The Concentration-Compactness principle

The Concentration-Compactness principle was designed to solve minimization problems in unbounded domains, where Sobolev embedding theorems cannot be used. The method was introduced by P. L. Lions in the articles “The Concentration-Compactness principle in the Calculus of Variations. The locally compact case, part 1 and part 2” 1984. We will analyze the general result for measures, present a general and heuristic way to use the principle and then discuss an application to some minimization problems in L^1(R^N), an example presented by P. L. Lions in the original paper.