{
  "id": "1809.05317",
  "version": "v1",
  "published": "2018-09-14T09:20:45.000Z",
  "updated": "2018-09-14T09:20:45.000Z",
  "title": "Uniqueness of the viscosity solution of a constrained Hamilton-Jacobi equation",
  "authors": [
    "Vincent Calvez",
    "King-Yeung Lam"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In quantitative genetics, viscosity solutions of Hamilton-Jacobi equations appear naturally in the asymptotic limit of selection-mutation models when the population variance vanishes. They have to be solved together with an unknown function I(t) that arises as the counterpart of a non-negativity constraint on the solution at each time. Although the uniqueness of viscosity solutions is known for many variants of Hamilton-Jacobi equations, the uniqueness for this particular type of constrained problem was not resolved, except in a few particular cases. Here, we provide a general answer to the uniqueness problem, based on three main assumptions: convexity of the Hamiltonian function H(I, x, p) with respect to p, monotonicity of H with respect to I, and BV regularity of I(t).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-14T09:20:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "viscosity solution",
      "constrained hamilton-jacobi equation",
      "hamilton-jacobi equations appear",
      "population variance vanishes",
      "unknown function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}