The equation of prescribed Gauss curvature without boundary conditions
The Gauss-Kronecker curvature K of the graph of a uniformly convex function \(u: {\mathbb{R}}^ n\supset \Omega \to {\mathbb{R}}\) satisfies the partial differential equation \[ \det D^ 2u=K(x)(1+| Du|^ 2)^{(n+2)/2}\quad (*) \] K\(>0\), and the estimate \(\int K=\omega_ n:=volume\) of unit ball \(\subset {\mathbb{R}}^ n\). The author proves the existence of convex regular solutions u of (*) for a given \(K>0\) with \(\int K=\omega_ n\) using the concept of generalized solutions introduced by Aleksandrov and Bakelman for the investigation of general Monge-Ampère equations (MAE). In fact, the proofs can be carried over to a wider class of MAEs.
scientific article; zbMATH DE number 3903469