{
  "id": "2210.12441",
  "version": "v1",
  "published": "2022-10-22T13:15:22.000Z",
  "updated": "2022-10-22T13:15:22.000Z",
  "title": "bounded weak solutions of degenerate p-poisson equations",
  "authors": [
    "Sullivan Francis MacDonald",
    "Scott Rodney"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper studies boundedness of weak solutions to $p$-Poisson type equations on a bounded domain $\\Omega\\subset\\mathbb{R}^n$. Given a symmetric, non-negative definite matrix-valued function $Q$ defined on $\\Omega$, a weight $v\\in L^1_\\mathrm{loc}(\\Omega)$ comparable to the operator norm of $Q$, and a suitable non-negative function $\\tau$, we show that weak solutions $(u,\\nabla u)\\in QH^{1,p}_0(\\Omega,v)$ to the Dirichlet problem \\begin{eqnarray} \\begin{array}{rccl} -\\mathrm{{div}}(\\left|\\sqrt{Q}\\nabla u\\right|^{p-2}Q\\nabla u)+\\tau\\left|u\\right|^{p-2}uv&=&fv&\\textrm{in }\\Omega\\nonumber u&= & 0&\\textrm{on }\\partial\\Omega \\end{array} \\end{eqnarray} are \\emph{a priori} bounded, provided a Sobolev inequality with gain $\\sigma>1$, $$\\left(\\int_\\Omega |\\varphi|^{p\\sigma}dx\\right)^{1/p\\sigma} \\leq C\\left(\\int_\\Omega |\\sqrt{Q}\\nabla\\varphi|^pdx\\right)^{1/p}, $$ holds for every Lipschitz $\\varphi$ with compact support in $\\Omega$ and the data function $f$ belongs to a weighted Orlicz space $L^\\Gamma(\\Omega,v)$. The Young function $\\Gamma$ is required to satisfy an integral condition that depends on the Sobolev gain factor $\\sigma$. In particular, we show that the weak solution satisfies the estimate $$ \\|u\\|_{L^\\infty(\\Omega,v)} \\leq C\\|f\\|_{L^\\Gamma(\\Omega,v)}^\\frac{1}{p-1}. $$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-22T13:15:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B45",
      "35D30",
      "35J25",
      "46E30"
    ],
    "keywords": [
      "degenerate p-poisson equations",
      "bounded weak solutions",
      "poisson type equations",
      "paper studies boundedness",
      "sobolev gain factor"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}