On the existence of $W_p^2$ solutions for fully nonlinear elliptic equations under relaxed convexity assumptions

Abstract

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 $|D^{2}v|\le K$, 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.

23 pages

Cited by in corpus (1)

• On the existence of smooth solutions for fully nonlinear parabolic equations with measurable "coefficients" without convexity assumptions