{
  "id": "2009.13677",
  "version": "v1",
  "published": "2020-09-28T23:04:58.000Z",
  "updated": "2020-09-28T23:04:58.000Z",
  "title": "Convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on the unknown function",
  "authors": [
    "Qinbo Chen"
  ],
  "comment": "26 pages",
  "categories": [
    "math.AP",
    "math.DS"
  ],
  "abstract": "Motivated by the vanishing discount problem, we study the convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on the unknown function. Let $H(x,p,u)$ be a continuous Hamiltonian which is strictly increasing in $u$, and is convex and coercive in $p$. For each parameter $\\lambda>0$, we denote by $u^\\lambda$ the unique viscosity solution of the H-J equation \\[H( x,Du(x),\\lambda u(x) )=c.\\] Under quite general assumptions, we prove that $u^\\lambda$ converges uniformly, as $\\lambda$ tends to zero, to a specific solution of the critical H-J equation $ H(x,Du(x),0)=c.$ We also characterize the limit solution in terms of Peierls barrier and Mather measures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-28T23:04:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B40",
      "49L25",
      "37J50"
    ],
    "keywords": [
      "hamilton-jacobi equations",
      "unknown function",
      "convergence",
      "unique viscosity solution",
      "vanishing discount problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}