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