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Krylov","submitted_at":"2017-05-05T21:18:58Z","abstract_excerpt":"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. 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