We apply new results on free boundary regularity to obtain a quantitative convergence rate for the shape optimizers of the first Dirichlet eigenvalue in periodic homogenization. We obtain a linear (with logarithmic factors) convergence rate for the optimizing eigenvalue. Large scale Lipschitz free boundary regularity of almost minimizers is used to apply the optimal $L^2$ homogenization theory in Lipschitz domains of Kenig et al. A key idea, to deal with the hard constraint on the volume, is a combination of a large scale almost dilation invariance with a selection principle argument.

中文翻译：

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主狄利克雷特征值形状优化器的定量均质化

我们应用自由边界正则性的新结果来获得周期性均匀化中第一狄利克雷特征值的形状优化器的定量收敛率。我们获得了优化特征值的线性（具有对数因子）收敛率。几乎最小化器的大规模 Lipschitz 自由边界正则性用于应用最优$L^2$Kenig 等人的 Lipschitz 域中的均质化理论。处理体积上的硬约束的一个关键思想是将大规模几乎膨胀不变性与选择原理论证相结合。