Calculus of Variations: Principles and Applications

This lecture covers the principles and applications of calculus of variations, focusing on the Young mean value theorem, compactness principle, and the concept of equi-integrability. The instructor explains the conditions for uniform boundedness and equi-integrability of Carathéodory integrands, providing insights into the decomposition of functions and the extension of convergence to bounded functions. The lecture also delves into the properties of positive Radon measures and the construction of compact sets for continuous functions. Additionally, it explores the convergence of sequences and the implications of equi-integrability in the context of Carathéodory functions.