Cocompact Imbeddings and Structure of Weakly Convergent Sequences

Abstract

The concentration compactness method is a powerful technique for establishing the existence of minimizers for inequalities and critical points of functionals. We give a functional-analytic formulation for the method in a Banach space. The key object is a dislocation space, i.e., a triple (X, F, D), where F is a convex functional defining a norm on a Banach space X and D is a group of isometries on X. Bounded sequences in dislocation spaces admit a decomposition into an asymptotic sum of “profiles” w(n) ∊ X dislocated by the actions of D. This decomposition allows to extend the weak convergence argument from variational problems with compactness to problems, where X is cocompactly (relatively to D) imbedded into a Banach space Y . We prove a general statement on the existence of minimizers in cocompact imbeddings that applies, in particular, to Sobolev imbeddings which lack compactness (an unbounded domain, a critical exponent) including the subelliptic Sobolev spaces and spaces over Riemannian manifolds.