{
  "id": "2301.13026",
  "version": "v1",
  "published": "2023-01-30T16:09:06.000Z",
  "updated": "2023-01-30T16:09:06.000Z",
  "title": "Sobolev embeddings and distance functions",
  "authors": [
    "Lorenzo Brasco",
    "Francesca Prinari",
    "Anna Chiara Zagati"
  ],
  "comment": "42 pages, 1 figure",
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "On a general open set of the euclidean space, we study the relation between the embedding of the homogeneous Sobolev space $\\mathcal{D}^{1,p}_0$ into $L^q$ and the summability properties of the distance function. We prove that in the superconformal case (i.e. when $p$ is larger than the dimension) these two facts are equivalent, while in the subconformal and conformal cases (i.e. when $p$ is less than or equal to the dimension) we construct counterexamples to this equivalence. In turn, our analysis permits to study the asymptotic behaviour of the positive solution of the Lane-Emden equation for the $p-$Laplacian with sub-homogeneous right-hand side, as the exponent $p$ diverges to $\\infty$. The case of first eigenfunctions of the $p-$Laplacian is included, as well. As particular cases of our analysis, we retrieve some well-known convergence results, under optimal assumptions on the open sets. We also give some new geometric estimates for generalized principal frequencies.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-30T16:09:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "35J92",
      "35P30"
    ],
    "keywords": [
      "distance function",
      "sobolev embeddings",
      "general open set",
      "well-known convergence results",
      "euclidean space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}