{
  "id": "2412.20958",
  "version": "v1",
  "published": "2024-12-30T13:55:18.000Z",
  "updated": "2024-12-30T13:55:18.000Z",
  "title": "The selection problem for a new class of perturbations of Hamilton-Jacobi equations and its applications",
  "authors": [
    "Qinbo Chen"
  ],
  "comment": "32 pages",
  "categories": [
    "math.AP",
    "math.DS"
  ],
  "abstract": "This paper studies a perturbation problem given by the equation: \\begin{equation*} H(x, d_xu_\\lambda, \\lambda u_\\lambda(x))+\\lambda V(x,\\lambda)=c \\quad \\text{in $M$}, \\end{equation*} where $M$ is a closed manifold and $\\lambda>0$ is a perturbation parameter. The Hamiltonian $H(x,p,u):T^*M\\times \\mathbb{R}\\to \\mathbb{R}$ satisfies certain convexity, superlinearity, and monotonicity conditions. $\\lambda V(\\cdot,\\lambda):M\\to\\mathbb{R}$ converges to zero as $\\lambda\\to 0$. First, we study the asymptotic behavior of the viscosity solution $u_\\lambda:M\\to\\mathbb{R}$ as $\\lambda$ approaches zero. This perturbation problem explores the combined effects of both the vanishing discount process and potential perturbations, leading to a new selection principle that extends beyond the classical vanishing discount approach. Additionally, we apply this principle to Hamilton-Jacobi equations with $u$-independent Hamiltonians, resulting in the introduction of a new solution operator. This operator provides new insights into the variational characterization of viscosity solutions and Mather measures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-30T13:55:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B40",
      "37J51",
      "49L25"
    ],
    "keywords": [
      "hamilton-jacobi equations",
      "selection problem",
      "viscosity solution",
      "applications",
      "perturbation problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}