Relations Between Young Measures and H-Measures

Abstract

By describing the L^∞(Ω) weak * limits of all continuous functions of a bounded sequence Uⁿ in L^∞(Ω;ℝ^p), the Young measures give a natural mathematical way to interpret the local (one-point) statistics of the values taken by Uⁿ without falling prey to the unrealistic fashion of imposing old probabilistic games which usually destroy the physical relevance, since nature behaves in a different way, which it is the role of scientists to discover. However, Young measures are limited and cannot discern any geometric pattern or take into account differential equations satisfied by Uⁿ; actually, the theory has no need for Ω to be an open subset of ℝ^N or a manifold, and it can be applied to a set endowed with a measure without atoms.