Analytic properties of conditional curvatures of convex hypersurfaces and the Dirichlet problem for the Monge–Ampére equation

A. Taskaraev

V. I. Lenin Tashkent State University

Abstract: 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.

UDC: 517

Received: 15.09.1996

DOI: 10.4213/mzm1452

 English version: Mathematical Notes, 1998, 64:5, 658–662

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