Existence of \(C^{\infty}\) local solutions for the Monge-Ampère equation
type: Item
instance of: scholarly article
title: Existence of \(C^{\infty}\) local solutions for the Monge-Ampère equation
zbMATH Open document ID: 0648.35016
DOI: 10.1007/BF01388988
author: Claude Zuily
author: Jiaxing Hong
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143497
review text: The authors prove the existence of \(C^{\infty}\) convex solutions to the Monge-Ampère equation \(\det (u_{ij})=f(y,u,\nabla u)\) in a neighbourhood of \(y^ 0\) in \({\mathbb{R}}^ n,\) assuming that \(f\geq 0\) near \((y^ 0,u^ 0,p^ 0)\) and that one of the following conditions is satisfied: (I) There exists k such that \((\partial^{\alpha}f)(y^ 0,u^ 0,p^ 0)=0\) if \(| \alpha | <k\), but \((\partial^{\beta}_ yf)(y^ 0,u^ 0,p^ 0)\neq 0\) for some \(\beta\) with \(| \beta | =k.\) (II) \(f(y,u,p)=K(y)g(y,u,p)\) with K(y \(0)=0\), \(K\geq 0\), \(g>0\), and \(K^{-1}(0)\) is contained, near \(y^ 0\), in the union of a finite number of \(C^ 1\) hypersurfaces. The authors use a Nash-Moser method together with Bony paradifferential calculus.
Mathematics Subject Classification ID: 35G20
Mathematics Subject Classification ID: 35A07
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 4058030
zbMATH Keywords: existence
zbMATH Keywords: convex solutions
zbMATH Keywords: Monge-Ampère equation
zbMATH Keywords: Nash-Moser method
zbMATH Keywords: Bony paradifferential calculus
reviewed by: Paul Godin
MaRDI profile type: Publication
cites work: Q5331410
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OpenAlex ID: W2045939215
description: scientific article; zbMATH DE number 4058030