On the Monge-Ampère equation

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2019-01-01 DOI:10.24033/ast.1092

Alessio FIGALLI

引用次数: 0

Abstract

where Ω ⊂ R is some open set, u : Ω → R is a convex function, and the function f : Ω× R× R → R is given. In other words, the Monge-Ampère equation prescribes the product of the eigenvalues of the Hessian of u, in contrast with the “model” elliptic equation ∆u = f which prescribes their sum. As we shall explain later, the convexity of the solution u is a necessary condition to make the equation degenerate elliptic, and therefore to hope for regularity results. The goal of this note is to give first a general overview of the classical theory, and then discuss some recent important developments on this beautiful topic. For our presentation of the classical theory, we follow the survey paper [25].

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关于蒙奇-安培方程

其中Ω∧R为某开集，u: Ω→R为凸函数，并给出函数f: Ω× rx R→R。换句话说，monge - ampontre方程规定了u的Hessian特征值的乘积，而“模型”椭圆方程∆u = f规定了它们的和。我们将在后面解释，解u的凸性是使方程退化为椭圆的必要条件，因此希望得到正则性结果。这篇笔记的目的是首先对经典理论进行总体概述，然后讨论这个美丽话题最近的一些重要发展。对于经典理论的介绍，我们遵循调查论文[25]。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

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