Orientation-preserving young measures

Koumatos K, Rindler F, Wiedemann E (2016)

Publication Type: Journal article

Publication year: 2016

Journal

Quarterly Journal of Mathematics Oxford University Press (OUP): Policy B

Book Volume: 67

Pages Range: 439-466

Journal Issue: 3

DOI: 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.

Authors with CRIS profile

Emil Wiedemann

Involved external institutions

Gran Sasso Science Institute IT Italy (IT) University of Warwick GB United Kingdom (GB) Rheinische Friedrich-Wilhelms-Universität Bonn DE Germany (DE)

How to cite

APA:

Koumatos, K., Rindler, F., & Wiedemann, E. (2016). Orientation-preserving young measures. Quarterly Journal of Mathematics, 67(3), 439-466. https://dx.doi.org/10.1093/qmath/haw019

MLA:

Koumatos, Konstantinos, Filip Rindler, and Emil Wiedemann. "Orientation-preserving young measures." Quarterly Journal of Mathematics 67.3 (2016): 439-466.

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