Motivated by a model for elastic membranes, we study the minimization of the Canham-Helfrich energy among closed surfaces of prescribed area that are trapped inside a given container. We show existence of minimizers in the class of immersed bubble trees and derive the Euler-Lagrange equations which involve a measure term that concentrates on the free boundary. This system can be transferred into a system of conservation laws with a Jacobian structure which allows us to conlcude the optimal regularity for solutions. This is joint work with M. Röger (Dortmund).