Characterization of homogeneous gradient Young measures in the case of arbitrary integrands

Abstract: "In the case of a continuous integrand L : R[superscript nm] -> R [union of] [infinity] and a probability measure v supported in R[superscript nm] we indicate conditions both necessary and sufficient for this measure to be generated as a homogeneous Young measure by gradients of piece-wise affine functions u[subscript k] [element of] l[subscript A] + W[superscript 1, infinity/subscript 0] ([omega]) with the property L(Du[subscript k]) - ̀[L;v] in L¹([omega]). Here A is the center of mass of v and l[subscript A] is a linear function with gradient equal to A everywhere. We show also that in the scalar case m = 1 any probability measure with finite action on L has this property. We provide elementary proofs of these results."