Smooth Elliptic Solutions of Monge-Ampere Equations

Abstract

In § 8 of Chapter 2 we presented in detail the classical Minkowski Theorem on the problem of existence and uniqueness of a closed convex hypersurface with prescribed Gaussian curvature K(η) in (n + l)-dimensional Euclidean space Eⁿ⁺¹. Here K(η) is a positive continuous function on the unit hypersphere Sⁿ ⊂ Eⁿ⁺¹, which is centered at the origin of Eⁿ⁺¹. The Minkowski problem is the problem of existence and uniqueness of a closed convex hypersurface F with Gaussian curvature K(η) at a point x with exterior unit normal η. Here we do not assume that F is a regular hypersurface. Therefore the Gaussian curvature of a hypersurface F at a point x ∈ F is defined as the limit of the ratio $\frac{{w\left( G \right)}}{{w\left( G \right)}}$ as domain G shrinks to the point x, where σ(G) is the area of G and ω(G) is the area of the spherical image of G. Both set functions σ(G) and ω(G) are defined in §§ 5, 8. This definition of Gaussian curvature does not assume the C^m-smoothness (m ≥ 2) of a convex hypersurface.