Geodesic
Weakly connected systems of Monge–Amper elliptic equations and the problem of existence of
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 3 (1996) no. 1, pp. 27-33 
B. E. Kantor; V. M. Vereshchagin. Weakly connected systems of Monge–Amper elliptic equations and the problem of existence of. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 3 (1996) no. 1, pp. 27-33. http://geodesic.mathdoc.fr/item/JMAG_1996_3_1_a2/
@article{JMAG_1996_3_1_a2,
     author = {B. E. Kantor and V. M. Vereshchagin},
     title = {Weakly connected systems of {Monge{\textendash}Amper} elliptic equations and the problem of existence of},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {27--33},
     year = {1996},
     volume = {3},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/JMAG_1996_3_1_a2/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

A surface $z^i=u^i(x,y)$, $i=1,\dots,k$, projected regularly onto a domain $\Omega$ of the $(x,y)$-plane is considered in a $(k+2)$-dimensional Euclidean space. We introduce natural unit vectors $\xi_i$ directed along the vectors $(u^i_x,u^i_y,0,\dots,0,-1,0,\dots)$, $i=1,\dots,k$, where $-1$ is in the $(2+i)$-coordinate place, and the Killing–Lipschitz curvatures $K^i (x, y)$ with respect to these normal vectors. The problem of construction of a surface with given positive functions $K^i(x,y)$ and a given boundary value $u^i|_{\partial\Omega}=\varphi^i(\sigma)$, where $\sigma$ is the parameter in the curve $\partial\Omega$, is solved.