Limit Shapes of Local Minimizers for the Alt–Caffarelli Energy Functional in Inhomogeneous Media

Feldman, William M.

doi:10.1007/s00205-021-01635-6

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2023

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“…For example, as seen in ref. [9,15,17,21], the homogenized problem associated with local minimizers of Bernoulli-type functionals can be of a completely different type compared to the effective problem for minimizers or almost minimizers. Thus this eigenvalue shape optimization problem is a rather special scenario where we can take advantage of the variational/almost minimizer property of the solutions in question.…”

Section: Literaturementioning

confidence: 99%

Quantitative homogenization of principal Dirichlet eigenvalue shape optimizers

Feldman

2023

Comm Pure Appl Math

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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 L2 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.

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Section: Literaturementioning

confidence: 99%