Nonlinear analysis on manifolds, Monge-Ampère equations
Nonlinear analysis on manifolds, Monge-Ampère equations
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DOI：
10.1007/978-1-4612-5734-9
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发表时间：
1982
影响因子：
3.8
通讯作者：
Thierry Aubin
中科院分区：
文献类型：
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作者：
Thierry Aubin
这卷的目的是让数学家和物理学家，特别是分析师，了解非线性问题出现在黎曼几何。黎曼流形的分析是一个正在迅速发展的领域。分析越来越被证明是解决几何问题的一种非常有力的手段。相反，几何学可以帮助我们解决分析中的某些问题。这个话题既困难又有趣，原因有几个。它非常大，几乎未被探索过。另一方面，几何问题在分析中往往导致已知问题的极限情况，有时甚至存在不止一种途径，现有的理论研究不足以解决它们。每个问题都有其特殊的困难。尽管如此，还是存在一些有用的标准方法，我们必须知道这些方法才能应用它们。人们不应忘记，我们的问题是由几何学激发的，几何论证可以简化所研究的问题。这样的例子还是太少了。这项工作既不是对一个数学领域的系统研究，也不是大量理论知识的呈现。相反，我尽量把课文限制在基本知识范围内。我定义尽可能少的概念，并只给出对我们的主题有用的基本定理。但我希望读者会发现这足以解决其他几何问题的分析。
This volume is intended to allow mathematicians and physicists, especially analysts, to learn about nonlinear problems which arise in Riemannian Geometry. Analysis on Riemannian manifolds is a field currently undergoing great development. More and more, analysis proves to be a very powerful means for solving geometrical problems. Conversely, geometry may help us to solve certain problems in analysis. There are several reasons why the topic is difficult and interesting. It is very large and almost unexplored. On the other hand, geometric problems often lead to limiting cases of known problems in analysis, sometimes there is even more than one approach, and the already existing theoretical studies are inadequate to solve them. Each problem has its own particular difficulties. Nevertheless there exist some standard methods which are useful and which we must know to apply them. One should not forget that our problems are motivated by geometry, and that a geometrical argument may simplify the problem under investigation. Examples of this kind are still too rare. This work is neither a systematic study of a mathematical field nor the presentation of a lot of theoretical knowledge. On the contrary, I do my best to limit the text to the essential knowledge. I define as few concepts as possible and give only basic theorems which are useful for our topic. But I hope that the reader will find this sufficient to solve other geometrical problems by analysis.