Calculus of Variations and Geometric Measure Theory

R. Rossi - G. Savaré

Tightness-Concentration Principles and Compactness for Evolution Problems in Banach Spaces

created on 02 May 2002
modified on 10 Jun 2003


Accepted Paper

Inserted: 2 may 2002
Last Updated: 10 jun 2003

Journal: Annali Scuola Normale Superiore - Pisa
Year: 2002


Compactness in the space $L^p(0,T;B)$, $B$ being a separable Banach space, has been deeply investigated by J.P. Aubin (1963), J.L. Lions (1961,1969), J. Simon (1987), and, more recently, by J.M. Rakotoson and R. Temam (2001), who have provided various criteria for relative compactness, which turn out to be crucial tools in the existence proof of solutions to many abstract time dependent problems related to evolutionary PDE's.

In the present paper, the problem is examined in view of Young measure theory: exploiting the underlying principles of ''tightness'' and ''concentration'', new necessary and sufficient conditions for compactness are given, unifying some of the previous contributions and showing that the Aubin-Lions condition is not only sufficient but also necessary for compactness. Furthermore, the related issue of compactness with respect to convergence in measure is studied and a general criterion is proved.

Keywords: Young Measures, concentration, Compactness, Tightness