Orientation-preserving Young measures

We prove a characterization result in the spirit of the Kinderlehrer–Pedregal Theorem for Young measures generated by gradients of Sobolev maps satisfying the orientation-preserving constraint, that is the pointwise Jacobian is positive almost everywhere. The argument to construct the appropriate generating sequences from such Young measures is based on a variant of convex integration in conjunction with an explicit lamination construction in matrix space. Our generating sequence is bounded in Lp for p less than the space dimension, a regime in which the pointwise Jacobian loses some of its important properties. On the other hand, for p larger than, or equal to, the space dimension the situation necessarily becomes rigid and a construction as presented here cannot succeed. Applications to relaxation of integral functionals, the theory of semiconvex hulls, and approximation of weakly orientation-preserving maps by strictly orientation-preserving ones in Sobolev spaces are given. MSC (2010): 49J45 (PRIMARY); 28B05, 46G10.