O Princípio de concentração e compacidade e aplicações

Abstract

Many issues in Physics-Mathematics or Differential Geometry involve minimization problems, where minimizing sequences are not automatically compressed by standar arguments of Functional Analysis. Typically, the loss of compactness is due to domain invariance, for example, by non-compact groups of translations and dilations; or by the loss of compact dives, often by not limiting the domain or by working with critical exponents. The main advance to deal with this loss of compactness was obtained by Lions, P. L. in four famous articles: The concentration-compactness principle in the Calculus of Variations. The locally compact case, part 1 and part 2 [PL1] and [PL2] The concentration-compacting principle in the Calculus Variation. The limit case, part 1 and part 2 [PL3] and [PL2] The first two present a method for solving minimization problems in non-limited domains. It is derived from a heuristic principle the equivalence between the compactness of all minimizing sequences in a sub-diactivity condition. The proof is based on the Lemma of Concentration and Compaction, which is obtained with the notion of concentration functions of a measure.