{
  "id": "2208.07175",
  "version": "v1",
  "published": "2022-08-15T13:22:31.000Z",
  "updated": "2022-08-15T13:22:31.000Z",
  "title": "Fourier methods for fractional-order operators",
  "authors": [
    "Gerd Grubb"
  ],
  "comment": "20 pages. Prepared for the Proceedings of the RIMS Symposium \"Harmonic Analysis and Nonlinear Partial Differential equations\", July 11-13, 2022, in the RIMS Kokyuroku Bessatsu series",
  "categories": [
    "math.AP",
    "math.SP"
  ],
  "abstract": "This is a survey on the use of Fourier transformation methods in the treatment of boundary problems for the fractional Laplacian $(-\\Delta)^a$ (0<a<1), and pseudodifferential generalizations P, over a bounded open set $\\Omega$ in $R^n$. Two points are explained in detail: 1) How the factor $d^a$, with $d(x)=dist(x,d\\Omega)$, comes into the picture, related to the fact that the precise solution spaces for the homogeneous Dirichlet problem are so-called a-transmission spaces. 2) The natural definition of a local nonhomogeneous Dirichlet condition $\\gamma_0(u/d^{a-1})=\\varphi$. We also give brief accounts of some further developments: Evolution problems (for $d_t u - r^+Pu = f(x,t)$) and resolvent problems (for $Pu-\\lambda u=f$), also with nonzero boundary conditions. Integration by parts, Green's formula.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-15T13:22:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35S15",
      "47G30",
      "35J25",
      "35K05",
      "60G51"
    ],
    "keywords": [
      "fourier methods",
      "fractional-order operators",
      "nonzero boundary conditions",
      "local nonhomogeneous dirichlet condition",
      "precise solution spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}