Uniqueness and Convexity in the Calculus of Variations

Explore the role of convexity in the calculus of variations through this mathematical physics lecture delivered at the Institute for Advanced Study. Delve into the fundamental importance of convexity in variational problems and examine the extensive range of generalizations that have been developed for vectorial problems, from classical observations to practical conditions. Learn about a recently emerged convexity concept that has appeared across different research projects and discover its significance in describing the stability of uniqueness of minimizers under a broad class of perturbations. Compare and contrast this new concept with other established convexity notions for vectorial problems, gaining insight into both their differences and areas of agreement. The presentation provides a comprehensive overview of current developments in variational analysis and the evolving understanding of convexity's role in ensuring solution uniqueness.