Barycenter and Young Measures

This lecture covers the concept of barycenter in LP spaces, weak convergence, and the uniqueness of limits. It explains the properties of sequences in LP spaces, the barycenter of a sequence, and the convergence of subsequences. The lecture also discusses the concept of weak convergence and pre-compactness in LP spaces, along with the application of the fundamental theorem. Furthermore, it delves into the equivalence of statements involving distances and mean values, as well as the role of Young measures in gradient analysis. The lecture concludes with the importance of equi-integrability and homogeneous gradient Young measures.