Geodesic
On passing to the limit in degenerate Bellman equations. I
Sbornik. Mathematics, Tome 34 (1978) no. 6, pp. 765-783 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper the author proves theorems on passage to the limit in nonlinear parabolic equations of the form $Fu=0$, arising in the theory of optimal control of random processes of diffusion type. Under the assumptions that i) the functions $u_n$ and $u$ have bounded Sobolev derivatives in $t$, ii) the $u_n$ and $u$ are convex downwards in $x$, iii) the $u_n$ are uniformly bounded in some domain $Q$, iv) $u_n\to u$ a.e. in $Q$, v) the coefficients of linear combinations of $F$ satisfy certain smoothness conditions, it is proved that $Fu_n=0$ on $Q$ for all $n$ implies $Fu=0$ on $Q$. The second derivatives of the $u_n$ and $u$ with respect to $x$ are understood in the generalized sense (as measures), and the equations $Fu_n=0$ and $Fu=0$ are considered in the lattice of measures. Bibliography: 10 titles. 
@article{SM_1978_34_6_a4,
     author = {N. V. Krylov},
     title = {On passing to the limit in degenerate {Bellman} {equations.~I}},
     journal = {Sbornik. Mathematics},
     pages = {765--783},
     year = {1978},
     volume = {34},
     number = {6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1978_34_6_a4/}
}
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N. V. Krylov. On passing to the limit in degenerate Bellman equations. I. Sbornik. Mathematics, Tome 34 (1978) no. 6, pp. 765-783. http://geodesic.mathdoc.fr/item/SM_1978_34_6_a4/

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