  A cost functional involving the eigenvalues of an elastic structure, that is
described by a multi-phase-field equation, is optimized. This allows us to
handle topology changes and multiple materials. We prove continuity and
differentiability of the eigenvalues and we establish the existence of a global
minimizer to our optimization problem. We further derive first-order necessary
optimality conditions for local minimizers. Moreover, an optimization problem
combining eigenvalue and compliance optimization is also discussed.
