On passing to the limit in degenerate bellman equations. I

In this paper the author proves theorems on passage to the limit in nonlinear parabolic equations of the form, arising in the theory of optimal control of random processes of diffusion type. Under the assumptions that i) the functions and have bounded Sobolev derivatives in, ii) the and are convex downwards in, iii) the are uniformly bounded in some domain, iv) a.e. in, v) the coefficients of linear combinations of satisfy certain smoothness conditions, it is proved that on for all implies on. The second derivatives of the and with respect to are understood in the generalized sense (as measures), and the equations and are considered in the lattice of measures. Bibliography: 10 titles.