{
  "abstract": "For a nonconstant planar $p$-harmonic function $u$, with $1 < p < \\infty$, and according to previous results, most notably Aronsson's fundamental analysis of the hodograph representation in a neighborhood of an isolated critical point, we introduce the flux-normalized frequency $$ N_*(x_0,r) = \\frac{r\\displaystyle\\int_{B_r(x_0)} |D u|^p\\,dx} {\\displaystyle\\int_{\\partial B_r(x_0)} |D u|^{p-2}(u-u(x_0))^2\\,dS}. $$ This ratio constitutes the natural counterpart of Almgren's frequency function in the nonlinear setting, as it recovers precisely the degree of homogeneity for every homogeneous $p$-harmonic profile. We further establish the corresponding gauge-corrected version of Almgren-type monotonicity and reinterpret, in terms of the frequency function, the classical planar unique continuation property. In addition, we derive a quantitative pinching estimate for the flux-normalized frequency and identify the specific obstruction that prevents a straightforward extension of the method to higher dimensions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30847",
  "author_total": 1,
  "authors": [
    {
      "name": "Yi Ru-Ya Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3cf39hawk7jah.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30847",
  "fulltext_url": null,
  "id": "3cf39hawk7jah",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30847",
  "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",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7204881172",
  "pdf_url": "https://arxiv.org/pdf/2608.30847",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3cf39hawk7jah.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30847",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30847"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "An Almgren-type formula for planar $p$-harmonic functions",
  "title_text": "An Almgren-type formula for planar p-harmonic functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/3cf39hawk7jah",
  "venue": "arXiv",
  "versions": []
}