  We consider an optimization problem for the eigenvalues of a multi-material
elastic structure that was previously introduced by Garcke et al. [Adv.
Nonlinear Anal. 11 (2022), no. 1, 159--197]. There, the elastic structure is
represented by a vector-valued phase-field variable, and a corresponding
optimality system consisting of a state equation and a gradient inequality was
derived. In the present paper, we pass to the sharp-interface limit in this
optimality system by the technique of formally matched asymptotics. Therefore,
we derive suitable Lagrange multipliers to formulate the gradient inequality as
a pointwise equality. Afterwards, we introduce inner and outer expansions,
relate them by suitable matching conditions and formally pass to the
sharp-interface limit by comparing the leading order terms in the state
equation and in the gradient equality. Furthermore, the relation between these
formally derived first-order conditions and results of Allaire and Jouve
[Comput. Methods Appl. Mech. Engrg., 194 (2005), pp. 3269--3290] obtained in
the framework of classical shape calculus is discussed. Eventually, we provide
numerical simulations for a variety of examples. In particular, we illustrate
the sharp-interface limit and also consider a joint optimization problem of
simultaneous compliance and eigenvalue optimization.
