{
  "id": "1009.0997",
  "version": "v3",
  "published": "2010-09-06T09:28:34.000Z",
  "updated": "2011-01-27T13:41:51.000Z",
  "title": "Spectral asymptotics for Robin problems with a discontinuous coefficient",
  "authors": [
    "Gerd Grubb"
  ],
  "comment": "20 pages, notation simplified. To appear in J. Spectral Theory",
  "journal": "J. Spectral Theory 1 (2011), 155-177",
  "doi": "10.4171/JST/7",
  "categories": [
    "math.AP",
    "math.SP"
  ],
  "abstract": "The spectral behavior of the difference between the resolvents of two realizations $\\tilde A_1$ and $\\tilde A_2$ of a second-order strongly elliptic symmetric differential operator $A$, defined by different Robin conditions $\\nu u=b_1\\gamma_0u$ and $\\nu u=b_2\\gamma_0u$, can in the case where all coefficients are $C^\\infty$ be determined by use of a general result by the author in 1984 on singular Green operators. We here treat the problem for nonsmooth $b_i$. Using a Krein resolvent formula, we show that if $b_1$ and $b_2$ are in $L_\\infty$, the s-numbers $s_j$ of $(\\tilde A_1 -\\lambda)^{-1}-(\\tilde A_2 -\\lambda)^{-1}$ satisfy $s_j j^{3/(n-1)}\\le C$ for all $j$; this improves a recent result for $A=-\\Delta $ by Behrndt et al., that $\\sum_js_j ^p<\\infty$ for $p>(n-1)/3$. A sharper estimate is obtained when $b_1$ and $b_2$ are in $C^\\epsilon$ for some $\\epsilon >0$, with jumps at a smooth hypersurface, namely that $s_j j^{3/(n-1)}\\to c$ for $j\\to \\infty$, with a constant $c$ defined from the principal symbol of $A$ and $b_2-b_1$. As an auxiliary result we show that the usual principal spectral asymptotic estimate for pseudodifferential operators of negative order on a closed manifold extends to products of pseudodifferential operators interspersed with piecewise continuous functions.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-01-27T13:41:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J40",
      "47G30",
      "58C40"
    ],
    "keywords": [
      "robin problems",
      "discontinuous coefficient",
      "usual principal spectral asymptotic estimate",
      "strongly elliptic symmetric differential operator"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.0997G"
    }
  }
}