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

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 L-1-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 W-1,W-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 W-1,W-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 W-1,W-1- and BV-case and contains its version for weakly converging sequences in W-1,W-1 as a special case. Finally, we give an application to a new lower semicontinuity theorem in BV.