PreprintCC BY 4.0arXiv
Quantitative Uniqueness and Rough Damping on 𝕋2
Abstract
Motivated by a conjecture of Burq and Gérard, we investigate quantitative uniqueness principles for functions on 𝕋2 whose Fourier spectra lie in fixed-width annuli. We obtain observability estimates uniform in the radius, under a mild Sobolev regularity condition, and the generalized geometric control condition (GGCC) on the damping function. This leads to exponential decay for the damped wave equation with rough damping. We also establish an analogous uncertainty principle for spectra near dilates of convex polygonal boundaries, where no Sobolev regularity of the damping is required beyond L∞.
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BibTeX
@misc{jaye2026quantitative,
title = {{Quantitative Uniqueness and Rough Damping on $\mathbb T^2$}},
author = {Benjamin Jaye and Rahul Sethi},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27544},
eprint = {2608.27544},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27544},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantitative Uniqueness and Rough Damping on 𝕋^2
AU - Benjamin Jaye
AU - Rahul Sethi
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.27544
UR - https://arxiv.org/abs/2608.27544
AB - Motivated by a conjecture of Burq and Gérard, we investigate quantitative uniqueness principles for functions on 𝕋^2 whose Fourier spectra lie in fixed-width annuli. We obtain observability estimates uniform in the radius, under a mild Sobolev regularity condition, and the generalized geometric control condition (GGCC) on the damping function. This leads to exponential decay for the damped wave equation with rough damping. We also establish an analogous uncertainty principle for spectra near dilates of convex polygonal boundaries, where no Sobolev regularity of the damping is required beyond L^∞.
ER -