{
  "id": "1609.07601",
  "version": "v1",
  "published": "2016-09-24T10:54:09.000Z",
  "updated": "2016-09-24T10:54:09.000Z",
  "title": "Globally Lipschitz minimizers for variational problems with linear growth",
  "authors": [
    "Lisa Beck",
    "Miroslav Bulíček",
    "Erika Maringová"
  ],
  "comment": "19 pages, 2 figures. Comments are welcome!",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the minimization of convex, variational integrals of linear growth among all functions in the Sobolev space $W^{1,1}$ with prescribed boundary values (or its equivalent formulation as a boundary value problem for a degenerately elliptic Euler--Lagrange equation). Due to insufficient compactness properties of these Dirichlet classes, the existence of solutions does not follow in a standard way by the direct method in the calculus of variations and in fact might fail, as it is well-known already for the non-parametric minimal surface problem. Assuming radial structure, we establish a necessary and sufficient condition on the integrand such that the Dirichlet problem is in general solvable, in the sense that a Lipschitz solution exists for any regular domain and all prescribed regular boundary values, via the construction of appropriate barrier functions in the tradition of Serrin's paper [19].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-24T10:54:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A01",
      "35B65",
      "35J70",
      "49N60"
    ],
    "keywords": [
      "globally lipschitz minimizers",
      "linear growth",
      "variational problems",
      "non-parametric minimal surface problem",
      "degenerately elliptic euler-lagrange equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}