The concentration-compactness principle in the Calculus of Variations
The concentration-compactness principle in the Calculus of Variations
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DOI：
10.4171/rmi/6
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发表时间：
1985-03
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影响因子：
--
通讯作者：
P. Lions
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文献类型：
--
作者：
P. Lions
在研究了具有平移不变性的变分问题的局部紧情形后，我们在这里研究了例如RN中的变分问题（具有约束），其中RN的不变性由群的不变性产生了一些可能的紧性损失。例如，这是与函数不等式中的极值函数的确定相关的所有问题的情况（例如Sobolev不等式）。我们在这里展示了如何集中紧凑性原则必须修改，以便能够处理这类问题，我们目前的应用功能分析，数学物理，微分几何和谐波分析。
After the study made in the locally compact case for variational problems with some translation invariance, we investigate here the variational problems (with constraints) for example in RN where the invariance of RN by the group of dilatations creates some possible loss of compactness. This is for example the case for all the problems associated with the determination of extremal functions in functional inequalities (like for example the Sobolev inequalities). We show here how the concentration-compactness principle has to be modified in order to be able to treat this class of problems and we present applications to Functional Analysis, Mathematical Physics, Differential Geometry and Harmonic Analysis.