Uniqueness and Stability of Optimizers for a Membrane Problem

We investigate a PDE-constrained optimization problem, with an intuitive interpretation in terms of the design of robust membranes made out of an arbitrary number of different materials. We prove existence and uniqueness of solutions for general smooth bounded domains, and derive a symmetry result for radial ones. We strengthen our analysis by proving that, for this particular problem, there are no non-global local optima. When the membrane is made out of two materials, the problem reduces to a shape optimization problem. We lay the preliminary foundation for computable analysis of this type of problem by proving stability of solutions with respect to some of the parameters involved.