{
  "id": "1008.1565",
  "version": "v1",
  "published": "2010-08-09T18:36:07.000Z",
  "updated": "2010-08-09T18:36:07.000Z",
  "title": "On the dependence of the reflection operator on boundary conditions for biharmonic functions",
  "authors": [
    "Tatiana Savina"
  ],
  "comment": "18 pages",
  "journal": "J. Math. Anal. Appl., 370 (2010), 716-725",
  "categories": [
    "math.AP"
  ],
  "abstract": "The biharmonic equation arises in areas of continuum mechanics including linear elasticity theory and the Stokes flows, as well as in a radar imaging problem. We discuss the reflection formulas for the biharmonic functions $u(x,y)\\in\\mathbb{R}^2$ subject to different boundary conditions on a real-analytic curve in the plane. The obtained formulas, generalizing the celebrated Schwarz symmetry principle for harmonic functions, have different structures. In particular, in the special case of the boundary, $\\Gamma_0 :=\\{y=0\\}$, reflections are point to point when the given on $\\Gamma_0$ conditions are $u=\\partial_nu=0$, $u=\\Delta u=0$ or $\\partial_nu=\\partial n\\Delta u=0$, and point to a continuous set when $u=\\partial_n\\Delta u=0$ or $\\partial_nu=\\Delta u=0$ on $\\Gamma_0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-08-09T18:36:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31B30",
      "35G05",
      "35J40"
    ],
    "keywords": [
      "boundary conditions",
      "biharmonic functions",
      "reflection operator",
      "dependence",
      "biharmonic equation arises"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.1565S"
    }
  }
}