Characterization of gradient Young measures under determinant constraints

In this talk, I will present a characterization result, in the spirit of Kinderlehrer & Pedregal, for Young measures generated by gradients of Sobolev maps which additionally satisfy various determinant constraints. Young measures are an important tool in nonlinear analysis as they allow one to characterise limits of nonlinear quantities depending on the sequence generating the Young measure. However, important determinant constraints in elasticity, e.g. the orientation-preserving condition, are non-convex and cannot be accounted for by the proof of Kinderlehrer & Pedregal. In joint work with Filip Rindler and Emil Wiedemann, we overcome this difficulty by employing a variant of convex integration to "correct" the Jacobian and provide generating sequences in Sobolev spaces of exponent less than the dimension where the determinant constraints prove to be soft. Several interesting applications are also presented.