Uniform boundary observability for periodic waves, I: up to the ε−1 eigenvalue scale
Abstract
Let ℒε=−div(A(x/ε)∇) be a scalar divergence-form operator on a bounded C3 domain, where A is symmetric, uniformly elliptic, Lipschitz continuous, and periodic. For initial data spectrally truncated at the eigenvalue level N, we establish whole-boundary observability and projected exact controllability, uniformly in the range N≤δε−1. This extends the cutoff N≲ε−1/2 obtained by Lin and Shen (J. Eur. Math. Soc. 2022). The key input is a two-sided boundary estimate for fixed-width spectral packets of √(ℒε) at frequencies that satisfy εμ2≪1. Its lower bound combines quantitative Dirichlet corrector estimates with a forcing-stable microscopic boundary Cauchy estimate. Miller's spectral packet criterion and the Hilbert uniqueness method then convert this packet estimate into the corresponding wave results, with constants independent of ε and N.
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BibTeX
@misc{geng2026uniform,
title = {{Uniform boundary observability for periodic waves, I: up to the $\varepsilon^-1$ eigenvalue scale}},
author = {Jun Geng and Shumao Wang and Wei Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33547},
eprint = {2610.33547},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33547},
}
RIS
TY - UNPB
TI - Uniform boundary observability for periodic waves, I: up to the ε^−1 eigenvalue scale
AU - Jun Geng
AU - Shumao Wang
AU - Wei Wang
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33547
UR - https://arxiv.org/abs/2610.33547
AB - Let ℒ_ε=−div(A(x/ε)∇) be a scalar divergence-form operator on a bounded C^3 domain, where A is symmetric, uniformly elliptic, Lipschitz continuous, and periodic. For initial data spectrally truncated at the eigenvalue level N, we establish whole-boundary observability and projected exact controllability, uniformly in the range N≤δε^−1. This extends the cutoff N≲ε^−1/2 obtained by Lin and Shen (J. Eur. Math. Soc. 2022). The key input is a two-sided boundary estimate for fixed-width spectral packets of √(ℒ_ε) at frequencies that satisfy εμ^2≪1. Its lower bound combines quantitative Dirichlet corrector estimates with a forcing-stable microscopic boundary Cauchy estimate. Miller's spectral packet criterion and the Hilbert uniqueness method then convert this packet estimate into the corresponding wave results, with constants independent of ε and N.
ER -