza and Mercado studied the system on an annulus $R_0 < |x| < R_1$ with inner boundary equation $\alpha z_{\Gamma,tt} - \beta \Delta_\Gamma z_\Gamma + \partial_\nu z = 0$, $\alpha, \beta > 0$, observed through the outer circle. They proved that the observability inequality holds in some time when $\alpha < \beta$, and fails in every time when $\alpha > \beta$. Here $\alpha = \beta = 1$: the bulk and surface waves have equal speeds.
Problem. Does there exist a finite $T_0$ such that for every $T > T_0$ there is $C_T < \infty$ with $$E(0) \le C_T \int_0^T \int_{\Gamma_0} |\partial_\nu z|^2 \, dS \, dt$$ for every smooth compatible solution? By density this is an energy-space observability question.
Source: Stated by AIM — Open Applied Problems, after Chorfi, Maniar and Morales (2025), §5.1, on equal bulk and surface wave speeds, as the case α = β = 1, R_0 = 1, R_1 = 2 of the annular system (1.3) of Baudouin, Dardé, Ervedoza and Mercado (2022).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, dynamic boundary conditions, observability, wave equation
References
@article{baudouin2022,
author = {Baudouin, Lucie and Dard{\'e}, J{\'e}r{\'e}mi and Ervedoza, Sylvain and Mercado, Alberto},
title = {A unified strategy for observability of waves in an annulus with various boundary conditions},
journal = {Math. Rep. (Bucur.)},
volume = {24(74)},
number = {1-2},
year = {2022},
pages = {59--112},
url = {https://imar.ro/journals/Mathematical_Reports/Pdfs/2022/1-2/5.pdf},
}
@misc{chorfi2025,
author = {Chorfi, S. E. and Maniar, L. and Morales, R.},
title = {Controllability and inverse problems for hyperbolic and dispersive equations with dynamic boundary conditions},
year = {2025},
eprint = {2505.14795},
archivePrefix = {arXiv},
}
@misc{aim-equal-speed-wentzell-wave-observability,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Observability of an acoustic shell at equal bulk and surface speeds}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/385-equal-speed-wentzell-wave-observability.md},
note = {Entry 385 at commit a153f46, 2026-10-13},
}
Resolution claims
We consider a wave equation in a planar annulus, with a Dirichlet condition on the outer circle and a dynamic wave equation on the inner circle. The bulk and surface propagation speeds are both one. For every finite observation time, we construct smooth real solutions with unit initial energy whose exterior normal-derivative observation tends to zero. The construction uses exact …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, paper/equal_speed_observability.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00746, "Observability of an acoustic shell at equal bulk and surface speeds", clopen.solutions.