Analytic properties of conditional curvatures of convex hypersurfaces and the Dirichlet problem for the Monge--Amp\'ere equation

A. Taskaraev

Matematičeskie zametki, Tome 64 (1998) no. 5, pp. 763-768

Résumé

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.