{
  "abstract": "J. Danskin proved in 1966 that the envelope of a family of continuous functions $\\mathrm F : \\mathbb R^n \\times \\mathrm K \\longrightarrow \\mathbb R$, parameterised by the points of a compact $\\mathrm K \\subseteq \\mathbb R^m$, given by \\[ f(u) := \\max_{k \\in \\mathrm K} \\mathrm F(u,k), \\ \\ \\ \\ f \\, : \\, \\mathbb R^n \\longrightarrow \\mathbb R, \\] is semi-differentiable on $\\mathbb R^n$, if $\\mathrm F$ is sufficiently regular. This result is of utmost importance in numerous applications. Extensions have been proved towards almost every direction, but none permits to replace ``max over $\\mathrm K$\" with ``essential sup over $\\mathrm K$\". We establish the semi-differentiability of generalised $\\mathrm L^\\infty$ envelopes when $ \\mathrm F$ is defined on the product of a Banach space with a measure space. As an application, we obtain a new foundational regularity result in the Calculus of Variations in $\\mathrm L^\\infty$, asserting that supremal functionals are semi-differentiable everywhere, with an explicit formula for their semi-derivative, despite generally being non-differentiable. Moreover, we establish a \\emph{variational characterisation} of (absolute) minimisers of general level-convex $\\mathrm L^\\infty$ functionals via their semi-differentials, revealing the genuine $\\mathrm L^\\infty$ counterpart of the Euler-Lagrange equations. Conventional Aronsson equations and related divergence PDE involving measures, deducible from $\\mathrm L^p$ approximations as $p\\to \\infty$, in general are not sufficient for minimality in $\\mathrm L^\\infty$. Our results dislodge the trademark necessity for extrinsic $\\mathrm L^p$-approximations as $p\\to \\infty$ to derive and study PDEs, reinterpreting the existing $\\mathrm L^\\infty$ theory. Most crucially, they provide a new powerful intrinsic $\\mathrm L^\\infty$ framework, evocative of the straightforward variational methods for integral functionals.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27918",
  "author_total": 1,
  "authors": [
    {
      "name": "Nikos Katzourakis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/76b9s8j0yfa3p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27918",
  "fulltext_url": null,
  "id": "76b9s8j0yfa3p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27918",
  "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"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7214211547",
  "pdf_url": "https://arxiv.org/pdf/2609.27918",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/76b9s8j0yfa3p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27918",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27918"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Semi-differentiability of generalised $L^\\infty$ envelopes with applications to supremal functionals",
  "title_text": "Semi-differentiability of generalised L^∞ envelopes with applications to supremal functionals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/76b9s8j0yfa3p",
  "venue": "arXiv",
  "versions": []
}