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

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     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://www.numdam.org/articles/10.1051/cocv/2013050/}
}

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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://www.numdam.org/articles/10.1051/cocv/2013050/

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