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A course on Young measures. (English) Zbl 0880.49013

The paper contains the lecture notes of a course given by the author in Italy on September 1993 about Young measures. This tool was introduced by L. C. Young to describe the behaviour of minimizing sequences of variational problems and since then there have been many generalizations to wider frameworks and applications to PDE and mechanics (see the wide list of references at the end of the paper).
Given a measure space \((\Omega,{\mathcal F},\mu)\) with \(\mu\) a nonnegative finite measure, we say that \(\nu\) is a Young measure on \(\mathbb{R}^m\) if it is a nonnegative measure on \(\Omega\times\mathbb{R}^m\) such that \(\nu(E\times\mathbb{R}^m)= \mu(E)\) for every \(E\in{\mathcal F}\). To any measurable function \(u:\Omega\to\mathbb{R}^m\), we can associate the Young measure \(\nu_u\) carried by the graph of \(u\), that is \[ \nu_u(E\times B)= \mu(E\cap u^{-1}(B))\qquad\forall E\in{\mathcal F},\;\forall B\in{\mathcal B}(\mathbb{R}^m); \] in this way, for every integrand \(f(x,s)\) on \(\Omega\times\mathbb{R}^m\) we have \[ \int_{\Omega\times\mathbb{R}^m} f(x,s)d\nu_u(x,s)= \int_\Omega f(x,u(x))d\mu(x). \] On the other hand, there are Young measures which are not associated to any function \(u\), as for instance it happens in general to weak limits of sequences \(\{\nu_{u_n}\}\).
Many properties of Young measures are presented here in detail, as for instance relaxation, disintegration, the narrow topology, lower semicontinuity, Prokhorov’s theorem, oscillations, and some applications to the calculus of variations.
Reviewer: G.Buttazzo (Pisa)

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
28A33 Spaces of measures, convergence of measures
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