On one generalization of a DiPerna and Majda theorem

Abstract

SUMMARYThe weak limits of sequences {f(u )} ∈N where u 's are vector-valued -measurable functions deÿned on a compact set and f is (possibly) discontinuous are investigated. As shown by the author (J. Conv. Anal. (to appear)), they are described in terms of integral formulae involving parametrized measures independent of f, similarly as in the classical theorem by Young and its generalization due to DiPerna and Majda. In the present paper we describe the supports of the involved parametrized measures.

on a certain compactification of an arbitrary subset of ℝm and its applications to DiPerna-Majda measures theory

“…To motivate the second of our main results, let us discuss two theorems which are of interest in the field of Calculus of Variations. The first is a variant of Young (DiPerna-Majda) Theorem for discontinuous integrands -Theorem 2.11 due to Kałamajska [18], see also [17,[19][20][21] for related results. The theorem shows a representation formula for the weak-limit for the sequences of compositions {f (u ν )dµ}, where f : R m → R is a continuous function on every set A i , i = 1, .…”

“…To proceed with such tasks we require a compactification γR m of R m -a target space for functions u ν -such that every function i x : p → f (x, p) ∈ R is continuously extendable to function defined on γR m and the space γR m is independent of x. However, in further analysis it is required that γR m is metric, separable space (see [17,[19][20][21][22]). Arranging the compactification for every function i x separately is too naïve for those purposes.…”