Weak convergence of the sequences of homogeneous Young measures associated with a class of oscillating functions

TL;DR

This paper investigates the weak convergence of homogeneous Young measures associated with oscillating functions, introducing densities and total slopes, and establishing conditions for convergence and closedness in measure spaces.

Contribution

It introduces the notion of density for Young measures, links strong closedness of measures to densities, and establishes weak convergence criteria based on total slopes of oscillating functions.

Findings

Density of Young measures is weakly sequentially closed.

Strong closedness of measures relates to densities in L1(K).

Monotonic total slopes imply weak convergence of Young measures.

Abstract

We take under consideration Young measures with densities. The notion of density of a Young measure is introduced and illustrated with examples. It is proved that the density of a Young measure is weakly sequentially closed set. In the case when density of a Young measure is a singleton (up to the set of null measure), it is shown that the strong closedness (in rca(K)) of the set of such measures, associated with Borel functions with values in the compact set K ? Rl, is equivalent with the strong closedness (in L1(K)) of the set of their densities, provided the set K is convex. For an m-oscillating function the notion of a total slope is proposed. It turns out, that if the total slopes of the elements of the sequence of oscillating functions form monotonic sequence, then the sequence of the respective (homogeneous) Young measures is weakly convergent in rca(K). The limit is a homogeneous Young measure with the density being the weak L1 sequential limit of the densities of the underlying Young measures.