{
  "id": "2108.08869",
  "version": "v1",
  "published": "2021-08-19T18:31:34.000Z",
  "updated": "2021-08-19T18:31:34.000Z",
  "title": "Partial regularity for local minimizers of variational integrals with lower order terms",
  "authors": [
    "Judith Campos Cordero"
  ],
  "comment": "25 pp",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider functionals of the form $$ \\mathcal{F}(u):=\\int_\\Omega\\!F(x,u,\\nabla u)\\,\\mathrm{d} x, $$ where $\\Omega\\subseteq\\mathbb{R}^n$ is open and bounded. The integrand $F\\colon\\Omega\\times\\mathbb{R}^N\\times\\mathbb{R}^{N\\times n}\\to\\mathbb{R}$ is assumed to satisfy the classical assumptions of a polynomial $p$-growth and strong quasiconvexity. In addition, $F$ is H\\\"older continuous in its first two variables uniformly with respect to the third variable, and bounded below by a quasiconvex function depending only on $z\\in\\mathbb{R}^{N\\times n}$. We establish that, for every $\\alpha\\in(0,1)$, strong local minimizers of $\\mathcal{F}$ are of class $\\mathrm{C}^{1,\\alpha}$ in a subset $\\Omega_0\\subseteq\\Omega$ with $\\Omega_0$ of full $n$-dimensional measure. This extends the partial regularity result for strong local minimizers by Kristensen and Taheri (2003) to the case where the integrand depends also on $u$. Furthermore, we provide a direct strategy for this result, in contrast to the blow-up argument used for the case of homogeneous integrands.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-19T18:31:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B65",
      "35J50",
      "35J60",
      "49N99"
    ],
    "keywords": [
      "lower order terms",
      "variational integrals",
      "strong local minimizers",
      "partial regularity result",
      "quasiconvex function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}