{
  "id": "1805.00282",
  "version": "v1",
  "published": "2018-05-01T12:03:28.000Z",
  "updated": "2018-05-01T12:03:28.000Z",
  "title": "The Helmholtz equation in random media: well-posedness and a priori bounds",
  "authors": [
    "O. R. Pembery",
    "E. A. Spence"
  ],
  "comment": "23 pages, 2 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove well-posedness results and a priori bounds on the solution of the Helmholtz equation $\\nabla\\cdot(A\\nabla u) + k^2 n u = -f$, posed either in $\\mathbb{R}^d$ or in the exterior of a star-shaped Lipschitz obstacle, for a class of random $A$ and $n,$ random data $f$, and for all $k>0$. The particular class of $A$ and $n$ and the conditions on the obstacle ensure that the problem is nontrapping almost surely. These are the first well-posedness results and a priori bounds for the stochastic Helmholtz equation for arbitrarily large $k$ and for $A$ and $n$ varying independently of $k$. These results are obtained by combining recent bounds on the Helmholtz equation for deterministic $A$ and $n$ and general arguments (i.e. not specific to the Helmholtz equation) presented in this paper for proving a priori bounds and well-posedness of variational formulations of linear elliptic stochastic PDEs. We emphasise that these general results do not rely on either the Lax-Milgram theorem or Fredholm theory, since neither are applicable to the stochastic variational formulation of the Helmholtz equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-01T12:03:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J05",
      "35R60",
      "60H15"
    ],
    "keywords": [
      "priori bounds",
      "random media",
      "linear elliptic stochastic pdes",
      "stochastic helmholtz equation",
      "stochastic variational formulation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}