{
  "id": "1806.10021",
  "version": "v1",
  "published": "2018-06-26T14:37:40.000Z",
  "updated": "2018-06-26T14:37:40.000Z",
  "title": "Limited regularity of solutions to fractional heat and Schrödinger equations",
  "authors": [
    "Gerd Grubb"
  ],
  "comment": "21 pages",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "When $P$ is the fractional Laplacian $(-\\Delta )^a$, $0<a<1$, or a pseudodifferential generalization thereof, the Dirichlet problem for the associated heat equation over a smooth set $\\Omega \\subset{\\Bbb R}^n$: $r^+Pu(x,t)+\\partial_tu(x,t)=f(x,t)$ on $\\Omega \\times \\,]0,T[\\,$, $u(x,t)=0$ for $x\\notin\\Omega $, $u(x,0)=0$, is known to be solvable in relatively low-order Sobolev or H\\\"older spaces. We now show that in contrast with differential operator cases, the regularity of $u$ in $x$ at $\\partial\\Omega $ when $f$ is very smooth cannot in general be improved beyond a certain estimate. An improvement requires the vanishing of a Neumann boundary value. --- There is a similar result for the Schr\\\"odinger Dirichlet problem $r^+Pv(x)+Vv(x)=g(x)$ on $\\Omega $, $v(x)=0$ for $x\\notin \\Omega $, with $V(x)\\in C^\\infty $. The proofs involve a precise description of the Dirichlet domains in terms of functions supported in $\\overline\\Omega$ and functions pulled back from boundary values.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-26T14:37:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K05",
      "35K35",
      "35S11",
      "47G30",
      "60G52"
    ],
    "keywords": [
      "schrödinger equations",
      "fractional heat",
      "limited regularity",
      "dirichlet problem",
      "differential operator cases"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}