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Shape optimization problems for metric graphs

Published online by Cambridge University Press:  29 August 2013

Giuseppe Buttazzo ,
Berardo Ruffini  and
Bozhidar Velichkov
Giuseppe Buttazzo
Affiliation:
Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy. buttazzo@dm.unipi.it
Berardo Ruffini
Affiliation:
Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italy; berardo.ruffini@sns.it; b.velichkov@sns.it
Bozhidar Velichkov
Affiliation:
Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italy; berardo.ruffini@sns.it; b.velichkov@sns.it

Abstract

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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 Hausdorffmeasure and𝒜is an admissible class of one-dimensional setsconnecting 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 theDirichlet energy of Γ defined through the Sobolev functions onΓ vanishing on the pointsDi. We analyze the existence of a solutionin both the families of connected sets and of metric graphs. At the end, several explicitexamples are discussed.

Keywords

Shape optimizationrectifiable setsmetric graphsquantum graphsDirichlet energy

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 1 , January 2014 , pp. 1 - 22
Copyright
© EDP Sciences, SMAI, 2013

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