Diff. Geom, Math. Phys., PDE Seminar: Emil Wiedemann

It is well-known that variational problems may fail to have a classical minimiser if the integrand is not convex. In the 1930s, L. C. Young suggested a relaxation of such problems, where the minimising map is allowed to be measure-valued. In physical applications (e.g. elasticity theory), one often looks at variational problems for gradients of vector fields. A crucial problem in the context of relaxation is to characterise those measure-valued maps that arise as limits of a sequence of gradients. While this was achieved by D. Kinderlehrer and P. Pedregal about 20 years ago, the question remained open whether a similar characterisation could be found under the additional constraint that the gradients have positive determinant, i.e. the underlying maps be orientation-preserving. I will present such a characterisation, recently obtained in joint work with K. Koumatos (Oxford) and F. Rindler (Warwick).