Shape optimization problems for metric graphs

We consider a shape optimization problem $$\min\big\{E(\Gamma):\ \Gamma\in\mathcal{A},\ \mathcal{H}^1(\Gamma)=l\ \big\},$$ where $\mathcal{A}$ is an admissible set of one dimensional objects (sets of finite Hausdorff measure in $\R^d$ or metric graphs) connecting some prescribed set of points $\mathcal{D}=\{D_1,\dots,D_k\}\subset\R^d$. The cost functional $E$ is the Dirichlet Energy of $\Gamma$ defined throughout the Sobolev functions on $\Gamma$ vanishing on the points $D_i$. We analyze the existence of a solution in both the family of rectifiable sets and that of metric graphs. Ar the end, several explicit examples are discussed.