Geodesic
On passing to the limit in degenerate Bellman equations. I
Sbornik. Mathematics, Tome 34 (1978) no. 6, pp. 765-783 
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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     author = {N. V. Krylov},
     title = {On passing to the limit in degenerate {Bellman} {equations.~I}},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] N. V. Krylov, Upravlyaemye protsessy diffuzionnogo tipa, izd-vo “Nauka”, Moskva, 1977 | MR

[2] S. N. Kruzhkov, “Obobschennye resheniya uravnenii Gamiltona–Yakobi tipa eikonala. I”, Matem. sb., 98(140) (1975), 450–493 | Zbl

[3] A. I. Volpert, S. I. Khudyaev, “O zadache Koshi dlya kvazilineinykh vyrozhdayuschikhsya parabolicheskikh uravnenii vtorogo poryadka”, Matem. sb., 78(120) (1969), 374–396 | MR

[4] N. V. Krylov, “Ob uravneniyakh minimaksnogo tipa v teorii ellipticheskikh i parabolicheskikh uravnenii na ploskosti”, Matem. sb., 81(123) (1970), 3–22 | MR | Zbl

[5] N. Danford, Dzh. T. Shvarts, Lineinye operatory, obschaya teoriya, IL, Moskva, 1962

[6] V. I. Smirnov, Kurs vysshei matematiki, t. 5, Fizmatgiz, Moskva, 1959 | MR

[7] N.,V. Krylov, “Nekotorye svoistva normalnogo izobrazheniya vypuklykh funktsii”, Matem. sb., 105(147) (1978), 180–191 | MR | Zbl

[8] S. N. Kruzhkov, Nelineinye uravneniya s chastnymi proizvodnymi (lektsii), ch. 2, Moskva, 1970, Uravneniya pervogo poryadka, rotaprint MGU

[9] M. M. Vainberg, Variatsionnyi metod i metod monotonnykh operatorov, izd-vo “Nauka”, Moskva, 1972 | MR

[10] Zh.-L. Lione, Nekotorye metody resheniya nelineinykh kraevykh zadach, izd-vo “Mir”, Moskva, 1972 | MR