Shape optimization problems for metric graphs

Giuseppe Buttazzo; Berardo Ruffini; Bozhidar Velichkov

ESAIM: Control, Optimisation and Calculus of Variations (2014)

  • Volume: 20, Issue: 1, page 1-22
  • ISSN: 1292-8119

Abstract

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Γ):Γ ∈ 𝒜, ℋ1(Γ) = l}, where ℋ1D1,...,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.

How to cite

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Buttazzo, Giuseppe, Ruffini, Berardo, and Velichkov, Bozhidar. "Shape optimization problems for metric graphs." ESAIM: Control, Optimisation and Calculus of Variations 20.1 (2014): 1-22. <http://eudml.org/doc/272872>.

@article{Buttazzo2014,
abstract = {Γ):Γ ∈ &#x1d49c;, ℋ1(Γ) = l\}, where ℋ1D1,...,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.},
author = {Buttazzo, Giuseppe, Ruffini, Berardo, Velichkov, Bozhidar},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {shape optimization; rectifiable sets; metric graphs; quantum graphs; Dirichlet energy},
language = {eng},
number = {1},
pages = {1-22},
publisher = {EDP-Sciences},
title = {Shape optimization problems for metric graphs},
url = {http://eudml.org/doc/272872},
volume = {20},
year = {2014},
}

TY - JOUR
AU - Buttazzo, Giuseppe
AU - Ruffini, Berardo
AU - Velichkov, Bozhidar
TI - Shape optimization problems for metric graphs
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2014
PB - EDP-Sciences
VL - 20
IS - 1
SP - 1
EP - 22
AB - Γ):Γ ∈ &#x1d49c;, ℋ1(Γ) = l}, where ℋ1D1,...,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.
LA - eng
KW - shape optimization; rectifiable sets; metric graphs; quantum graphs; Dirichlet energy
UR - http://eudml.org/doc/272872
ER -

References

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  1. [1] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Math. Monogr. Clarendon Press, Oxford (2000). Zbl0957.49001MR1857292
  2. [2] L. Ambrosio and P. Tilli, Topics on Analysis in Metric Spaces. Oxford Lect. Ser. Math. Appl. Oxford University Press, Oxford (2004) Zbl1080.28001MR2039660
  3. [3] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal.9 (1999) 428–517. Zbl0942.58018MR1708448
  4. [4] L. Friedlander, Extremal properties of eigenvalues for a metric graph. Ann. Inst. Fourier55 (2005) 199–211. Zbl1074.34078MR2141695
  5. [5] S. Gnutzmann and U. Smilansky, Quantum graphs: Applications to quantum chaos and universal spectral statistics. Adv. Phys.55 (2006) 527–625. 
  6. [6] P. Kuchment, Quantum graphs: an introduction and a brief survey, in Analysis on graphs and its applications. AMS Proc. Symp. Pure. Math. 77 (2008) 291–312. Zbl1210.05169MR2459876
  7. [7] F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems. Cambridge University Press, Cambridge (2012). Zbl1255.49074MR2976521

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