{
  "id": "1206.2257",
  "version": "v2",
  "published": "2012-06-02T21:19:19.000Z",
  "updated": "2012-09-06T10:34:45.000Z",
  "title": "Ultrafunctions and generalized solutions",
  "authors": [
    "Vieri Benci"
  ],
  "comment": "34 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "The theory of distributions provides generalized solutions for problems which do not have a classical solution. However, there are problems which do not have solutions, not even in the space of distributions. As model problem you may think of -\\Deltau=u^{p-1}, u>0, p\\geq(2N)/(N-2) with Dirichlet boundary conditions in a bounded open star-shaped set. Having this problem in mind, we construct a new class of functions called ultrafunctions in which the above problem has a (generalized) solution. In this construction, we apply the general ideas of Non Archimedean Mathematics and some techniques of Non Standard Analysis. Also, some possible applications of ultrafunctions are discussed.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-09-06T10:34:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26E30",
      "26E35",
      "35D99",
      "81Q99"
    ],
    "keywords": [
      "generalized solutions",
      "ultrafunctions",
      "dirichlet boundary conditions",
      "non archimedean mathematics",
      "non standard analysis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.2257B"
    }
  }
}