Geodesic
On estimates of $\max|u_x|$ for solutions of quasilinear elliptic and parabolic equations of the general type and on some existence theorems
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 16, Tome 138 (1984), pp. 90-107
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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. 
@article{ZNSL_1984_138_a6,
     author = {O. A. Ladyzhenskaya and N. N. Ural'tseva},
     title = {On estimates of $\max|u_x|$ for solutions of quasilinear elliptic and parabolic equations of the general type and on some existence theorems},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {90--107},
     year = {1984},
     volume = {138},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1984_138_a6/}
}
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O. A. Ladyzhenskaya; N. N. Ural'tseva. On estimates of $\max|u_x|$ for solutions of quasilinear elliptic and parabolic equations of the general type and on some existence theorems. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 16, Tome 138 (1984), pp. 90-107. http://geodesic.mathdoc.fr/item/ZNSL_1984_138_a6/