Shape optimization problems for metric graphs
ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 1, p. 1-22

We consider the shape optimization problem $min\left\{ℰ\left(\Gamma \right):\Gamma \in 𝒜,{ℋ}^{1}\left(\Gamma \right)=l\right\}$, 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 $D=\left\{{D}_{1},...,{D}_{k}\right\}\subset {ℝ}^{d}$. The cost functional $ℰ\left(\Gamma \right)$ is the Dirichlet energy of $\Gamma$ defined through the Sobolev functions on $\Gamma$ 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.

DOI : https://doi.org/10.1051/cocv/2013050
Classification:  49R05,  49Q20,  49J45,  81Q35
Keywords: 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},
publisher = {EDP-Sciences},
volume = {20},
number = {1},
year = {2014},
pages = {1-22},
doi = {10.1051/cocv/2013050},
zbl = {1286.49050},
mrnumber = {3182688},
language = {en},
url = {http://www.numdam.org/item/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, Volume 20 (2014) no. 1, pp. 1-22. doi : 10.1051/cocv/2013050. http://www.numdam.org/item/COCV_2014__20_1_1_0/

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