Quantitative homogenization of principal Dirichlet eigenvalue shape optimizers

TL;DR

This paper establishes a quantitative convergence rate for shape optimizers of the first Dirichlet eigenvalue in periodic homogenization, utilizing free boundary regularity and homogenization theory.

Contribution

It introduces a novel approach combining free boundary regularity and homogenization techniques to achieve a linear convergence rate with logarithmic factors.

Findings

Linear convergence rate for eigenvalue optimization

Application of free boundary regularity in homogenization

Use of Lipschitz free boundary regularity in periodic media

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.