{
  "abstract": "We give a proof of the $Γ$-liminf inequality for the Maz'ya--Shaposhni\\-kova limit, as $s\\to0^+$, of the fractional Sobolev seminorm, based on spherical rearrangements. The key point is that the rearrangements of a weakly convergent sequence are monotone radial profiles, which converge strongly by Helly's theorem; since the energy does not increase under rearrangement (in the limit), this reduces the liminf inequality to a pointwise convergence statement for radially decreasing functions. As an application, we treat truncated energies, in which the interaction is restricted to a ball of radius $r_s\\to\\infty$. Assuming $r_s^{-2s}\\toλ\\in[0,1]$, we prove that the scaled energies $Γ$-converge, with respect to weak $L^2$-convergence, to $$ (1-λ)Nω_N\\|u\\|_{L^2}^2, $$ where $ω_N$ is the measure of the unit ball of $\\mathbb R^N$. This identifies the contribution of the long-range interactions and gives a continuous interpolation between the classical Maz'ya--Shaposhnikova limit ($λ=0$) and the vanishing-energy regime ($λ=1$).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05499",
  "author_total": 3,
  "authors": [
    {
      "name": "Andrea Braides"
    },
    {
      "name": "Anneliese Defranceschi"
    },
    {
      "name": "Enrico Vitali"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3ed84rmdq0hx9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05499",
  "fulltext_url": null,
  "id": "3ed84rmdq0hx9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05499",
  "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|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7220733390",
  "pdf_url": "https://arxiv.org/pdf/2610.05499",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3ed84rmdq0hx9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05499",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05499"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Rearrangements and the Maz'ya-Shaposhnikova limit: $Γ$-convergence for truncated interactions",
  "title_text": "Rearrangements and the Maz'ya-Shaposhnikova limit: Γ-convergence for truncated interactions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/3ed84rmdq0hx9",
  "venue": "arXiv",
  "versions": []
}