Course Description: In broad terms, free boundaries appear when the solution to a problem consists of a pair: a function (often satisfying a partial differential equation), and a set associated to this function. Two central issues in the study of free boundary problems are: 

• What is the optimal regularity of the solution? 

• How smooth is the free boundary? 

We will learn tools, such as monotonicity formulas, that will allow us to investigate these questions.

Lecture notes.