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arXiv · 2209.01446

Quantitative homogenization of principal Dirichlet eigenvalue shape optimizers

William M Feldman
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Abstract

We apply new results on free boundary regularity of one-phase almost minimizers in periodic media 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, Lin and Shen. 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.

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William M Feldman. 2022-09-03. Quantitative homogenization of principal Dirichlet eigenvalue shape optimizers. https://arxiv.org/abs/2209.01446

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