On the existence of $W_p^{1,2}$ solutions for fully nonlinear parabolic equations under either relaxed or no convexity assumptions

Abstract

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 $|D^{2}v|\le K$, 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. We also establish the solvability without almost any conditions on $H$, apart from ellipticity, but of a "cut-off" version of the equation ${\partial }_{t}u+H(v,Dv,D^{2}v,t,x)=0$.

30 pages, a few errors corrected

References in corpus (1)

• On the existence of $W_p^2$ solutions for fully nonlinear elliptic equations under either relaxed or no convexity assumptions

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• Recent progress in the $L_p$ theory for elliptic and parabolic equations with discontinuous coefficients
• Convex semigroups on $L^p$-like spaces