{
  "id": "1908.11186",
  "version": "v1",
  "published": "2019-08-29T12:50:48.000Z",
  "updated": "2019-08-29T12:50:48.000Z",
  "title": "Well-posedness of renormalized solutions for a stochastic $p$-Laplace equation with $L^1$-initial data",
  "authors": [
    "Niklas Sapountzoglou",
    "Aleksandra Zimmermann"
  ],
  "comment": "43 pages, preprint",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider a $p$-Laplace evolution problem with stochastic forcing on a bounded domain $D\\subset\\mathbb{R}^d$ with homogeneous Dirichlet boundary conditions for $1<p<\\infty$. The additive noise term is given by a stochastic integral in the sense of It\\^{o}. The technical difficulties arise from the merely integrable random initial data $u_0$ under consideration. Due to the poor regularity of the initial data, estimates in $W^{1,p}_0(D)$ are available with respect to truncations of the solution only and therefore well-posedness results have to be formulated in the sense of generalized solutions. We extend the notion of renormalized solution for this type of SPDEs, show well-posedness in this setting and study the Markov properties of solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-29T12:50:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K55",
      "35R60",
      "60H15",
      "35D99"
    ],
    "keywords": [
      "renormalized solution",
      "laplace equation",
      "stochastic",
      "homogeneous dirichlet boundary conditions",
      "laplace evolution problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}