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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
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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 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
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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