Shape optimization problems for metric graphs
ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 1, pp. 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: 10.1051/cocv/2013050
Classification: 49R05,  49Q20,  49J45,  81Q35
Keywords: shape optimization, rectifiable sets, metric graphs, quantum graphs, Dirichlet energy
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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/articles/10.1051/cocv/2013050/

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