余紧嵌入与弱收敛序列的结构
Cocompact imbeddings and structure of weakly convergent sequences
摘要：
集中紧性方法是证明不等式极小解存在性和泛函临界点存在性的有力方法。本文给出了一个功能分析制定的方法在Banach空间中，广义希尔伯特空间的情况下，阐述了\cite{ccbook}。主要对象是一个位错空间-一个三元组$(X,F,D)$，其中$F$是一个凸泛函，定义了Banach空间$X$上的范数，$D$是$X$上的一组等距。位错空间中的有界序列允许分解成由$D$的作用位错的渐近和“轮廓”$w^{(n)}\in X$，即形式为$\sum_ng^{(n)}_kw^{(n)}$，$g^{(n)}_k\in D$的和，而余项在任何序列$g_k\in D$的作用下弱收敛（{ \em $D$ -弱收敛}）。这种分解允许扩展弱收敛的论点从变分问题与紧的问题，其中$X$是{ \em cocomplex}（相对于组$D$）嵌入到一个Banach空间$Y$，也就是说，当每个序列$D$ -弱收敛在$X$是收敛的规范$Y$。我们证明了一个一般的声明存在的极小在cocompact嵌入适用，特别是索伯列夫嵌入缺乏紧凑性（无界域，临界指数），包括次椭圆索伯列夫空间和空间的黎曼流形。
英文摘要：
Concentration compactness method is a powerful techniques for establishing existence of minimizers for inequalities and of critical points of functionals in general. The paper gives a functional-analytic formulation for the method in Banach space, generalizing the Hilbert space case elaborated in \cite{ccbook}. The key object is a dislocation space - a triple $(X,F,D)$, where $F$ is a convex functional that defines a norm on Banach space $X$, and $D$ is a group of isometries on $X$. Bounded sequences in dislocation spaces admit a decomposition into an asymptotic sum "profiles" $w^{(n)}\in X$ dislocated by actions of $D$, that is, a sum of the form $\sum_ng^{(n)}_kw^{(n)}$, $g^{(n)}_k\in D$, while the remainder term converges weakly under actions of any sequence $g_k\in D$ ({\em $D$-weak convergence}). This decomposition allows to extend the weak convergence argument from variational problems with compactness to problems where $X$ is {\em cocompactly} (relatively to the group $D$) imbedded into a Banach space $Y$, that is, when every sequence $D$-weakly convergent in $X$ is convergent in the norm of $Y$. We prove a general statement on existence of minimizers in cocompact imbeddings that applies, in particular to Sobolev imbeddings which lack compactness (unbounded domain, critical exponent) including the subelliptic Sobolev spaces and spaces over Riemannian manifolds.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
