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.