A note on Monge-Ampère equation in \(\mathbb{R}^2\)
type: Item
instance of: scholarly article
title: A note on Monge-Ampère equation in \(\mathbb{R}^2\)
zbMATH Open document ID: 1470.35155
author: C. Enache
author: Giovanni Porru
published in: Results in Mathematics
publication date: 2021-06-23T00:00:00Z
review text: Let \(\Omega \subset \mathbb{R}^2\) be a bounded convex domain and \(f:\mathbb{R}\rightarrow (0,\infty)\) be a smooth function. Moreover, let \(u\) be a strictly convex solution of the Monge-Ampère equation \[\text{det}(D^2 u)=f(u)\] and let \(M:\Omega \rightarrow \mathbb{R}\) be the function defined by \[M=u_{yy}u_x^2-2u_{xy}u_xu_y+u_{xx}u_y^2-2\int_{\min_\Omega u}^uf(t)dt.\] In this paper, the authors prove that: \begin{itemize} \item[1)] if \(f'\geq 0\), then \(M\) assumes its maximum value on \(\partial \Omega\); \item[2)] if \(f'\leq 0\), then \(M\) assumes its maximum value either on \(\partial \Omega\) or at a point \((x_0,y_0)\) such that \(u_x(x_0,y_0)=u_y(x_0,y_0)=0\). \end{itemize} In addiction, the authors exhibit an example which shows that the conclusion in \(1)\) may fail if the condition \(f'\geq 0\) is not satisfied. The authors also point out that when solution \(u\) satisfies the boundary condition \(u=0\) on \(\partial \Omega\), then as application of the above result, one get the inequality \[2\int_{\min_\Omega u}^uf(t)dt\leq \max_{\partial \Omega}(K|\nabla u|^3)\] where \(K\) is the curvature of \(\partial \Omega\).
reviewed by: Giovanni Anello
Mathematics Subject Classification ID: 35J30
Mathematics Subject Classification ID: 35J96
Mathematics Subject Classification ID: 35B50
zbMATH DE Number: 7362295
zbMATH Keywords: fully nonlinear equations
zbMATH Keywords: Monge-Ampère problems
zbMATH Keywords: maximum principles
MaRDI profile type: Publication
full work available at URL: s00025-020-01339-5
OpenAlex ID: W3119418946
cites work: Q5484066
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cites work: A Remark on Linear Elliptic Differential Equations of Second Order
cites work: SOME MAXIMUM PRINCIPLES AND SYMMETRY RESULTS FOR A CLASS OF BOUNDARY VALUE PROBLEMS INVOLVING THE MONGE-AMPÈRE EQUATION
cites work: Maximum principles and their applications
DOI: 10.1007/S00025-020-01339-5
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description: scientific article; zbMATH DE number 7362295