On the measures of DiPerna and Majda. (English) Zbl 0902.28009

Kružík, Martin; Roubíček, Tomáš

Math. Bohem. 122, No. 4, 383-399 (1997).

Summary: R. J. DiPerna and A. J. Majda [Commun. Math. Phys. 108, 667-689 (1987; Zbl 0626.35059)] generalized Young measures [L. C. Young, C. R. Soc. Sci. Varsovie 30, 212-234 (1937; Zbl 0019.21901)] so that it is possible to describe “in the limit” oscillation as well as concentration effects of bounded sequences in \(L^p\)-spaces. Here the complete description of all such measures is stated, showing that the “energy” put at “infinity” by concentration effects can be described in the limit basically by an arbitrary positive Radon measure. Moreover, it is shown that concentration effects are intimately related to rays (in a suitable locally convex geometry) in the set of all DiPerna-Majda measures. Finally, a complete characterization of extreme points and extreme rays is established.

MSC:

28C15	Set functions and measures on topological spaces (regularity of measures, etc.)
46N10	Applications of functional analysis in optimization, convex analysis, mathematical programming, economics

Keywords:

bounded sequences in Lebesgue spaces; oscillations; concentrations; Young measures; DiPerna and Majda measures; rays; extreme points; extreme rays

Citations:

Zbl 0626.35059; Zbl 0019.21901