Quantitative homogenization of principal Dirichlet eigenvalue shape optimizers

2022/09/03 by Feldman, William M
#35B27 #49Q10 #49R05 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2209.01446

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 L² 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.

Related

• Homogenization of eigenvalues for problems with high-contrast inclusions
• Lectures on Periodic Homogenization of Elliptic Systems
• Large scale regularity of almost minimizers of the one-phase problem in periodic media
• Regularity for Shape Optimizers: The Nondegenerate Case
• Lipschitz regularity of homogenization with continuous coefficients: Dirichlet problem