Local solutions to a class of Monge-Ampère equations of mixed type
type: Item
instance of: scholarly article
title: Local solutions to a class of Monge-Ampère equations of mixed type
zbMATH Open document ID: 1116.35098
author: Qing Han
published in: Duke Mathematical Journal
publication date: 2007-03-12T00:00:00Z
review text: The paper presents a study of Monge-Ampère type equations which is motivated mainly by the geometric question of the existence of surfaces with prescribed-Gauss curvature. The general Monge-Ampère type equation is \[ {1\over \det(g)} \text{det}(\nabla^2_g u)= K(x, u,\nabla_g u),\tag{1} \] where \(g\) is Riemannian metric in some domain \(\Omega\subset\mathbb{R}^n\) and \(K\) is sufficiently smooth function in \(\Omega\times\mathbb{R}\times \mathbb{R}^n\). Equation (1) can also be viewed as special case of \(\sigma_k\)-equations. Instead of the determinant, it is possible to consider the \(\sigma_k\), the \(k\)th symmetric function of eigenvalues of the Hessian in the left-hand side of (1), as the determinant is the \(\sigma_n\). To present the idea clearly, equation (1) is considered corresponding only to the case when \(g\) is the trivial metric, that is, \(\text{det}(D^2 u)= K(x,u,Du)\) and for a class \(K\) of mixed sign, i.e. the equation \[ \text{det}(D^2 u)= \eta(\varphi(x)) h(x,u,Du),\tag{2} \] where \(\eta= \eta(t)\) is a suffciently smooth, strictly increasing function with \(\eta(0)= 0\), \(\varphi\) is a sufficiently smooth function in a neighborhood \(\Omega\subset\mathbb{R}^n\) of the origin satisfying \(\varphi(0)= 0\), \(D_\varphi(0)\neq 0\) and \(h:\Omega\times\mathbb{R}\times \mathbb{R}^n\to\mathbb{R}\) is a sufficiently smooth function with \(h(0,0,0)\neq 0\). The existence of a local solution in a neighborhood of \(0\in\mathbb{R}^n\) is under consideration. A typical example for \(\eta\) is given by \(\eta(t)= t^q\) for some odd integer \(q\) . The implicit function theorem implies that \(\varphi^{-1}(0)\) is an hypersurface in a neighbood of the origin. An example is given by \(\varphi(x)= x_n\). In general, there exists a hypersurface in a neighborhood of the origin given by \(\{\varphi= 0\}\) such that (2) is of mixed type, elliptic in one side of this hypersurface, where \(\varphi> 0\), and hyperbolic in another side, where \(\varphi< 0\). The…
reviewed by: Elena Gavrilova
Mathematics Subject Classification ID: 35M10
Mathematics Subject Classification ID: 35A07
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53C44
zbMATH DE Number: 5132698
zbMATH Keywords: Monge-Ampère type equations
zbMATH Keywords: mixed type equations
zbMATH Keywords: existence of a local solution
zbMATH Keywords: elliptic hyperbolic
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5132698