On the classical solvability of the Dlrichlet problem for the Monge–Ampère equation

It is proved that the problem $\det(u_{xx})=f(x, u, u_x)\geqslant\nu>0$, $u|_{\partial\Omega}=\phi(x)$ is solvable in $C^{k+2+\alpha}(\bar\Omega)$, $k\geqslant2$, $0<\alpha<1$ if the natural connection between $\partial\Omega$-curvature and $|p|$-growth of $f(x, u, p)$ is valid.