Mathematics

Many imbeddings of functional spaces lack compactness because of the presence of a non-compact invariance, such as translation or scale invariance. Loss of compactness for bounded sequences can be effectively described with the help of this group: any bounded sequence has a subsequence consisting of a sum of decoupled "bubbles" (by group action) and a convergent remainder. This representation, called profile decomposition, exists on the functional-analytic level, and the hard analysis is involved only in the question what is the best norm for which an absence of bubbles guarantees convergence. Successor of the classical concentration compactness, theory of profile decompositions in its present state has been applied to concentration analysis in dispersive equations (Terence Tao), yields a necessary and sufficient condition for a symmetry on a manifold to define a compact Sobolev imbedding, and shows that Moser-Trudinger functional is weakly continuous on a unit ball $B$ in the Sobolev norm with an exception only for some  sequences on a single three-dimensional surface contained in the boundary of the unit ball.