Frida Fejne: An introduction to the obstacle problem

The classical obstacle problem seeks to model the shape of an elastic membrane which is attached to a fixed wire and forced to stay above some obstacle, affecting its shape. Mathematically this is done by minimizing the Dirichlet integral over all functions that lie above the given obstacle subject to some boundary conditions. The boundary of the contact set between the obstacle and the solution of the minimization problem is called the free boundary. In this talk we will go through some classical regularity theory for the obstacle problem and discuss the infinitesemal behaviour of the solution around different points on the free boundary, through so called blow-ups.