S. Chanillo et al., Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes, COMM MATH P, 214(2), 2000, pp. 315-337
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.