We prove some refinements of the concentration compactness principle for Sobolev space W^{1, n} on a smooth compact Riemannian manifold of dimension n. As an application, we extend Aubin's theorem for functions on with zero first-order moments of the area element to the higher-order moments case. Our arguments are very flexible and can be easily modified for functions satisfying various boundary conditions or belonging to higher-order Sobolev spaces. © 2020 Wiley Periodicals LLC.

中文翻译：

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论临界维集中紧致原则

我们在维数为n的光滑紧致黎曼流形上证明了 Sobolev 空间W ^{1, n}的浓度紧致性原理的一些改进。作为一个应用，我们将 Aubin 函数的面积元一阶矩为零的定理扩展到高阶矩的情况。我们的论点非常灵活，可以很容易地修改为满足各种边界条件或属于高阶 Sobolev 空间的函数。© 2020 威利期刊有限责任公司。