Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes

We consider the following eigenvalue optimization problem: Given a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ and numbers $\alpha \ge 0$, $A\in [0,\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }]$, find a subset $D\subset \mathrm{\Omega }$ of area $A$ for which the first Dirichlet eigenvalue of the operator $-\mathrm{\Delta }+\alpha {\chi }_D$ 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 $\mathrm{\Omega }$; on the other hand, for convex $\mathrm{\Omega }$ reflection symmetries are preserved. Also, we present numerical results and formulate some conjectures suggested by them.