{
  "abstract": "We study global compact finite propagation (GCFP) for the wave equation $u_{tt}=\\operatorname{div}(A(x)\\nabla u)$ on $\\mathbb{R}^d$, that is, the property that a disturbance starting in a compact set cannot reach spatial infinity in finite time. For smooth strictly positive matrices $A$ we show that GCFP implies $\\int_1^\\infty a_*(r)^{-1/2}\\,dr=\\infty$, where $a_*(r)$ is the minimum over the sphere of radius $r$ of $\\langle Aω,ω\\rangle$; the proof uses the equivalence of GCFP with completeness of the Riemannian metric $A^{-1}$, of which we prove the necessary direction for the cosine family of the Friedrichs extension. We also prove that the analogous divergence condition with $a^*(r)$, the maximum over the sphere, is sufficient, and that this holds for measurable coefficients $A\\in L^1_{\\mathrm{loc}}$ with $A\\ge I$. Finally, we construct a locally bounded measurable matrix with $a_*=a^*$ for which GCFP holds although the integral converges, so the necessary condition is not stable under the loss of continuity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00876",
  "author_total": 1,
  "authors": [
    {
      "name": "M. A. Perelmuter"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7ar0qcp6ns372.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00876",
  "fulltext_url": null,
  "id": "7ar0qcp6ns372",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00876",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00876",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7ar0qcp6ns372.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00876",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00876"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Radial conditions for global compact finite propagation for divergence-form elliptic operators",
  "title_text": "Radial conditions for global compact finite propagation for divergence-form elliptic operators",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:46Z",
  "url": "https://research.draconitic.com/paper/7ar0qcp6ns372",
  "venue": "arXiv",
  "versions": []
}