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Archive for Rational Mechanics and Analysis

, Volume 197, Issue 2, pp 539–598 | Cite as

Characterization of Generalized Gradient Young Measures Generated by Sequences in W1,1 and BV

  • Jan Kristensen
  • Filip Rindler
Article

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 developing 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.

Keywords

Radon Lower Semicontinuity Radon Measure Lipschitz Domain Young Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK

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