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.

### Shape optimization problems for metric graphs

#### Abstract

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.
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Buttazzo, Giuseppe; Ruffini, B.; Velichkov, B.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11568/227536