## Abstract

We establish the existence and uniqueness of solutions of fully nonlinear elliptic second-order equations like H(v, Dv, D^{2}v, x) = 0 in smooth domains 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 {pipe}D^{2}v{pipe} ≤K, where K is any given constant. For large {pipe}D^{2}v{pipe} some kind of relaxed convexity assumption with respect to D^{2}v mixed with a VMO condition with respect to x are still imposed. The solutions are sought in Sobolev classes.

Original language | English (US) |
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Pages (from-to) | 687-710 |

Number of pages | 24 |

Journal | Communications in Partial Differential Equations |

Volume | 38 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2013 |

### Bibliographical note

Funding Information:The author was partially supported by NSF Grant DMS-1160569.

## Keywords

- Bellman's equations
- Finite differences
- Fully nonlinear elliptic equations

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