Boundedly nonhomogeneous elliptic and parabolic equations in a domain

In this paper the Dirichlet problem is studied for equations of the form 0 = F(u_xi,_xj, u_xi, u, 1, x) and also the first boundary value problem for equations of the form u, = F(U_xi_xj u_xi, u, 1, t, x), where F(u_ij, u_i, u, β, x) and F(u_ij, u_i, u, β, t, x) are positive homogeneous functions of the first degree in (u_ij, u_i, u, β), convex upwards in (u_ij), that satisfy a uniform strict ellipticity condition. Under certain smoothness conditions on F and when the second derivatives of F with respect to (u_ij, u_i, u, x) are bounde above, the C^2+α solvability of these problems in smooth domains is proved. In the course of the proof, a priori estimates in C^2+α on the boundary are constructed, and convexity and restrictions on the second derivatives of F are not used in the derivation.