Shape optimization problems for metric graphs

Buttazzo, Giuseppe; Ruffini, Berardo; Velichkov, Bozhidar

doi:10.1051/cocv/2013050

Buttazzo, Giuseppe ; Ruffini, Berardo ; Velichkov, Bozhidar

ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 1, pp. 1-22.

Résumé

We consider the shape optimization problem $min {ℰ (Γ) : Γ \in 𝒜, ℋ^{1} (Γ) = l}$ , where $ℋ^{1}$ is the one-dimensional Hausdorff measure and $𝒜$ is an admissible class of one-dimensional sets connecting some prescribed set of points $D = {D_{1}, ..., D_{k}} \subset ℝ^{d}$ . The cost functional $ℰ (Γ)$ is the Dirichlet energy of $Γ$ defined through the Sobolev functions on $Γ$ vanishing on the points $D_{i}$ . 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.

MR Zbl

DOI : 10.1051/cocv/2013050

Classification : 49R05, 49Q20, 49J45, 81Q35
Mots-clés : shape optimization, rectifiable sets, metric graphs, quantum graphs, Dirichlet energy

@article{COCV_2014__20_1_1_0,
     author = {Buttazzo, Giuseppe and Ruffini, Berardo and Velichkov, Bozhidar},
     title = {Shape optimization problems for metric graphs},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1--22},
     publisher = {EDP-Sciences},
     volume = {20},
     number = {1},
     year = {2014},
     doi = {10.1051/cocv/2013050},
     mrnumber = {3182688},
     zbl = {1286.49050},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2013050/}
}

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
SP  - 1
EP  - 22
VL  - 20
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2013050/
DO  - 10.1051/cocv/2013050
LA  - en
ID  - COCV_2014__20_1_1_0
ER  -

%0 Journal Article
%A Buttazzo, Giuseppe
%A Ruffini, Berardo
%A Velichkov, Bozhidar
%T Shape optimization problems for metric graphs
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2014
%P 1-22
%V 20
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2013050/
%R 10.1051/cocv/2013050
%G en
%F COCV_2014__20_1_1_0

Buttazzo, Giuseppe; Ruffini, Berardo; Velichkov, Bozhidar. Shape optimization problems for metric graphs. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 1, pp. 1-22. doi : 10.1051/cocv/2013050. http://archive.numdam.org/articles/10.1051/cocv/2013050/

Bibliographie
Cité par

[1] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Math. Monogr. Clarendon Press, Oxford (2000). | MR | Zbl

[2] L. Ambrosio and P. Tilli, Topics on Analysis in Metric Spaces. Oxford Lect. Ser. Math. Appl. Oxford University Press, Oxford (2004) | MR | Zbl

[3] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9 (1999) 428-517. | MR | Zbl

[4] L. Friedlander, Extremal properties of eigenvalues for a metric graph. Ann. Inst. Fourier 55 (2005) 199-211. | Numdam | MR | Zbl

[5] S. Gnutzmann and U. Smilansky, Quantum graphs: Applications to quantum chaos and universal spectral statistics. Adv. Phys. 55 (2006) 527-625.

[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. | MR | Zbl

[7] F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems. Cambridge University Press, Cambridge (2012). | MR | Zbl

Cité par Sources :