The equation of prescribed Gauss curvature without boundary conditions
type: Item
instance of: scholarly article
title: The equation of prescribed Gauss curvature without boundary conditions
zbMATH Open document ID: 0566.53013
published in: Journal of Differential Geometry
publication date: 1984-01-01T00:00:00Z
review text: 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.
reviewed by: Helmut Reckziegel
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 35Q99
zbMATH DE Number: 3903469
zbMATH Keywords: Gauss-Kronecker curvature
zbMATH Keywords: convex function
zbMATH Keywords: convex regular solutions
zbMATH Keywords: Monge-Ampère equations
author: John Urbas
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214439282
OpenAlex ID: W1498783924
Wikidata QID: Q115187524
DOI: 10.4310/JDG/1214439282
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description: scientific article; zbMATH DE number 3903469