On the existence of $W^{2}_{p}$ solutions for fully nonlinear elliptic equations under either relaxed or no convexity assumptions

TL;DR

This paper proves the existence of solutions for fully nonlinear elliptic equations in Sobolev spaces without strict convexity assumptions, using relaxed conditions and VMO assumptions on the equations.

Contribution

It demonstrates existence results for fully nonlinear elliptic equations without requiring convexity or concavity of the operator, broadening the class of solvable equations.

Findings

Existence of solutions without convexity assumptions.

Solutions in Sobolev spaces under relaxed conditions.

Solvability of a cut-off version of the equations.

Abstract

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