Shape Optimization Problems for Metric Graphs

Abstract

We consider the shape optimization problem $$\min \{\mathcal{E}(\Gamma ):\Gamma \in \mathcal{A},{\mathcal{H}}^1(\Gamma )=l\},$$ 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.

23 pages, 11 figures, ESAIM Control Optim. Calc. Var., (to appear)

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