{
  "abstract": "This paper introduces a new extrapolation framework within the Calculus of Variations. We develop a robust quantitative theory of the $Γ$-convergence approach of De Giorgi by combining it with Gagliardo's work on relative completions in interpolation theory. A central new tool is the extrapolation of \\emph{variational $K$-functionals}, which associates to an arbitrary family of functionals a dynamic penalty method that simultaneously characterizes their $Γ$-limits and yields explicit rates of convergence for minimizers. As a technical foundation we prove compactness extrapolation theorems also involving Gagliardo completions. To illustrate the mechanism, we apply the technique to variational models arising in image processing, notably the Rudin--Osher--Fatemi (ROF) and Chan--Esedoglu models. Our main result in this setting is a contribution to the \\emph{inverse regularity problem} associated to discrete approximations of the ROF model: we show that the required regularity of the observed image that guarantees a prescribed optimal rate of convergence is measured in terms of the decay of variational $K$-functionals. Surprisingly, our characterization is explicitly captured by using the recently introduced spaces of Brezis, Van Schaftingen, and Yung. Moreover, our methods unify, and considerably extend, classical pointwise convergence and $Γ$-versions of the Bourgain--Brezis--Mironescu and Maz'ya--Shaposhnikova formulae, as well as related results in a variety of settings that involve BMO-type functionals, heat kernels, Sobolev spaces on metric measure spaces, ... By bridging De Giorgi's variational convergence with Gagliardo's completion through variational $K$-functionals and extrapolation of compactness, this work establishes a quantitative approach to the Calculus of Variations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06645",
  "author_total": 2,
  "authors": [
    {
      "name": "Oscar Dominguez"
    },
    {
      "name": "Mario Milman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/agwc3nv0y4py9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06645",
  "fulltext_url": null,
  "id": "agwc3nv0y4py9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06645",
  "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",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.AP"
    ],
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      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
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  "openalex": "W7220696002",
  "pdf_url": "https://arxiv.org/pdf/2610.06645",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/agwc3nv0y4py9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06645",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06645"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Quantitative $Γ$-Convergence: A Variational Extrapolation Framework",
  "title_text": "Quantitative Γ-Convergence: A Variational Extrapolation Framework",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/agwc3nv0y4py9",
  "venue": "arXiv",
  "versions": []
}