Thin films of diblock copolymers: existence and two-dimensional regularity of minimizing configurations.

We introduce a variational model for the study of thin films of diblock copolymers. We establish existence of minimizing configurations via relaxation of the energy involved. Moreover, we prove partial regularity results for mass-constrained minimizers in two space dimensions. To show this last result, we develop a two-dimensional regularity theory for partitions with quasi-minimal surface area, subject to an additional graph constraint.