Nonlinear nonuniformly elliptic second-order equations

A priori estimates of the first and second derivatives for solutions of nonuniformly elliptic equations of the form $\mathcal F(x, u, \mathcal Du, \mathcal D^2u)=0$ without the suggesting on the convexity $\mathcal F(x, p, z, r)$ in $r$ are investigated. These estimates permit to generalize the results of Krylov, Evans and Trudinger on the classical solvability of the Diriclet problem for fully nonlinear, uniformly elliptic, convex in $\mathcal D^2u$ equations to a more broader classes of nonlinear equations.