How does an elastic membrane lie on a given obstacle? This naive looking question hides a beautiful mathematical theory integrating powerful tools with far-reaching applications. In this mini-course we will discuss the obstacle problem as a free boundary model. All necessary tools will be carefully constructed from scratch. We will mainly focus on: optimal regularity of solutions, non-degeneracy estimates, weak geometric-measure properties of the free boundary, classification of global profiles, and differentiability of the free boundary.
Eduardo Teixeira is a Full Professor in the Department of Mathematics at the University of Central Florida, USA. His research interests include free boundary problems and regularity theory for nonlinear partial differential equations. Among many distinctions, Prof. Teixeira received the 2013 Mathematical Congress of the Americas Prize and the 2017 ICTP-IMU Ramanujan Prize. He was elected permanent fellow of the Brazilian National Academy of Sciences in 2015 and awarded honorary citizenship of the State of Ceará in 2021.