Monge-Ampère equations on Riemannian manifolds
type: Item
instance of: scholarly article
title: Monge-Ampère equations on Riemannian manifolds
zbMATH Open document ID: 0866.58067
author: Bo Guan
author: Yan Yan Li
published in: Journal of Differential Equations
publication date: 1996-12-01T00:00:00Z
review text: This article studies the Dirichlet problem: \[ (g^{-1} \text{det} \nabla_i \nabla_j u)^{1/n} =\psi (x,u,\nabla u) \text{ in } \Omega, \quad u= \varphi\text{ on } \partial \Omega, \] where \(\Omega\) is a compact domain with smooth boundary \(\partial\Omega\) in a smooth \(n\)-dimensional Riemannian manifold. Here \(g\) is the determinant of the metric tensor and \(\nabla\) denotes covariant differentiation. Assuming the right hand side \(\psi\) is convex with respect to its last variable and \(\psi_u\geq 0\), the authors give existence and uniqueness theorems for this problem. In the course of their proof, they give an interior estimate for second derivatives (under some convexity assumptions on \(u\) and \(\psi)\): \[ \bigl|\nabla^2 u(x) \bigr|\leq Cd (x)^{-N}, \] where \(C\) and \(N\) are constants and \(d(x)\) is the distance of \(x\in \Omega\) to the boundary \(\partial\Omega\). As a corollary they obtain the following theorem (already known under various additional assumptions): if \(\Omega\) is a strictly convex domain and \(\psi\) depends on \(x\) only, then the above problem admits a unique locally strictly convex solution which is smooth up to the boundary of \(\Omega\).
Mathematics Subject Classification ID: 58J60
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 952553
zbMATH Keywords: Monge-Ampère equation
zbMATH Keywords: nonlinear elliptic equation
zbMATH Keywords: Dirichlet problem
reviewed by: François Rouvière
MaRDI profile type: Publication
full work available at URL: jdeq.1996.0174
OpenAlex ID: W1989633637
DOI: 10.1006/JDEQ.1996.0174
description: scientific article; zbMATH DE number 952553