The sharp decay rate for waves damped on a disk
Let $\mathbb{T}^2 = \mathbb{R}^2 / (2\pi\mathbb{Z})^2$, let $D = \{(x, y) : x^2 + y^2 < 1\}$ in the fundamental square $[-\pi, \pi)^2$, and let $W = \mathbf{1}_D$. For real $\lambda \ne 0$ let $$P_\lambda = -\Delta + i\lambda W - \lambda^2, \qquad D(P_\lambda) = H^2(\mathbb{T}^2),$$ the stationary operator of the damped wave equation $\partial_t^2 u - \Delta u + W \partial_t u = 0$ on $\mathbb{T}^2$; the circle is a jump of the damping, not a boundary. By Borichev and Tomilov, a bound $\|P_\lambda^{-1}\|_{L^2 \to L^2} \le C|\lambda|^{1/\alpha - 1}$ for large $|\lambda|$ is equivalent to the decay $E(u, t)^{1/2} \le C t^{-\alpha} (\|u_0\|_{H^2} + \|u_1\|_{H^1})$ of the energy for initial data in $H^2 \times H^1$, and Anantharaman and Léautaud's rate $t^{-1/2}$ for every nonzero damping gives $\|P_\lambda^{-1}\|_{L^2 \to L^2} \le C|\lambda|$ here. Sun proved that a damping positive exactly on a strictly convex set and comparable to the $\beta$-th power of the distance to its boundary, with $\beta > 4$ and further regularity, gives the rate $t^{-1 + 2/(2\beta + 7)}$, which cannot be improved for some such dampings; this is resolvent growth $|\lambda|^{2/(2\beta + 5)}$, and $2/5$ is its value at $\beta = 0$.
Problem. Is $|\lambda|^{2/5}$ the sharp high-frequency growth of the resolvent: are there $C, \lambda_0 > 0$ with $$\|P_\lambda^{-1}\|_{L^2 \to L^2} \le C|\lambda|^{2/5} \quad \text{for all } |\lambda| \ge \lambda_0,$$ and is $$\limsup_{\lambda \to +\infty} \lambda^{-2/5} \|P_\lambda^{-1}\|_{L^2 \to L^2} > 0\,?$$
Source: Kleinhenz (2026), Question 1, at the CIRM conference on mathematical aspects of the physics with non-self-adjoint operators; Sun (2023), Remark 1.5, had asked about indicators of strictly convex sets. Stated by AIM — Open Applied Problems with the lower bound as a resolvent bound.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- SP
- Keywords
- AIM Open Applied Problems, damped wave equations, energy decay, resolvent estimates
References
@misc{kleinhenz2026,
author = {Kleinhenz, Perry},
title = {Geometry of rough damping},
howpublished = {Open problem, CIRM conference ``Mathematical aspects of the physics with non-self-adjoint operators'', Marseille},
year = {2026},
url = {https://nsa.fjfi.cvut.cz/problems/07_CIRM_2026/2026-01/2026-01_CIRM_Kleinhenz.pdf},
note = {Question 1},
}
@article{sun2023,
author = {Sun, Chenmin},
title = {Sharp decay rate for the damped wave equation with convex-shaped damping},
journal = {Int. Math. Res. Not. IMRN},
volume = {2023},
number = {7},
year = {2023},
pages = {5905--5973},
doi = {10.1093/imrn/rnac022},
eprint = {2106.11782},
archivePrefix = {arXiv},
}
@article{borichev2010,
author = {Borichev, Alexander and Tomilov, Yuri},
title = {Optimal polynomial decay of functions and operator semigroups},
journal = {Math. Ann.},
volume = {347},
number = {2},
year = {2010},
pages = {455--478},
doi = {10.1007/s00208-009-0439-0},
}
@article{anantharaman2014,
author = {Anantharaman, Nalini and L{\'e}autaud, Matthieu},
title = {Sharp polynomial decay rates for the damped wave equation on the torus},
journal = {Anal. PDE},
volume = {7},
number = {1},
year = {2014},
pages = {159--214},
doi = {10.2140/apde.2014.7.159},
eprint = {1210.6879},
archivePrefix = {arXiv},
note = {With an appendix by St{\'e}phane Nonnenmacher},
}
@misc{aim-rough-disk-damping,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The sharp decay rate for waves damped on a disk}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/215-rough-disk-damping.md},
note = {Entry 215 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-00468, "The sharp decay rate for waves damped on a disk", clopen.solutions.