Skip to main content
×
Home
    • Aa
    • Aa

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.

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.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

J. Cheeger , Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9 (1999) 428517.

L. Friedlander , Extremal properties of eigenvalues for a metric graph. Ann. Inst. Fourier 55 (2005) 199211.

S. Gnutzmann and U. Smilansky , Quantum graphs: Applications to quantum chaos and universal spectral statistics. Adv. Phys. 55 (2006) 527625.

P. Kuchment , Quantum graphs: an introduction and a brief survey, in Analysis on graphs and its applications. AMS Proc. Symp. Pure. Math. 77 (2008) 291312.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

ESAIM: Control, Optimisation and Calculus of Variations
  • ISSN: 1292-8119
  • EISSN: 1262-3377
  • URL: /core/journals/esaim-control-optimisation-and-calculus-of-variations
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 1 *
Loading metrics...

Abstract views

Total abstract views: 33 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 19th September 2017. This data will be updated every 24 hours.