On the Existence of Wp2 Solutions for Fully Nonlinear Elliptic Equations Under Relaxed Convexity Assumptions

N. V. Krylov

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16 Scopus citations

Abstract

We establish the existence and uniqueness of solutions of fully nonlinear elliptic second-order equations like H(v, Dv, D2v, 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}D2v{pipe} ≤K, where K is any given constant. For large {pipe}D2v{pipe} some kind of relaxed convexity assumption with respect to D2v mixed with a VMO condition with respect to x are still imposed. The solutions are sought in Sobolev classes.

Original languageEnglish (US)
Pages (from-to)687-710
Number of pages24
JournalCommunications in Partial Differential Equations
Volume38
Issue number4
DOIs
StatePublished - 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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