Symmetry in rearrangement optimization problems

This article concerns two rearrangement optimization problems. The first problem is motivated by a physical experiment in which an elastic membrane is sought, built out of several materials, fixed at the boundary, such that its frequency is minimal. We capture some features of the optimal solutions, and prove a symmetry property. The second optimization problem is motivated by the physical situation in which an ideal fluid flows over a seamount, and this causes vortex formation above the seamount. In this problem we address existence and symmetry.