{
  "id": "2303.14477",
  "version": "v1",
  "published": "2023-03-25T14:09:59.000Z",
  "updated": "2023-03-25T14:09:59.000Z",
  "title": "A primer on quasi-convex functions in nonlinear potential theories",
  "authors": [
    "Kevin R. Payne",
    "Davide F. Redaelli"
  ],
  "comment": "96 pages, 2 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "We present a self-contained introduction to the fundamental role that quasi-convex functions play in general (nonlinear second order) potential theories, which concerns the study of generalized subharmonics associated to a suitable closed subset of the space of 2-jets. Quasi-convex functions build a bridge between classical and viscosity notions of solutions of the natural Dirichlet problem in any potential theory. Moreover, following a program initiated by Harvey and Lawson in 2009, a potential theoretic-approach is widely being applied for treating nonlinear partial differential equations (PDEs). This viewpoint revisits the conventional viscosity approach to nonlinear PDEs under a more geometric prospective inspired by Krylov and takes much insight from classical pluripotential theory. The possibility of a symbiotic and productive relationship between general potential theories and nonlinear PDEs relies heavily on the class of quasi-convex functions, which are themselves the protagonists of a particular (and important) pure second order potential theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-25T14:09:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35D40",
      "35B51",
      "35J60",
      "35J70",
      "26B05",
      "26B25"
    ],
    "keywords": [
      "quasi-convex functions",
      "nonlinear potential theories",
      "pure second order potential theory",
      "nonlinear pdes",
      "treating nonlinear partial differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 96,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}