{
  "abstract": "Let $(\\mathcal{M},g)$ be a compact connected $W^{2,\\infty}$ 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 $\\partial\\mathcal{M}\\times(-s_0,s_0)$ of positive surface measure. Moreover, we apply the Dirichlet spectral inequality to establish the boundary observability estimate \\[ \\|u(T)\\|_{L^2(\\mathcal{M})}^2 \\le C \\int_J \\|\\partial_{ν_g}u(x,t)\\|^2 \\,\\mathrm{d}S_g\\,\\mathrm{d}t . \\] where $J\\subset\\partial\\mathcal{M}\\times(0,T)$ is measurable and has positive surface--time measure. A localized theorem requiring $W^{2,\\infty}$ 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 $W^{2,\\infty}$ 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 $s/2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.21039",
  "author_total": 3,
  "authors": [
    {
      "name": "Lu Chen"
    },
    {
      "name": "Haolin Liu"
    },
    {
      "name": "Hongyu Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bmkcyn5430928.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.21039",
  "fulltext_url": null,
  "id": "bmkcyn5430928",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.21039",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7204137782",
  "pdf_url": "https://arxiv.org/pdf/2608.21039",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bmkcyn5430928.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2608.21039",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.21039"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Boundary Spectral Inequalities from Measurable Sets on $W^{2,\\infty}$ manifolds and their applications",
  "title_text": "Boundary Spectral Inequalities from Measurable Sets on W^2,∞ manifolds and their applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/bmkcyn5430928",
  "venue": "arXiv",
  "versions": []
}