Differential inclusions and Young measures involving prescribed Jacobians

In elasticity theory, one naturally requires that the Jacobian determinant of the deformation is positive or even a-priori prescribed (e.g. for incompressibility). However, such strongly non-linear and non-convex constraints are difficult to deal with in mathematical models. This short note, which is based on joint work with K. Koumatos and E. Wiedemann, presents 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, we give a characterization theorem for Young measures under this side constraint. This is in the spirit of the celebrated Kinderlehrer–Pedregal Theorem and based on convex integration and “geometry” in matrix space. Finally, applications to approximation in Sobolev spaces and to the failure of lower semicontinuity for certain integral functionals with “realistic” growth conditions are given. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)