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Regularity for Shape Optimizers: The Nondegenerate Case
Author(s) -
Kriventsov Dennis,
Lin Fanghua
Publication year - 2018
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21743
Subject(s) - mathematics , bernoulli's principle , boundary (topology) , eigenvalues and eigenvectors , dimension (graph theory) , dirichlet distribution , function (biology) , dirichlet boundary condition , hausdorff space , set (abstract data type) , hausdorff dimension , mathematical analysis , pure mathematics , boundary value problem , programming language , physics , quantum mechanics , evolutionary biology , computer science , engineering , biology , aerospace engineering
Abstract We consider minimizers of F ( λ 1 ( Ω ) , … , λ N ( Ω ) ) + | Ω | , where F is a function strictly increasing in each parameter, andλ k ( Ω )is the k th Dirichlet eigenvalue of Ω. Our main result is that the reduced boundary of the minimizer is composed of C 1,α graphs and exhausts the topological boundary except for a set of Hausdorff dimension at most n – 3. We also obtain a new regularity result for vector‐valued Bernoulli‐type free boundary problems.© 2018 Wiley Periodicals, Inc.