{
  "id": "2205.01615",
  "version": "v1",
  "published": "2022-05-03T16:48:54.000Z",
  "updated": "2022-05-03T16:48:54.000Z",
  "title": "Global semiconcavity of solutions to first-order Hamilton-Jacobi equations with state constraints",
  "authors": [
    "Yuxi Han"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We focus on the global semiconcavity of solutions to first-order Hamilton--Jacobi equations with state constraints, especially for the Hamiltonian $H(x, \\beta):=|\\beta|^p-f(x)$ with $p \\in (1, 2]$. We first show that the solution is locally semiconcave, and the semiconcavity constant at each point depends on the first time a corresponding minimizing curve emanating from this point hits the boundary. Then, with appropriate conditions on $Df$, we prove that for any such minimizing curve, the time it takes to hit the boundary of the domain is $+\\infty$, and as a consequence, the solution is globally semiconcave. Moreover, the condition on $Df$ is essentially optimal with examples in one-dimensional space. The proofs employ the Euler-Lagrange equations and techniques in weak KAM theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-03T16:48:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B65",
      "35D40",
      "35F20",
      "49L25"
    ],
    "keywords": [
      "first-order hamilton-jacobi equations",
      "global semiconcavity",
      "state constraints",
      "weak kam theory",
      "minimizing curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}