{
  "abstract": "We study the Dirichlet problem for an infinite series of $p$-Laplacians, $$-\\sum_{p=2}^{\\infty}a_{p}Δ_{p}u=f \\ \\text{ in } Ω, \\qquad u=g \\ \\text{ on } \\partialΩ,$$ where $\\{a_{p}\\}$ is a sequence of nonnegative numbers whose power series has radius of convergence $σ\\in(0,\\infty]$. The operator is formally $-\\operatorname{div}( {A}(|\\nabla u|)\\nabla u)$ with $ {A}$ singular at $|\\nabla u|=σ$; model cases are the mean curvature operator in Minkowski space and the Born-Infeld operator of nonlinear electrostatics. The radius of convergence forces the gradient constraint $\\|\\nabla u\\|_\\infty\\leσ$, so the natural variational problem is constrained. A unique minimizer of the associated energy always exists (for boundary data compatible with the constraint) and always solves a variational inequality, and we identify the saturation flux $Λ:=\\sum_{p\\ge2}a_pσ^{p-1}$ as the quantity governing solvability of the equation: if $Λ<\\infty$ a weak solution exists only if $|\\int_E f|\\leΛP(E)$ for every set of finite perimeter $E\\SubsetΩ$, so for $f\\equivλ$ no solution exists once $λ>Λh(Ω)$, $h(Ω)$ the Cheeger constant of $Ω$. The threshold is sharp on balls, where the minimizer has a saturation region $\\{|\\nabla u|=σ\\}$ of positive measure. When $Λ=\\infty$, as for Born-Infeld type operators, no such obstruction is present, and the minimizer solves the equation whenever a Lipschitz bound below $σ$ is available; for $f\\equiv0$ we obtain such a bound, uniform in the truncations of the series, under a bounded slope condition on the boundary datum. As applications we obtain an isoperimetric bound on the charge densities supported by a nonlinear electrostatics with saturating displacement, and a quantitative convergence rate for the weak field expansion of the Born-Infeld model.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28288",
  "author_total": 4,
  "authors": [
    {
      "name": "Haydar Abdel Hamid"
    },
    {
      "name": "Juan Carlos Chata Ortiz"
    },
    {
      "name": "Francesco Petitta"
    },
    {
      "name": "Julio Daniel Rossi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6h6vjna99et7p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28288",
  "fulltext_url": null,
  "id": "6h6vjna99et7p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28288",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7214227268",
  "pdf_url": "https://arxiv.org/pdf/2609.28288",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6h6vjna99et7p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2609.28288",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28288"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Gradient constraints, Born-Infeld, and maximal surfaces via superposition of infinitely many $p$-Laplacians",
  "title_text": "Gradient constraints, Born-Infeld, and maximal surfaces via superposition of infinitely many p-Laplacians",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/6h6vjna99et7p",
  "venue": "arXiv",
  "versions": []
}