Isolated singularities of elliptic Monge-Ampère equations in dimensions \(n\geq 2\)
type: Item
instance of: scholarly article
title: Isolated singularities of elliptic Monge-Ampère equations in dimensions \(n\geq 2\)
zbMATH Open document ID: 0819.35039
DOI: 10.1007/BF01188573
author: Ralf Beyerstedt
published in: Archiv der Mathematik
publication date: 1995-08-24T00:00:00Z
review text: Let \(u\) be a twice continuously differentiable solution of the Monge- Ampère equation \[ \text{det } D^ 2 u(x)= f(x,u(x), Du(x)) \quad \text{ in the domain } \Omega=\{x\in \mathbb{R}^ n:\;0<| x-x_ 0|< R\},\;n\geq 2, \] where \(Du\) and \(D^ 2 u\) denote the gradient and the Hessian respectively of \(u\) and the function \(f\in C^ 2 (\Omega\cup \{x_ 0\} \times \mathbb{R}\times \mathbb{R}^ n)\) is positive. If \(D^ 2 u\) is positive definite in \(\Omega\) and \(\gamma Du\) has a continuous extension to \(\Omega\cup \{x_ 0\}\) for some unit vector \(\gamma\in \mathbb{R}^ n\), we prove that the isolated singularity of \(u\) at \(x_ 0\) is removable, i.e., \(u\) has a twice continuously differentiable extension to \(\Omega\cup \{x_ 0\}\).
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B60
zbMATH DE Number: 756422
zbMATH Keywords: elliptic Monge-Ampère equations
zbMATH Keywords: removable singularity
zbMATH Keywords: isolated singularity
MaRDI profile type: Publication
cites work: The behaviour of solutions to elliptic Monge-Ampère equations at singular points
cites work: The dirichlet problem for nonlinear second-order elliptic equations I. Monge-ampégre equation
cites work: Q5187597
cites work: Isolated singularities of Monge-Ampère equations
cites work: Harmonische Abbildungen und die Differentialgleichung \(rt-s^2=1\)
cites work: Über nichtlineare, konkave elliptische Differentialgleichungen. (On nonlinear concave elliptic differential equations)
cites work: Regularity of generalized solutions of Monge-Ampère equations
description: scientific article; zbMATH DE number 756422