Symmetry Breaking and Other Phenomena in the Optimization of Eigenvalues for Composite Membranes

We consider the following eigenvalue optimization problem: Given a bounded domain Ω ⊂ R and numbers α ≥ 0, A ∈ [0, |Ω|], find a subset D ⊂ Ω of area A for which the first Dirichlet eigenvalue of the operator −∆+ αχD is as small as possible. We prove existence of solutions and investigate their qualitative properties. For example, we show that for some symmetric domains (thin annuli and dumbbells with narrow handle) optimal solutions must possess fewer symmetries than Ω; on the other hand, for convex Ω reflection symmetries are preserved. Also, we present numerical results and formulate some conjectures suggested by them. 1. Problem and Main Results We study qualitative properties of solutions of a certain eigenvalue optimization problem. In physical terms, the problem can be stated as follows: Problem (P) Build a body of prescribed shape out of given materials (of varying densities) in such a way that the body has a prescribed mass and so that the basic frequency of the resulting membrane (with fixed boundary) is as small as possible. In fact, we will consider a more general problem, which we now state in mathematical terms: Given a domain Ω ⊂ Rn (bounded, connected, with Lipschitz boundary) and numbers α > 0, A ∈ [0, |Ω|] (with | · | denoting volume). For any measurable subset D ⊂ Ω let χD be its characteristic function and λΩ(α,D) the lowest eigenvalue λ of the problem −∆u+ αχDu = λu on Ω u = 0 on ∂Ω. (1)