Short course

Karen Yeresian
Introduction to the Regularity in Free Boundary Problems

In the last decades a beautiful theory has been developed for the regularity (smoothness) of free boundary problems and in this course our aim is to introduce the regularity theory of the free boundary in the well known obstacle problem. We will discuss the variational problems and in particular variational inequalities leading to free boundaries, classical obstacle problem, its optimal regularity, monotonicity formulas and regularity of the free boundary near regular points.
Examples of free boundary problems: Consider an elastic membrane stretched on an object. Where the membrane is not touching the object, it is minimizing the elastic energy and thus solving a partial differential equation. There is a boundary around where the membrane is touching the object. This boundary is a-priori unknown and is part of the configuration. A similar phenomenon is the boundary between oil and water in a mixture of these two liquids. And another example is the surface of melting ice as a boundary between ice and water. 

One may find the lecture notes at: