Geodesic
Nonlinear nonuniformly elliptic second-order equations
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 16, Tome 138 (1984), pp. 35-64

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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. 
@article{ZNSL_1984_138_a3,
     author = {A. V. Ivanov},
     title = {Nonlinear nonuniformly elliptic second-order equations},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {35--64},
     publisher = {mathdoc},
     volume = {138},
     year = {1984},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1984_138_a3/}
}
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A. V. Ivanov. Nonlinear nonuniformly elliptic second-order equations. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 16, Tome 138 (1984), pp. 35-64. http://geodesic.mathdoc.fr/item/ZNSL_1984_138_a3/