Characterization of generalized gradient young measures generated by sequences in W<sup>1,1</sup> and BV

Author: 

Kristensen, J
Rindler, F

Publication Date: 

1 January 2010

Journal: 

Archive for Rational Mechanics and Analysis

Last Updated: 

2019-08-18T14:08:54.16+01:00

Issue: 

2

Volume: 

197

DOI: 

10.1007/s00205-009-0287-9

page: 

539-598

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. © 2009 Springer-Verlag.

Symplectic id: 

65330

Submitted to ORA: 

Not Submitted

Publication Type: 

Journal Article