Some Variational Problems

Abstract

This chapter gathers a collection of problems for which the analysis does not involve any differential constraint, or if it does it is in a somewhat elementary way. It is a good way of practicing with the general ideas we will pursue for more complicated situations in subsequent chapters. For this reason we do not pretend to give the sharpest hypotheses under which theorems can be proved or improved, but rather focus on understanding the main techniques in each example. Some formal proofs are left to the reader as exercises. The same principle explains why in some of the problems we do not pursue the proof of all the steps and lemmas used when they are not relevant to our discussion. Three of the examples refer to variational principles or optimization. The last one does not. This has been included with the sole purpose of providing an illustration of how some analysis in terms of parametrized measures can also be helpful and provide some insight even though the problem is not directly related to variational principles but it is placed in a completely different context: large time behavior of complicated turbulent systems.