The concentration-compactness principle in the Calculus of Variations. The locally compact case, part 1.

Abstract

We present here a new method for solving minimization problems in unbounded domains. We first derive a general principle showing the equivalence between the compactness of all minimizing sequences and some strict sub-additivity conditions. The proof is based upon a compactness lemma obtained with the help of the notion of concentration function of a measure. We give various applications to problems arising in Mathematical Physics.