{
  "id": "2208.09528",
  "version": "v1",
  "published": "2022-08-19T19:28:36.000Z",
  "updated": "2022-08-19T19:28:36.000Z",
  "title": "The fractional $p\\,$-biharmonic systems: optimal Poincaré constants, unique continuation and inverse problems",
  "authors": [
    "Manas Kar",
    "Jesse Railo",
    "Philipp Zimmermann"
  ],
  "comment": "31 pages",
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "This article investigates nonlocal, fully nonlinear generalizations of the classical biharmonic operator $(-\\Delta)^2$. These fractional $p$-biharmonic operators appear naturally in the variational characterization of the optimal fractional Poincar\\'e constants in Bessel potential spaces. We study the following basic questions for anisotropic fractional $p$-biharmonic systems: existence and uniqueness of weak solutions to the associated interior source and exterior value problems, unique continuation properties (UCP), monotonicity relations, and inverse problems for the exterior Dirichlet-to-Neumann maps. Furthermore, we show the UCP for the fractional Laplacian in all Bessel potential spaces $H^{t,p}$ for any $t\\in \\mathbb{R}$, $1 \\leq p < \\infty$ and $s \\in \\mathbb{R}_+ \\setminus \\mathbb{N}$: If $u\\in H^{t,p}(\\mathbb{R}^n)$ satisfies $(-\\Delta)^su=u=0$ in a nonempty open set $V$, then $u\\equiv 0$ in $\\mathbb{R}^n$. This property of the fractional Laplacian is then used to obtain a UCP for the fractional $p$-biharmonic systems and plays a central role in the analysis of the associated inverse problems. Our proofs use variational methods and the Caffarelli-Silvestre extension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-19T19:28:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R30",
      "26A33",
      "42B37"
    ],
    "keywords": [
      "biharmonic systems",
      "inverse problems",
      "bessel potential spaces",
      "fractional laplacian",
      "optimal fractional poincare constants"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}