Geodesic
Analytic properties of conditional curvatures of convex hypersurfaces and the Dirichlet problem for the Monge–Ampére equation
Matematičeskie zametki, Tome 64 (1998) no. 5, pp. 763-768 
A. Taskaraev. Analytic properties of conditional curvatures of convex hypersurfaces and the Dirichlet problem for the Monge–Ampére equation. Matematičeskie zametki, Tome 64 (1998) no. 5, pp. 763-768. http://geodesic.mathdoc.fr/item/MZM_1998_64_5_a12/
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     author = {A. Taskaraev},
     title = {Analytic properties of conditional curvatures of convex hypersurfaces and the {Dirichlet} problem for the {Monge{\textendash}Amp\'ere} equation},
     journal = {Matemati\v{c}eskie zametki},
     pages = {763--768},
     year = {1998},
     volume = {64},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1998_64_5_a12/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The existence and uniqueness of a surface with given geometric characteristics is one of the important topical problems of global differential geometry. By stating this problem in terms of analysis, we arrive at second-order elliptic and parabolic partial differential equations. In the present paper we consider generalized solutions of the Monge–Ampére equation $\|z_{ij}\|=\varphi(x,z,p)$ in $\Lambda^n$, where $z=z(x_1,\dots,x_n)$ is a convex function, $p=(p_1,\dots,p_n)= (\partial z/\partial x_1,\dots,\partial z/\partial x_n)$, $z_{ij}=\partial^2z/\partial x_i\partial x_j$. We consider the Cayley–Klein model of the space $\Lambda^n$ and use a method based on fixed point principle for Banach spaces.

[1] Bakelman I. Ya., Verner A. L., Kantor B. E., Vvedenie v differentsialnuyu geometriyu “v tselom”, Nauka, M., 1973 | Zbl

[2] Taskaraev A., “Suschestvovanie vypuklykh giperpoverkhnostei, uslovnaya krivizna kotorykh est summa integralnykh uslovnykh krivizn razlichnykh poryadkov”, Algebra, geometriya i diskretnaya matematika v nelineinykh zadachakh, Izd-vo MGU, M., 1991, 176–185 | Zbl

[3] Verner A. L., “Vosstanovlenie polnoi vypukloi poverkhnosti v prostranstvakh Lobachevskogo po ee integralnoi vneshnei krivizne”, Uchenye zap. LGPI im. A. I. Gertsena, 218 (1960)