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

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 |D ² v| ≤K, where K is any given 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 x are still imposed. The solutions are sought in Sobolev classes.