On the existence of W² _p solutions for fully nonlinear elliptic equations under either relaxed or no convexity assumptions

We establish the existence 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 |D2v|≤ K, where K is any given constant. For large |D2v| 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. We also establish the solvability without almost any conditions on H, apart from ellipticity, but of a "cut-off" version of the equation H(v,Dv,D2v,x) = 0.