In elasticity theory, one naturally requires that the Jacobian determinant of the deformation is positive or even a-priori prescribed (for example incompressibility). However, such strongly non-linear and non-convex constraints are difficult to deal with in mathematical models. In this talk, which is based on joint work with K. Koumatos (Oxford) and E. Wiedemann (UBC/PIMS), I will present various recent results on how this constraint can be manipulated in subcritical Sobolev spaces, where the integrability exponent is less than the dimension.

In particular, I will give a characterization theorem for Young measures under this side constraint, which are widely used in the Calculus of Variations to model limits of nonlinear functions of weakly converging "generating" sequences. This is in the spirit of the celebrated Kinderlehrer--Pedregal Theorem and based on convex integration and "geometry" in matrix space.

Finally, applications to the minimization of integral functionals, the theory of semiconvex hulls, incompressible extensions, and approximation of weakly orientation-preserving maps by strictly orientation-preserving ones in Sobolev spaces are given.