{
  "id": "2211.14185",
  "version": "v1",
  "published": "2022-11-25T15:39:37.000Z",
  "updated": "2022-11-25T15:39:37.000Z",
  "title": "Stability for the Sobolev inequality: existence of a minimizer",
  "authors": [
    "Tobias König"
  ],
  "comment": "17 pages, comments welcome!",
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "We prove that the stability inequality associated to Sobolev's inequality and its set of optimizers $\\mathcal M$ and given by \\[ \\frac{\\|\\nabla f\\|_{L^2(\\mathbb R^d)}^2 - S_d \\|f\\|_{L^\\frac{2d}{d-2}(\\mathbb R^d)}^2}{ \\inf_{h \\in \\mathcal M} \\|\\nabla (f - h)\\|_{L^2(\\mathbb R^d)}^2 } \\geq c_{BE} > 0 \\qquad \\text{ for every } f \\in \\dot{H}^1(\\mathbb R^d),\\] which is due to Bianchi and Egnell, admits a minimizer for every $d \\geq 3$. Our proof consists in an appropriate refinement of a classical strategy going back to Brezis and Lieb. As a crucial ingredient, we establish the strict inequality $c_{BE} < 2 - 2^\\frac{d-2}{d}$, which means that a sequence of two asymptotically non-interacting bubbles cannot be minimizing. Our arguments cover in fact all dimensions $d \\geq 1$ and fractional exponents $s \\in (0, d/2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-25T15:39:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sobolev inequality",
      "sobolevs inequality",
      "fractional exponents",
      "appropriate refinement",
      "stability inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}