{
  "id": "1705.02400",
  "version": "v1",
  "published": "2017-05-05T21:18:58.000Z",
  "updated": "2017-05-05T21:18:58.000Z",
  "title": "On the existence of $W^{1,2}_{p}$ solutions for fully nonlinear parabolic equations under either relaxed or no convexity assumptions",
  "authors": [
    "N. V. Krylov"
  ],
  "comment": "30 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We establish the existence of solutions of fully nonlinear parabolic second-order equations like $\\partial_{t}u+H(v,Dv,D^{2}v,t,x)=0$ in smooth cylinders without requiring $H$ to be convex or concave with respect to the second-order derivatives. Apart from ellipticity nothing is required of $H$ at points at which $|D^{2}v|\\leq K$, where $K$ is any fixed constant. For large $|D^{2}v|$ some kind of relaxed convexity assumption with respect to $D^{2}v$ mixed with a VMO condition with respect to $t,x$ are still imposed. The solutions are sought in Sobolev classes. We also establish the solvability without almost any conditions on $H$, apart from ellipticity, but of a \"cut-off\" version of the equation $\\partial_{t}u+H(v,Dv,D^{2}v,t,x)=0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-05T21:18:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K55",
      "35K20"
    ],
    "keywords": [
      "fully nonlinear parabolic equations",
      "convexity assumption",
      "fully nonlinear parabolic second-order equations",
      "smooth cylinders",
      "second-order derivatives"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}