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Shape optimization problems for metric graphs

  • Giuseppe Buttazzo (a1), Berardo Ruffini (a2) and Bozhidar Velichkov (a2)
Abstract

We consider the shape optimization problem \hbox{$\min\big\{\E(\Gamma)\ :\ \Gamma\in\A,\ \H^1(\Gamma)=l\ \big\},$}min{ℰ(Γ):Γ ∈ 𝒜, ℋ1(Γ) = l}, where ℋ1 is the one-dimensional Hausdorff measure and 𝒜 is an admissible class of one-dimensional sets connecting some prescribed set of points \hbox{$\D=\{D_1,\dots,D_k\}\subset\R^d$}𝒟 =  { D1,...,Dk }  ⊂ Rd. The cost functional ℰ(Γ) is the Dirichlet energy of Γ defined through the Sobolev functions on Γ vanishing on the points Di. We analyze the existence of a solution in both the families of connected sets and of metric graphs. At the end, several explicit examples are discussed.

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ESAIM: Control, Optimisation and Calculus of Variations
  • ISSN: 1292-8119
  • EISSN: 1262-3377
  • URL: /core/journals/esaim-control-optimisation-and-calculus-of-variations
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