{
  "id": "2111.14763",
  "version": "v2",
  "published": "2021-11-29T18:15:33.000Z",
  "updated": "2022-08-16T13:57:22.000Z",
  "title": "Resolvents for fractional-order operators with nonhomogeneous local boundary conditions",
  "authors": [
    "Gerd Grubb"
  ],
  "comment": "47 pages. Small improvements here and there",
  "categories": [
    "math.AP",
    "math.FA",
    "math.SP"
  ],
  "abstract": "For $2a$-order strongly elliptic operators $P$ generalizing $(-\\Delta )^a$, $0<a<1$, the treatment of the homogeneous Dirichlet problem on a bounded open set $\\Omega \\subset R^n$ by pseudodifferential methods, has been extended in a recent joint work with Helmut Abels to nonsmooth settings, showing regularity theorems in $L_q$-Sobolev spaces $H_q^s$ for $1<q<\\infty $, when $\\Omega $ is $C^{\\tau +1}$ with a finite $\\tau >2a$. Presently, we study the $L_q$-Dirichlet realizations of $P$ and $P^*$, showing invertibility or Fredholmness, finding smoothness results for the kernels and cokernels, and establishing similar results for $P-\\lambda I$, $\\lambda \\in C$. The solution spaces equal $a$-transmission spaces $H_q^{a(s+2a)}(\\bar\\Omega)$. Similar results are shown for nonhomogeneous Dirichlet problems, prescribing the local Dirichlet trace $(u/d^{a-1})|_{\\partial\\Omega }$, $d(x)=dist(x,\\partial\\Omega)$. They are solvable in the larger spaces $H_q^{(a-1)(s+2a)}(\\bar\\Omega)$. Moreover, the nonhomogeneous problem with a spectral parameter $\\lambda \\in C$, $$ Pu-\\lambda u = f \\text { in }\\Omega ,\\quad u=0 \\text { in }R^n\\setminus \\Omega ,\\quad (u/d^{a-1 })|_{\\partial\\Omega }=\\varphi \\text{ on }\\partial\\Omega , $$ is for $q<(1-a)^{-1}$ shown to be uniquely resp. Fredholm solvable when $\\lambda $ is in the resolvent set resp. the spectrum of the $L_2$-Dirichlet realization. Finally, we show solvability results for evolution problems $Pu+d_tu= f(x,t)$ in $L_2$ and $L_q$-based spaces over $C^{1+\\tau}$-domains, including nonhomogeneous local boundary conditions.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-08-16T13:57:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35S15",
      "47G30",
      "35J25",
      "35K05",
      "60G51"
    ],
    "keywords": [
      "nonhomogeneous local boundary conditions",
      "fractional-order operators",
      "similar results",
      "dirichlet realization",
      "dirichlet problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}