On p-Quasiconvex Hulls of Matrix Sets

We present some basic properties and equivalent definitions of the p-quasiconvex hull of a given set of matrices. In particular, we completely characterize the p-quasiconvex hull in terms of the W ^1,p -gradient Young measures studied by D. Kinderlehrer and P. Pedregal ["Gradient Young measures generated by sequences in Sobolev spaces", J. Geom. Anal. 4 (1994) 59–90] and establish an important relationship with the weak convergence in Sobolev spaces. We also give some simple characterization of the p-quasiconvex hulls for certain special sets