# Shape optimization problems for graphs

##
Berardo Ruffini
(Institut Montpelliérain Alexander Grothendieck, Université de Montpellier)

created by magnani on 11 May 2012

modified by ruffini on 24 Jun 2012

23 may 2012 -- 17:00

Sala Seminari, Department of Mathematics, Pisa University

**Abstract.**

In this seminar we will discuss a shape optimization problem of the form
\[
\min\big\{J(\Gamma)\ :\ \Gamma\in\mathcal{A},\ \mathcal{H}^1(\Gamma)=l\ \big\},
\]
where $\mathcal{A}$ is a suitable set of graphs immersed in $\mathbb R^d$ with set of vertices $E(\Gamma)$ containing some prescribed set of points $\mathcal{D}=\{D_1,\dots,D_k\}$, and $J$ depends on $\Gamma$ via some differential operator on $\Gamma$ (for example, the Dirichlet Laplacian). We will analyze the existence of a solution in the class $\mathcal{A}$ and show some techniques useful to obtain explicit examples. Joint work with Giuseppe Buttazzo and Bozhidar Velichkov.