{
  "id": "1611.03532",
  "version": "v1",
  "published": "2016-11-10T22:05:13.000Z",
  "updated": "2016-11-10T22:05:13.000Z",
  "title": "On the strict monotonicity of the first eigenvalue of the $p$-Laplacian on annuli",
  "authors": [
    "T. V. Anoop",
    "Vladimir Bobkov",
    "Sarath Sasi"
  ],
  "comment": "19 pages",
  "categories": [
    "math.AP",
    "math.OC"
  ],
  "abstract": "Let $B_1$ be a ball in $\\mathbb{R}^N$ centred at the origin and $B_0$ be a smaller ball compactly contained in $B_1$. For $p\\in(1, \\infty)$, using the shape derivative method, we show that the first eigenvalue of the $p$-Laplacian in annulus $B_1\\setminus \\overline{B_0}$ strictly decreases as the inner ball moves towards the boundary of the outer ball. The analogous results for the limit cases as $p \\to 1$ and $p \\to \\infty$ are also discussed. Using our main result, further we prove the nonradiality of the eigenfunctions associated with the points on the first nontrivial curve of the Fu\\v{c}ik spectrum of the $p$-Laplacian on bounded radial domains.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-10T22:05:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J92",
      "35P30",
      "35B06",
      "49R05"
    ],
    "keywords": [
      "first eigenvalue",
      "strict monotonicity",
      "first nontrivial curve",
      "inner ball moves",
      "shape derivative method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}