*Published Paper*

**Inserted:** 21 sep 2012

**Last Updated:** 5 jun 2017

**Journal:** ESAIM Control Optim. Calc. Var.

**Volume:** 20

**Number:** 01

**Pages:** 1--22

**Year:** 2014

**Abstract:**

We consider the shape optimization problem \[\min\big\{\mathcal E(\Gamma)\ :\ \Gamma\in\mathcal A,\ \mathcal H^1(\Gamma)=l\ \big\},\] where $\mathcal H^1$ is the one-dimensional Hausdorff measure and $\mathcal A$ is an admissible class of one-dimensional sets connecting some prescribed set of points $\mathcal D=\{D_1,\dots,D_k\}\subset\mathbb R^d$. The cost functional $\mathcal E(\Gamma)$ 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.

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