, Volume 4, Issue 1, pp 59-90

Gradient Young measures generated by sequences in Sobolev spaces

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Abstract

Oscillatory properties of a weak convergent sequence of functions bounded inL p , 1 ≤p ≤ ∞, may be summarized by the parametrized measure it generates. When such a measure is generated by the gradients of a sequence of functions bounded inH 1,p , it must have special properties. The purpose of this paper is to characterize such parametrized measures as the ones that obey Jensen’s inequality for all quasiconvex functions with the appropriate growth at infinity. We have found subtle differences between the casesp < ∞ andp = ∞. A consequence is that any measure determined by biting convergence is in fact generated by a sequence convergent in a stronger sense. We also give a few applications.

Research groupTransitions and Defects in Ordered Materials, funded by the NSF, the AFOSR, and the ARO. The work of the second author is also supported by DGICYT (Spain) through “Programa de Perfeccionamiento y Movilidad del Personal Investigador” and through Grant PB90-0245.