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

We consider the shape optimization problem min { ( Γ ) : Γ 𝒜 , 1 ( Γ ) = l } , where 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 } d . The cost functional ( 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.

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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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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