Abstract
One establishes a priori estimates for the first and second derivatives of the solutions of nonuniformly elliptic equations of the form F(x,u,Du,Dzu) without the assumptions of the convexity of the function F(x,z,p,r) with respect to r. These estimates allow us to extend the results of N. V. Krylov, L. C. Evans, and N. S. Trudinger on the classical solvability of the Dirichlet problem for essential nonlinear uniformly elliptic equations, convex with respect to D2, to wider classes of nonlinear equations.
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