Abstract
Generalized Young measures as introduced by DiPerna and Majda (Commun Math Phys 108:667–689, 1987) provide a quantitative tool for studying the one-point statistics of oscillation and concentration in sequences of functions. In this work, after develo** a functional-analytic framework for such measures, including a compactness theorem and results on the generation of such Young measures by L1-bounded sequences (or even by sequences of bounded Radon measures), we turn to investigation of those Young measures that are generated by bounded sequences of W1,1-gradients or BV-derivatives. We provide several techniques to manipulate such measures (including shifting, averaging and approximation by piecewise-homogeneous Young measures) and then establish the main new result of this work, the duality characterization of the set of (BV- or W1,1-)gradient Young measures in terms of Jensen-type inequalities for quasiconvex functions with linear growth at infinity. This result is the natural generalization of the Kinderlehrer–Pedregal Theorem (Arch Ration Mech Anal 115:329–365, 1991; J Geom Anal 4:59–90, 1994) for classical Young measures to the W1,1- and BV-case and contains its version for weakly converging sequences in W1,1 as a special case. Finally, we give an application to a new lower semicontinuity theorem in BV.
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Communicated by G. Dal Maso
An erratum to this article can be found online at http://dx.doi.org/10.1007/s00205-011-0477-0
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Kristensen, J., Rindler, F. Characterization of Generalized Gradient Young Measures Generated by Sequences in W1,1 and BV. Arch Rational Mech Anal 197, 539–598 (2010). https://doi.org/10.1007/s00205-009-0287-9
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DOI: https://doi.org/10.1007/s00205-009-0287-9