{
  "id": "2112.12011",
  "version": "v2",
  "published": "2021-12-22T16:38:46.000Z",
  "updated": "2023-05-10T14:01:59.000Z",
  "title": "Game-theoretic approach to Hölder regularity for PDEs involving eigenvalues of the Hessian",
  "authors": [
    "Pablo Blanc",
    "Jeongmin Han",
    "Mikko Parviainen",
    "Eero Ruosteenoja"
  ],
  "doi": "10.1007/s11118-022-10037-6",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove a local H\\\"{o}lder estimate with an exponent $0<\\delta<\\frac 12$ for solutions of the dynamic programming principle $$u^\\varepsilon (x) =\\sum_{j=1}^n \\alpha_j\\inf_{\\dim(S)=j}\\sup_{\\substack{v\\in S\\\\ |v|=1}}\\frac{ u^\\varepsilon (x + \\varepsilon v) + u^\\varepsilon (x - \\varepsilon v)}{2}.$$ The proof is based on a new coupling idea from game theory. As an application, we get the same regularity estimate for viscosity solutions of the PDE $$\\sum_{i=1}^n \\alpha_i\\lambda_i(D^2u)=0,$$ where $\\lambda_1(D^2 u)\\leq\\cdots\\leq \\lambda_n(D^2 u)$ are the eigenvalues of the Hessian.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-05-10T14:01:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "91A05",
      "91A15",
      "35D40",
      "35B65"
    ],
    "keywords": [
      "hölder regularity",
      "game-theoretic approach",
      "eigenvalues",
      "dynamic programming principle",
      "viscosity solutions"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}