{
  "id": "2403.13539",
  "version": "v1",
  "published": "2024-03-20T12:20:27.000Z",
  "updated": "2024-03-20T12:20:27.000Z",
  "title": "The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth",
  "authors": [
    "Simone Ciani",
    "Eurica Henriques",
    "Igor i. Skrypnik"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this work we prove that the non-negative functions $u \\in L^s_{loc}(\\Omega)$, for some $s>0$, belonging to the De Giorgi classes \\begin{equation}\\label{eq0.1} \\fint\\limits_{B_{r(1-\\sigma)}(x_{0})} \\big|\\nabla \\big(u-k\\big)_{-}\\big|^{p}\\, dx \\leqslant \\frac{c}{\\sigma^{q}} \\,\\Lambda\\big(x_{0}, r, k\\big)\\bigg(\\frac{k}{r}\\bigg)^{p}\\bigg(\\frac{\\big|B_{r}(x_{0})\\cap\\big\\{u\\leqslant k\\big\\}\\big|}{|B_{r}(x_{0})|}\\bigg)^{1-\\delta}, \\end{equation} under proper assumptions on $\\Lambda$, satisfy a weak Harnack inequality with a constant depending on the $L^s$-norm of $u$. Under suitable assumptions on $\\Lambda$, the minimizers of elliptic functionals with generalized Orlicz growth belong to De Giorgi classes satisfying \\eqref{eq0.1}; thus this study gives a wider interpretation of Harnack-type estimates derived to double-phase, degenerate double-phase functionals and functionals with variable exponents.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-20T12:20:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B40",
      "35B45",
      "35B65"
    ],
    "keywords": [
      "weak harnack inequality",
      "elliptic functionals",
      "unbounded minimizers",
      "giorgi classes",
      "degenerate double-phase functionals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}