Orientation-preserving young measures - Koumatos, Konstantinos; Rindler, Filip; Wiedemann, Emil (Quarterly Journal of Mathematics, 2016)
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 behaves flexibly, as is illustrated by our results. 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.
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