Abstract. For a positive integer N, the N-membranes problem describes the equilibrium position of N ordered elastic membranes subject to forcing and boundary conditions. If the heights of the membranes are described by real functions u_1, u_2,...,u_N, then the problem can be understood as a system of N-1 coupled obstacle problems with interacting free boundaries which can cross each other. When N=2 there is only one free boundary and the problem is equivalent to the classical obstacle problem. In my talk I will review the free boundary regularity in obstacle-type problems and discuss some recent work in collaboration with Hui Yu on the multiple membrane problem.

Bio. Savin received his Ph.D. in mathematics from the University of Texas at Austin in 2003. He was a Miller postdoctoral fellow at UC Berkeley and in 2006 he joined the faculty of Columbia University where he is currently a professor of mathematics. Savin's research is in the area of regularity theory for nonlinear PDEs and free boundary problems. He is well known for his proof of De Giorgi's conjecture and his contributions to the theory of the Monge-Ampere equation. Savin was an invited speaker at the International Congress of Mathematics in 2006 and was awarded the Stampacchia Medal in 2012. He is an editor of several journals including Advances in Mathematics, Calculus of Variations and PDEs, Communications in Analysis and Geometry.