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. 11:159–197, 2022). 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. Eng. 194:3269–3290, 2005) 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.

我们考虑 Garcke 等人先前介绍的多材料弹性结构特征值的优化问题。（高级非线性分析。11:159–197, 2022）。其中，弹性结构由矢量值相场变量表示，并推导了相应的由状态方程和梯度不等式组成的最优系统。在本文中，我们通过形式匹配渐近技术达到了该最优系统中的尖锐界面极限。因此，我们推导出合适的拉格朗日乘子来将梯度不等式表示为逐点等式。然后，我们引入内扩展和外扩展，通过适当的匹配条件将它们联系起来，并通过比较状态方程和梯度等式中的前导项来正式传递到锐界面极限。此外，还讨论了这些正式导出的一阶条件与 Allaire 和 Jouve (Comput.Methods Appl.Mech.Eng.194:3269–3290, 2005)在经典形状微积分框架中获得的结果之间的关系。最终，我们提供了各种示例的数值模拟。特别是，我们说明了尖锐界面限制，并考虑了同时合规性和特征值优化的联合优化问题。