Eugene Stepanov

Optimal Transportation Problems with Free Dirichlet Regions

A Dirichlet region for an optimal mass transportation problem is, roughly speaking, a zone in which the transportation cost is vanishing. We study the optimal transportation problem with an unknown Dirichlet region Σ which varies in the class of closed connected subsets having prescribed 1-dimensional Hausdorff measure. We show the existence of an optimal Σ opt and study some of its geometrical properties. We also present numerical computations which show the shape of Σ opt in some model examples.

Optimal urban networks via mass transportation

Problem setting.- Optimal connected networks.- Relaxed problem and existence of solutions.- Topological properties of optimal sets.- Optimal sets and geodesics in the two-dimensional case.

Decomposition of acyclic normal currents in a metric space

Abstract We prove that every acyclic normal one-dimensional real Ambrosio–Kirchheim current in a Polish (i.e. complete separable metric) space can be decomposed in curves, thus generalizing the analogous classical result proven by S. Smirnov in Euclidean space setting. The same assertion is true for every complete metric space under a suitable set-theoretic assumption.

Optimal transportation networks as flat chains

We provide a model of optimization of transportation networks (e.g. urban traffic lines, subway or railway networks) in a geographical area (e.g. a city) with given density of population and that of services and/or workplaces, the latter being the destinations of everyday movements of the former. The model is formulated in terms of the Federer‐Fleming theory of currents, and allows us to get both the position and the necessary capacity of the optimal network. Existence and some qualitative properties of solutions to the relevant optimization problem are studied. Also, in an important particular case it is shown that the model proposed is equivalent to another known model of optimization of a transportation network, the latter not using the language of currents.

On Regularity of Transport Density in the Monge--Kantorovich Problem

We show that the optimal regularity result for the transport density in the classical Monge-Kantorovich optimal mass transport problem, with the measures having summable densities, is a Sobolev differentiability along transport rays.

An example of an infinite Steiner tree connecting an uncountable set

Abstract We construct an example of a Steiner tree with an infinite number of branching points connecting an uncountable set of points. Such a tree is proven to be the unique solution to a Steiner problem for the given set of points. As a byproduct we get the whole family of explicitly defined finite Steiner trees, which are unique connected solutions of the Steiner problem for some given finite sets of points, and with growing complexity (i.e. the number of branching points).

Variational methods for a class of nonlocal functionals

Abstract The paper gives on overview of recent developments and presents some new results for variational problems involving functionals with nonlocal transformation of argument, in particular, argument deviation.

Weak convergence of inner superposition operators

The equivalence of the weak (pointwise) and strong convergence of a sequence of inner superposition operators is proved as well as the criteria for such convergence are provided. Besides, the problems of continuous weak convergence of such operators and of representation of a limit operator are studied.

Self-contracted curves have finite length

A curve

Regularity for the optimal compliance problem with length penalization

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