Cocompact imbeddings and structure of weakly convergent sequences

TL;DR

This paper develops a functional-analytic framework for the concentration compactness method in Banach spaces, introducing dislocation spaces and D-weak convergence to analyze minimizers in cocompact embeddings, especially in Sobolev spaces.

Contribution

It generalizes the concentration compactness method to Banach spaces using dislocation spaces and D-weak convergence, extending applicability to non-compact Sobolev embeddings.

Findings

Decomposition of bounded sequences into profiles with D-weak convergence.

Extension of weak convergence arguments to cocompact embeddings.

Application to Sobolev spaces on unbounded domains and manifolds.

Abstract

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.