{
  "id": "1907.05769",
  "version": "v1",
  "published": "2019-07-12T14:37:20.000Z",
  "updated": "2019-07-12T14:37:20.000Z",
  "title": "Herglotz' variational principle and Lax-Oleinik evolution",
  "authors": [
    "Piermarco Cannarsa",
    "Wei Cheng",
    "Liang Jin",
    "Kaizhi Wang",
    "Jun Yan"
  ],
  "categories": [
    "math.AP",
    "math.DS"
  ],
  "abstract": "We develop an elementary method to give a Lipschitz estimate for the minimizers in the problem of Herglotz' variational principle proposed in \\cite{CCWY2018} in the time-dependent case. We deduce Erdmann's condition and the Euler-Lagrange equation separately under different sets of assumptions, by using a generalized du Bois-Reymond lemma. As an application, we obtain a representation formula for the viscosity solution of the Cauchy problem for the Hamilton-Jacobi equation \\begin{align*} D_tu(t,x)+H(t,x,D_xu(t,x),u(t,x))=0 \\end{align*} and study the related Lax-Oleinik evolution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-12T14:37:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "variational principle",
      "deduce erdmanns condition",
      "elementary method",
      "cauchy problem",
      "viscosity solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}