Obstacle problems and free boundaries: an overview
arXiv:1707.00992
Abstract
Free boundary problems are those described by PDEs that exhibit a priori unknown (free) interfaces or boundaries. These problems appear in Physics, Probability, Biology, Finance, or Industry, and the study of solutions and free boundaries uses methods from PDEs, Calculus of Variations, Geometric Measure Theory, and Harmonic Analysis. The most important mathematical challenge in this context is to understand the structure and regularity of free boundaries. In this paper we provide an invitation to this area of research by presenting, in a completely non-technical manner, some classical results as well as some recent results of the author.
This expository paper originates from my lecture at the XXV CEDYA conference, where I was awarded the 2017 Antonio Valle Prize from the Sociedad Española de Matemática Aplicada (SeMA)