Orientation-preserving Young measures

TL;DR

This paper characterizes Young measures generated by orientation-preserving Sobolev maps, using convex integration and lamination, highlighting differences in behavior depending on the integrability exponent relative to the space dimension.

Contribution

It provides a novel construction of generating sequences for orientation-preserving Young measures in the subcritical integrability regime, extending the understanding of such measures.

Findings

Constructs bounded generating sequences in $L^p$ for $p$ less than the space dimension.

Shows rigidity and non-existence of such sequences for $p$ greater than or equal to the space dimension.

Applications include relaxation of functionals and approximation of maps in Sobolev spaces.

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 $L^p$ 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.