{
  "abstract": "In this paper, we study the sharp Sobolev-type inequality associated with the Born--Infeld energy on the lattice graph \\(\\mathbb Z^N\\), \\(N\\geq3\\): \\[ \\frac12\\sum_{x\\in\\mathbb Z^N}\\sum_{y\\sim x} \\left(1-\\sqrt{1-|\\nabla_{xy}u|^2}\\right) \\geq C_{N,α} \\left(\\sum_{x\\in\\mathbb Z^N}|u(x)|^α\\right)^{\\frac{N}{N+α}}, \\] for \\(u\\in D^{1,2}(\\mathbb Z^N)\\cap\\ell^α(\\mathbb Z^N) \\) satisfying \\(|\\nabla_{xy}u|\\leq1\\) for every \\(x\\sim y\\), where \\(C_{N,α}\\) denotes the optimal constant. We determine the exact positivity threshold and prove that \\(C_{N,α}>0\\) if and only if \\(α\\geq2^*:=2N/(N-2)\\). At the Sobolev critical exponent \\(α=2^*\\), we identify the optimal constant as \\[ C_{N,2^*}=\\frac12\\mathcal S_2, \\] where \\(\\mathcal S_2\\) is the optimal discrete Sobolev constant, and show that it is not attained. In the supercritical regime, there exists \\(\\varepsilon_0=\\varepsilon_0(N)>0\\) such that \\(C_{N,α}\\) is attained for every \\(2^*<α<2^*+\\varepsilon_0\\), with a nonnegative Schwarz symmetric extremal function. The main compactness difficulties stem from the lack of a suitable scaling on \\(\\mathbb Z^N\\) and from the nonhomogeneity of the Born--Infeld energy. We overcome them by combining discrete Schwarz rearrangement with a \\(Q_1\\) discrete-to-continuum comparison and by establishing a strict separation as the \\(\\ell^α\\)-norm tends to infinity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18142",
  "author_total": 2,
  "authors": [
    {
      "name": "Chao Ji"
    },
    {
      "name": "Kai Sheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cg4g36d7s4cpg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18142",
  "fulltext_url": null,
  "id": "cg4g36d7s4cpg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18142",
  "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": "W7213529528",
  "pdf_url": "https://arxiv.org/pdf/2609.18142",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cg4g36d7s4cpg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18142",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18142"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp Born--Infeld Sobolev inequality and extremal functions on the lattice graph $\\mathbb Z^N$",
  "title_text": "Sharp Born--Infeld Sobolev inequality and extremal functions on the lattice graph ℤ^N",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/cg4g36d7s4cpg",
  "venue": "arXiv",
  "versions": []
}