{
  "id": "2305.16672",
  "version": "v1",
  "published": "2023-05-26T06:42:04.000Z",
  "updated": "2023-05-26T06:42:04.000Z",
  "title": "Strict monotonicity of the first $q$-eigenvalue of the fractional $p$-Laplace operator over annuli",
  "authors": [
    "K Ashok Kumar",
    "Nirjan Biswas"
  ],
  "comment": "14 pages, 2 figures",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "math.OC"
  ],
  "abstract": "Let $B, B'\\subset \\mathbb{R}^d$ with $d\\geq 2$ be two balls such that $B'\\subset \\subset B$ and the position of $B'$ is varied within $B$. For $p\\in (1, \\infty ),$ $s\\in (0,1)$, and $q \\in [1, p^*_s)$ with $p^*_s=\\frac{dp}{d-sp}$ if $sp < d$ and $p^*_s=\\infty $ if $sp \\geq d$, let $\\lambda ^s_{p,q}(B\\setminus \\overline{B'})$ be the first $q$-eigenvalue of the fractional $p$-Laplace operator $(-\\Delta _p)^s$ in $B\\setminus \\overline{B'}$ with the homogeneous non-local Dirichlet boundary conditions. We prove that $\\lambda ^s_{p,q}(B\\setminus \\overline{B'})$ strictly decreases as the inner ball $B'$ moves towards the outer boundary $\\partial B$. To obtain the strict monotonicity, we establish a strict Faber-Krahn type inequality for $\\lambda _{p,q}^s(\\cdot )$ under polarization. This extends some monotonicity results obtained by Djitte-Fall-Weth (Calc. Var. Partial Differential Equations (2021), 60:231) in the case of $(-\\Delta )^s$ and $q=1, 2$ to $(-\\Delta _p)^s$ and $q\\in [1, p^*_s).$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-26T06:42:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R11",
      "49Q10",
      "35B51",
      "35B06",
      "47J10"
    ],
    "keywords": [
      "laplace operator",
      "strict monotonicity",
      "fractional",
      "eigenvalue",
      "strict faber-krahn type inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}