On estimates of $\max|u_x|$ for solutions of quasilinear elliptic and parabolic equations of the general type and on some existence theorems

A priori estimates of $\max|u_x|$ solutions of nondivergent quasilinear uniformly elliptic and parabolic equations are derived only under “natural” restrictions on the functions $a_{ij}$, on their first partial derivatives and on the function $a$. Some known existence theorems for the Dirichlet problem for elliptic equations are generalized.