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

We consider the following eigenvalue optimization problem: Given a bounded domain Omega subset of R and numbers alpha > 0, A is an element of [0, \Ome ga\], find a subset D subset of Omega of area A for which the first Dirichl et eigenvalue of the operator -Delta + alpha chi (D) is as small as possibl e, We prove existence of solutions and investigate their qualitative propertie s. For example, we show that for some symmetric domains (thin annuli and du mbbells with narrow handle) optimal solutions must possess fewer symmetries than R; on the other hand, for convex Omega reflection symmetries are pres erved. Also, we present numerical results and formulate some conjectures suggested by them.