Abstract: In the previous lecture, I explained two techniques for studying the existence and computing minimizers of functionals: a classical method based on solving the Euler–Lagrange equation coming from the first variation of the functional, and a modern approach based on extracting convergent sub-sequences from minimizing sequences of the functional under consideration. 

In this lecture, I will explain another direct technique: instead of looking at the original functional, one can look at a nice perturbation of the original functional and extract information about the minimizers of the original functional by studying the perturbed functional. Such a perturbation is known as a Gamma-perturbation. My plan is to explain the notion of Gamma-perturbation with examples and demonstrate with examples how one can recover some minimizers of a given functional via its gamma-perturbations.

Book for further reading on these topics: Gianni Dai Maso, An Introduction to Gamma-Convergence

Here are the details SLIDES and Video Recording from Lecture 02.

Abstract: In this talk, I introduce the audience to a beautiful piece of Mathematics, the Calculus of Variations. The Calculus of Variations deals with minimizing functionals (possibly non-linear) defined on Banach spaces (possibly infinite-dimensional). To motivate the theory,  I start with real-life examples to show how solutions to specific problems in life boil down to minimizing certain real-valued functionals defined on Banach spaces. Using the knowledge of the audience from the undergraduate analysis, I then in a heuristic way develop techniques that help to investigate the existence and computability of minimizers of a class of functionals. I explain both classical and direct methods (modern) to minimize functionals.

Here are the details SLIDES from Lecture 01. 

Homework: Here is homework that you can try. Please.

Book for further reading on these topics: Direct Methods in the Calculus of Variations | Bernard Dacorogna | download (b-ok.cc)