A Microlocal Version of Concentration-Compactness

Abstract

The concentration-compactness principle has been developed by P.-L. Lions [10] in order to study minimizing sequences of elliptic variational problems displaying some lack of compactness. It is closely related to the general problem of describing the sequences of functions on R^d which violate locally the compactness of the Sobolev imbedding

$$ {H^S}{\mkern 1mu} \left( {{\mathbb{R}^d}} \right){\mkern 1mu} \to {\mkern 1mu} {L^P}{\mkern 1mu} \left( {{\mathbb{R}^d}} \right),{\mkern 1mu} 0 < {\mkern 1mu} s < {\mkern 1mu} \frac{d}{2},{\mkern 1mu} \frac{1}{p}{\mkern 1mu} = {\mkern 1mu} \frac{1}{2}{\mkern 1mu} - {\mkern 1mu} \frac{s}{d}. $$

((1))