Geodesic
On equations of minimax type in the theory of elliptic and parabolic equations in the plane
Sbornik. Mathematics, Tome 10 (1970) no. 1, pp. 1-19 
N. V. Krylov. On equations of minimax type in the theory of elliptic and parabolic equations in the plane. Sbornik. Mathematics, Tome 10 (1970) no. 1, pp. 1-19. http://geodesic.mathdoc.fr/item/SM_1970_10_1_a0/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The existence and uniqueness of the solution in Sobolev spaces $W_p^2$ ($W_p^{2,1}$) is proved for the first boundary value problem for elliptic (parabolic) equations of the form $$ \lambda u-\inf_{\alpha\in\mathfrak U}\sup_{\beta\in\mathfrak B(\alpha)}(L_{\alpha\beta}u+f_{\alpha\beta})=f. $$ Here $L_{\alpha\beta}u=a_{ij}^{\alpha\beta}D_{ij}u+b_i^{\alpha\beta}u_{x_i}-c^{\alpha\beta}u$ and $D_{ij}u=u_{x_ix_j}$ in the elliptic case, $D_{ij}u=u_{x_ix_j}-\delta_{ij}u_t$ in the parabolic case. The subscript $p$ takes any values close to two. Bibliography: 10 titles.

[1] O. A. Ladyzhenskaya, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, Moskva, 1964 | MR

[2] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, Moskva, 1967

[3] I. Ya. Bakelman, Geometricheskie metody resheniya ellipticheskikh uravnenii, Nauka, Moskva, 1965 | MR

[4] I. V. Girsanov, “Minimaksnye zadachi v teorii diffuzionnykh protsessov”, DAN SSSR, 136:4 (1961), 761–764 | MR | Zbl

[5] W. H. Fleming, “Some markovian optimization problems”, J. Math. and Mech., 12:1 (1963), 131–140 | MR | Zbl

[6] H. V. Krylov, “Ob odnom klasse nelineinykh uravnenii v prostranstve izmerimykh funktsii”, Matem. sb., 80(122) (1969), 253–265 | MR | Zbl

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

[8] S. Agmon, A. Dutlis, L. Nirenberg, Otsenki vblizi granitsy reshenii ellipticheskikh uravnenii v chastnykh proizvodnykh pri obschikh granichnykh usloviyakh, t. I, IL, Moskva, 1962

[9] A. D. Aleksandrov, “Mazhorirovanie reshenii lineinykh uravnenii vtorogo poryadka”, Vestnik LGU, 1:1 (1966), 5–26

[10] E. B. Dynkin, Markovskie protsessy, Fizmatgiz, Moskva, 1963 | MR