Boundary Spectral Inequalities from Measurable Sets on W2,∞ manifolds and their applications
Abstract
Let (ℳ,g) be a compact connected W2,∞ Riemannian manifold with nonempty boundary and a uniformly elliptic Lipschitz metric. We establish quantitative boundary spectral inequalities for both Dirichlet and Neumann eigenfunctions from arbitrary measurable subsets of ∂ℳ×(−s0,s0) of positive surface measure. Moreover, we apply the Dirichlet spectral inequality to establish the boundary observability estimate |u(T)|L2(ℳ)2 ≤ C ∫J |∂νgu(x,t)|2 dSg dt . where J⊂∂ℳ×(0,T) is measurable and has positive surface--time measure. A localized theorem requiring W2,∞ regularity only near the observed boundary patch is retained as a separate consequence. The proof of the observability inequality combines quantitative continuation from positive-measure boundary sets with low-frequency spectral concentration. This result shows that boundary observability of solutions to the heat equation can also be achieved in W2,∞ domains. Finally, as an application, we prove that measurements of the boundary normal derivative on J uniquely determine the initial state of heat equation. Under an a priori bound in D(−Δg, Ds/2), the recovery satisfies a logarithmic stability estimate of optimal order s/2.
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BibTeX
@misc{chen2026boundary,
title = {{Boundary Spectral Inequalities from Measurable Sets on $W^2,\infty$ manifolds and their applications}},
author = {Lu Chen and Haolin Liu and Hongyu Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.21039},
eprint = {2608.21039},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.21039},
}
RIS
TY - UNPB
TI - Boundary Spectral Inequalities from Measurable Sets on W^2,∞ manifolds and their applications
AU - Lu Chen
AU - Haolin Liu
AU - Hongyu Liu
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2608.21039
UR - https://arxiv.org/abs/2608.21039
AB - Let (ℳ,g) be a compact connected W^2,∞ Riemannian manifold with nonempty boundary and a uniformly elliptic Lipschitz metric. We establish quantitative boundary spectral inequalities for both Dirichlet and Neumann eigenfunctions from arbitrary measurable subsets of ∂ℳ×(−s_0,s_0) of positive surface measure. Moreover, we apply the Dirichlet spectral inequality to establish the boundary observability estimate |u(T)|_L^2(ℳ)^2 ≤ C ∫_J |∂_ν_gu(x,t)|^2 dS_g dt . where J⊂∂ℳ×(0,T) is measurable and has positive surface--time measure. A localized theorem requiring W^2,∞ regularity only near the observed boundary patch is retained as a separate consequence. The proof of the observability inequality combines quantitative continuation from positive-measure boundary sets with low-frequency spectral concentration. This result shows that boundary observability of solutions to the heat equation can also be achieved in W^2,∞ domains. Finally, as an application, we prove that measurements of the boundary normal derivative on J uniquely determine the initial state of heat equation. Under an a priori bound in D(−Δ_g, D^s/2), the recovery satisfies a logarithmic stability estimate of optimal order s/2.
ER -