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Generalized Young Measures

  • Filip RindlerEmail author
Chapter
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Part of the Universitext book series (UTX)

Abstract

In this chapter we continue the study of the integral functional
$$ \mathscr {F}[u] := \int _\varOmega f(x, \nabla u(x)) \;\mathrm{d}x + \int _\varOmega f^\# \biggl ( x, \frac{\mathrm{d}D^s u}{\mathrm{d}|D^s u|}(x) \biggr ), \qquad u \in \mathrm {BV}(\varOmega ;\mathbb {R}^m), $$
for a Carathéodory integrand \(f :\varOmega \times \mathbb {R}^{m \times d} \rightarrow \mathbb {R}\) with linear growth. In contrast to the preceding chapter, however, here we proceed in a more abstract way: We first introduce the theory of generalized Young measures, which extends the standard theory of Young measures developed in Chapter  4. Besides quantifying oscillations (like classical Young measures), this theory crucially allows one to quantify concentrations as well, thus providing a rich toolbox for investigating linear-growth functionals. While the (generalized) Young measure approach requires a fair bit of abstract theory, the initial effort is rewarded with a robust general framework that has become a core tool in the calculus of variations with applications way beyond the lower semicontinuity theory of integral functionals.

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

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