Calculus of Variations and Geometric Measure Theory

G. Buttazzo - B. Ruffini - B. Velichkov

Shape Optimization Problems for Metric Graphs

created by ruffini on 21 Sep 2012
modified on 05 Jun 2017


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


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.