We consider an energy model for N ordered elastic membranes subject to forcing and boundary conditions. The heights of the membranes are described by real functions u_1, u_2,...,u_N, which minimize an energy functional involving the Dirichlet integral and a potential term depending on the cardinality of the set {u_1,..,u_N}. The potential term corresponds to the physical situation when consecutive membranes are glued to each other on their coincidence set. The problem can be understood as a system of N-1 coupled one-phase free boundary problems with interacting free boundaries. I will review the known results in the scalar case, and discuss the free boundary regularity when dealing with 3 or more membranes (joint with D. De Silva.)