# Ovidiu Savin

## Columbia University

### Free boundary regularity for the 3 membranes problem

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 discuss some recent work in collaboration with Hui Yu about the case when $$N=3$$ and there are two interacting free boundaries.