{
  "abstract": "We give an affirmative answer to the planar local smoothing problem in Question~1.7 of D.~Ntalampekos and positive and negative answers to the basic approximation and level-set parts of his higher-dimensional Question~1.8. In every dimension $n\\ge2$, each continuous Lebesgue monotone function in $W^{1,p}$ on a bounded open set admits uniform and strong $W^{1,p}$ approximation by monotone $C^{1,α}_{\\loc}$ functions, with unchanged Sobolev boundary values and no increase of the $p$-Dirichlet energy, for $1<p<\\infty$. The energy can be made strictly smaller unless the original function is $p$-harmonic. In the plane we obtain smooth local replacement at every isolated $p$-harmonic critical point, with arbitrarily small $C^1$ and Sobolev error. A point of gradient index $-m$ can be resolved into exactly $m$ nondegenerate saddles. Together with Ntalampekos's planar theorem, this gives smooth monotone density with fixed boundary values; at a nonsmooth $p$-harmonic minimizer the energy increase is unavoidable but can tend to zero. In dimensions $n\\ge3$, an explicit Lipschitz monotone function has a nonmanifold point on every level in an interval, although it admits smooth monotone approximation. A product extension of a planar homogeneous $7$-harmonic function also rules out a general discrete exceptional set under the exact boundary and energy constraints. The answer to Question~1.8 thus distinguishes the five basic approximation properties from the stronger topological and exceptional-set conclusions. Applications include constrained integral functionals, strict $BV$ convergence of superlevel sets, nonlinear flux convergence, and stability of persistence diagrams under tameness assumptions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34295",
  "author_total": 1,
  "authors": [
    {
      "name": "Deguang Zhong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/89f1z10a5wyga.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34295",
  "fulltext_url": null,
  "id": "89f1z10a5wyga",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34295",
  "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"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7214718340",
  "pdf_url": "https://arxiv.org/pdf/2609.34295",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/89f1z10a5wyga.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.34295",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34295"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Monotone Sobolev functions: approximation, critical points, and level sets",
  "title_text": "Monotone Sobolev functions: approximation, critical points, and level sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/89f1z10a5wyga",
  "venue": "arXiv",
  "versions": []
}