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Orientation-preserving young measures
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Koumatos, K., Rindler, Filip and Wiedemann, E. (2016) Orientation-preserving young measures. Quarterly Journal of Mathematics, 67 (3). pp. 439-466. doi:10.1093/qmath/haw019 ISSN 0033-5606.
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Official URL: http://dx.doi.org/10.1093/qmath/haw019
Abstract
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.
| Item Type: | Journal Article | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
| Journal or Publication Title: | Quarterly Journal of Mathematics | ||||||||
| Publisher: | Oxford University Press | ||||||||
| ISSN: | 0033-5606 | ||||||||
| Official Date: | 1 September 2016 | ||||||||
| Dates: |
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| Volume: | 67 | ||||||||
| Number: | 3 | ||||||||
| Page Range: | pp. 439-466 | ||||||||
| DOI: | 10.1093/qmath/haw019 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||
| Date of first compliant deposit: | 9 June 2016 | ||||||||
| Date of first compliant Open Access: | 30 June 2017 | ||||||||
| Related URLs: |
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