{
  "abstract": "We extend the theory of De Giorgi-Nash-Moser to elliptic equations with symmetric coefficients $\\mathbf{a}(x)$ which are possibly degenerate and unbounded but satisfy a coarse ellipticity condition. We prove local upper bounds for weak subsolutions, a weak Harnack inequality for nonnegative supersolutions, and a Harnack inequality for nonnegative solutions. The coarse ellipticity hypothesis requires spatial moments of coarse-grained matrices, suitably discounted and summed across scales, to be finite. In particular, it holds if, for some $α,β\\geq0$ and $1<p,q<\\infty$, $$\\mathbf{a}\\in W^{-α,p}\\cap L^1, \\quad \\mathbf{a}^{-1}\\in W^{-β,q}\\cap L^1 \\quad\\text{and}\\quad \\frac{α+β}{2}+\\frac{d-1}{2}\\Big(\\frac1p+\\frac1q\\Big)<1.$$ We show that the coefficient range is sharp, including its boundary, in every dimension $d\\geq3$, for $α=β=0$ and for the corresponding Besov-type quasi-norms of negative order when $α,β>0$. For $α=β=0$, it corresponds to the results of Bella and Schäffner [8].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11879",
  "author_total": 4,
  "authors": [
    {
      "name": "Scott Armstrong"
    },
    {
      "name": "Benny Avelin"
    },
    {
      "name": "Tuomo Kuusi"
    },
    {
      "name": "Aatu Turpeinen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d6fh2v0bqh66k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11879",
  "fulltext_url": null,
  "id": "d6fh2v0bqh66k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11879",
  "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",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7221150062",
  "pdf_url": "https://arxiv.org/pdf/2610.11879",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d6fh2v0bqh66k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11879",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11879"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Coarse ellipticity and De Giorgi-Nash-Moser theory in the optimal range",
  "title_text": "Coarse ellipticity and De Giorgi-Nash-Moser theory in the optimal range",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/d6fh2v0bqh66k",
  "venue": "arXiv",
  "versions": []
}