{
  "id": "2306.10913",
  "version": "v1",
  "published": "2023-06-19T13:22:45.000Z",
  "updated": "2023-06-19T13:22:45.000Z",
  "title": "Semilinear fractional elliptic PDEs with gradient nonlinearities on open balls: existence of solutions and probabilistic representation",
  "authors": [
    "Guillaume Penent",
    "Nicolas Privault"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:2110.09941, arXiv:2106.12127",
  "categories": [
    "math.NA",
    "cs.NA",
    "math.AP",
    "math.PR"
  ],
  "abstract": "We provide sufficient conditions for the existence of viscosity solutions of fractional semilinear elliptic PDEs of index $\\alpha \\in (1,2)$ with polynomial gradient nonlinearities on $d$-dimensional balls, $d\\geq 2$. Our approach uses a tree-based probabilistic representation based on $\\alpha$-stable branching processes, and allows us to take into account gradient nonlinearities not covered by deterministic finite difference methods so far. Numerical illustrations demonstrate the accuracy of the method in dimension $d=10$, solving a challenge encountered with the use of deterministic finite difference methods in high-dimensional settings.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-19T13:22:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J15",
      "35J25",
      "35J60",
      "35J61",
      "35R11",
      "35B65",
      "60J85",
      "60G51",
      "60G52",
      "65C05",
      "33C05"
    ],
    "keywords": [
      "semilinear fractional elliptic pdes",
      "gradient nonlinearities",
      "probabilistic representation",
      "open balls",
      "deterministic finite difference methods"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}