Calculus of Variations and Geometric Measure Theory

G. Buttazzo - E. Stepanov

Minimization problems for average distance functionals

created on 24 Mar 2004
modified by stepanov on 15 Oct 2006


Published Paper

Inserted: 24 mar 2004
Last Updated: 15 oct 2006

Journal: Calculus of Variations: Topics from the Mathematical Heritage of Ennio De Giorgi, D. Pallara (ed.), Quaderni di Matematica, Seconda Universit\`{a} di Napoli, Caserta
Volume: 14
Pages: 47-83
Year: 2004


The paper is concerned with problems of finding minima among the set of all compact connected subsets $\Sigma\subset \Omega$ ($\Omega\subset *R*^n$ being a given compact ambient set) of the functional $$ I(\Sigma):= \int\Omega A(dist(x,\Sigma))\, d\varphi(x) $$ subject to some penalization on the length ${\cal H}(\Sigma)$ of $\Sigma$, where $A$: $*R*^+\to *R*^+$ is a given nondecreasing function and $dist(x,\Sigma)$ stands for the distance between $x$ and $\Sigma$ defined with the use of the geodesic distance $d$ relative to $\Omega$ in the usual way, i.e. $$ dist(x, \Sigma) := \inf{z\in\Sigma} d(x,z). $$ We provide various applications which are sources of similar problems, and give some results concerning qualitative properties of minimizers in the case $\Omega$ is a convex set (with $d$ a Euclidean distance).

Keywords: Monge-Kantorovich problem, average distance functional, transportation network