Geodesic
The integral method of barrier functions and the Dirichlet problem for equations with operators of Monge–Ampère type
Sbornik. Mathematics, Tome 40 (1981) no. 2, pp. 179-192 
N. M. Ivochkina. The integral method of barrier functions and the Dirichlet problem for equations with operators of Monge–Ampère type. Sbornik. Mathematics, Tome 40 (1981) no. 2, pp. 179-192. http://geodesic.mathdoc.fr/item/SM_1981_40_2_a3/
@article{SM_1981_40_2_a3,
     author = {N. M. Ivochkina},
     title = {The integral method of barrier functions and the {Dirichlet} problem for equations with operators of {Monge{\textendash}Amp\`ere} type},
     journal = {Sbornik. Mathematics},
     pages = {179--192},
     year = {1981},
     volume = {40},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1981_40_2_a3/}
}
TY  - JOUR
AU  - N. M. Ivochkina
TI  - The integral method of barrier functions and the Dirichlet problem for equations with operators of Monge–Ampère type
JO  - Sbornik. Mathematics
PY  - 1981
SP  - 179
EP  - 192
VL  - 40
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1981_40_2_a3/
LA  - en
ID  - SM_1981_40_2_a3
ER  - 
%0 Journal Article
%A N. M. Ivochkina
%T The integral method of barrier functions and the Dirichlet problem for equations with operators of Monge–Ampère type
%J Sbornik. Mathematics
%D 1981
%P 179-192
%V 40
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1981_40_2_a3/
%G en
%F SM_1981_40_2_a3

Voir la notice de l'article provenant de la source Math-Net.Ru

A priori boundedness of the solution of the Dirichlet problem is proved for the equation $F(m;u)=f(x,u,u_x)$, where $F(m;u)$ is the sum of all principal minors of order $m$ in the Hessian $\det(u_{xx})$. The boundedness in question is relative to the $C^2(\Omega)$-norm and is demonstrated by combining the methods of integral inequalities and barrier functions. Bibliography: 7 titles.

[1] O. A. Ladyzhenskaya, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, izd-vo “Nauka”, Moskva, 1973 | MR

[2] J. Serrin, “The problem of Dirichlet for quasilinear elliptic differential equations with many independent varibles”, Phil. Trans. Roy. Soc., 264 (1969), 413–496 | DOI | MR | Zbl

[3] A. V. Pogorelov, Mnogomernaya problema Minkovskogo, izd-vo “Nauka”, Moskva, 1975 | MR

[4] N. M. Ivochkina, “Postroenie apriornykh otsenok dlya vypuklykh reshenii uravneniya Monzha–Ampera integralnym metodom”, Ukr. matem. zh., 30:1 (1978), 45–53 | MR | Zbl

[5] A. D. Aleksandrov, “K teorii smeshannykh ob'emov vypuklykh tel”, Matem. sb., 2(44) (1937), 947–972 ; 1205–1238 ; 3(45) (1938), 27–46 ; 227–251 | Zbl | Zbl | Zbl | Zbl

[6] S. N. Bernshtein, Sobranie sochinenii, t. III (uravneniya v chastnykh proizvodnykh), izd-vo AN SSSR, 1960

[7] N. M. Ivochkina, “O differentsialnykh uravneniyakh vtorogo poryadka $d$-ellipticheskimi operatorami”, Trudy Matem. in-ta im. V. A. Steklova, CXLVII, 1979, 40–56 | MR | Zbl