Free boundary problems naturally arise in several applications in mathematical physics, biology and engineering. They are characterized by different sets of differential relations on distinct domains which are not assigned apriori but are among the unknowns of the problem.
This course will deal with the non-parametric thin obstacle problem for minimal surfaces, which is a prototypical example of a free boundary problem with higher co-dimension free boundary. This problem consists in minimizing the area of a graph with prescribed boundary conditions and with a unilateral constrain imposed on a hyperplane.
In the lectures I will present some of the key-ideas that have been introduced to treat this and similar problems, with a special focus (time permitting) on the new variational and measure theoretic techniques recently developed.