Sharp observation time for wave equations with an inverse-square boundary potential
Abstract
We determine the sharp observation-time threshold for the autonomous, pure-potential wave equation with an inverse-square boundary singularity on the unit ball. For every fixed κ<0, full-boundary observability holds on (−T,T) whenever T>1 and fails when T<1. This resolves, for this model, the optimal-time question raised by Enciso, Shao, and Vergara [J. Eur. Math. Soc. 23 (2021), 3459--3495] and extends their parameter range −1/2<κ<0 to all κ<0, including the limit-point regime κ≤−1/2. Our approach is based on the Ingham--Beurling theory of nonharmonic Fourier series. Spherical harmonic decomposition reduces the problem to one-dimensional singular operators, for which we establish frequency separation, local counting estimates, and lower bounds for the boundary norming constants, uniformly in the angular parameter. A quantitative Ingham--Beurling inequality then yields the sharp-time observability estimate. As a by-product, this spectral approach provides an independent proof of boundary observability and an explicit functional characterization of its maximal L2 observation domain. The latter is described by a weighted ℓ2 condition on the spectral coefficients and consists precisely of energy solutions whose distributional Neumann trace belongs to L2. It contains the boundary-admissible energy solutions considered in Enciso--Shao--Vergara's original work.
Related
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BibTeX
@misc{wei2026sharp,
title = {{Sharp observation time for wave equations with an inverse-square boundary potential}},
author = {Longben Wei},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39475},
eprint = {2610.39475},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39475},
}
RIS
TY - UNPB
TI - Sharp observation time for wave equations with an inverse-square boundary potential
AU - Longben Wei
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39475
UR - https://arxiv.org/abs/2610.39475
AB - We determine the sharp observation-time threshold for the autonomous, pure-potential wave equation with an inverse-square boundary singularity on the unit ball. For every fixed κ<0, full-boundary observability holds on (−T,T) whenever T>1 and fails when T<1. This resolves, for this model, the optimal-time question raised by Enciso, Shao, and Vergara [J. Eur. Math. Soc. 23 (2021), 3459--3495] and extends their parameter range −1/2<κ<0 to all κ<0, including the limit-point regime κ≤−1/2. Our approach is based on the Ingham--Beurling theory of nonharmonic Fourier series. Spherical harmonic decomposition reduces the problem to one-dimensional singular operators, for which we establish frequency separation, local counting estimates, and lower bounds for the boundary norming constants, uniformly in the angular parameter. A quantitative Ingham--Beurling inequality then yields the sharp-time observability estimate. As a by-product, this spectral approach provides an independent proof of boundary observability and an explicit functional characterization of its maximal L^2 observation domain. The latter is described by a weighted ℓ^2 condition on the spectral coefficients and consists precisely of energy solutions whose distributional Neumann trace belongs to L^2. It contains the boundary-admissible energy solutions considered in Enciso--Shao--Vergara's original work.
ER -