{
  "id": "1201.3809",
  "version": "v1",
  "published": "2012-01-18T15:03:04.000Z",
  "updated": "2012-01-18T15:03:04.000Z",
  "title": "Maximal $L^2$ regularity for Dirichlet problems in Hilbert spaces",
  "authors": [
    "Giuseppe Da Prato",
    "Alessandra Lunardi"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the Dirichlet problem $\\lambda U - {\\mathcal{L}}U= F$ in \\mathcal{O}, U=0 on $\\partial \\mathcal{O}$. Here $F\\in L^2(\\mathcal{O}, \\mu)$ where $\\mu$ is a nondegenerate centered Gaussian measure in a Hilbert space $X$, $\\mathcal{L}$ is an Ornstein-Uhlenbeck operator, and $\\mathcal{O}$ is an open set in $X$ with good boundary. We address the problem whether the weak solution $U$ belongs to the Sobolev space $W^{2,2}(\\mathcal{O}, \\mu)$. It is well known that the question has positive answer if $\\mathcal{O} = X$; if $\\mathcal{O} \\neq X$ we give a sufficient condition in terms of geometric properties of the boundary $\\partial \\mathcal{O}$. The results are quite different with respect to the finite dimensional case, for instance if \\mathcal{O} is the ball centered at the origin with radius $r$ we prove that $U\\in W^{2,2}(\\mathcal{O}, \\mu)$ only for small $r$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-01-18T15:03:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dirichlet problem",
      "hilbert space",
      "regularity",
      "nondegenerate centered gaussian measure",
      "finite dimensional case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.3809D"
    }
  }
}