Prof. Dr. Filippo Santambrogio
Technische Universität München
Telefon: +49.89.289.18334John-von-Neumann-Professor Zentrum Mathematik, M8 Boltzmannstraße 3 85748 Garching bei München Raum: 02.08.035 santambrogiomth.univ-lyon1.fr |

The Steiner problem, the irrigation problem, and branched transport, together with their angle laws. In all these problems the unknown is a 1D network and the cost and the constraints involve its length and its suitability to reach a given target of points.

Some bases about positive measures and their convergence. Definition of k-dimensional Hausdorff measures. Densities and covering theorems.

The notion of Hausdorff convergence on the set of closed sets. Proof of the fact that the H^1 measure is l.s.c. for this convergence in the class of connected sets.

Application of the Golab's Theorem. Absence of loops. Some ideas about the blow-up of optimal sets around their points and about their regularity.

Xia's definition as an optimal flow problem under divergence constraints with a cost proportional to the mass to the power α. Finiteness of the distance if α>1-1/d.

The Beckmann's minimal flow problem for the Monge cost and the distance W_1. Quantitative comparison between branched transport distances and W_1. A relaxed Kantorovich formulation (Maddalena-Solimini) and a Benamou-Brenier one. Comparison with W_{1/α}.

Minimization among measures on curves (Maddalena-Morel-Solimini). Various definitions of multiplicities. The case of a single source.

Geophysical models for river basins and erosion. Optimal channel networks and landscape functions in a discrete setting. Definition in a continuous setting and comparison with the Kantorovich potential.

Lower semicontinuity. Holder continuity under lower bounds on the target measure. Maximal Slope in the direction of the network. Application to variational problems, shape optimization.

Basic definitions and properties of Γ-convergence. The example of Modica-Mortola as an approximation of the perimeter. Application to the Steiner problems to connect points on the boundary of a 2D convex polygon

Minimization of a convex-concave energy among smooth vector fields under divergence constraints: Γ-convergence

Weighted distance functions. Minimization of a Modica-Mortola term with weighted distance penalization. Convergence to a minimizer.

MA 5925 Geometric Measure Theory and Applications

MA 2501 Algorithmische Diskrete Mathematik

After successful completion of the module students are able to model classical transport or network problems as a rigorous variational problem and to study the mathematical questions that they bring (existence, regularity, approximation...) by handling the main relevant analysis tools.

The module is offered as lectures. In the lectures, the contents will be presented in a talk with demonstrative examples, as well as through discussion with the students. The lectures should animate the students to carry out their own analysis of the themes presented and to independently study the relevant literature. Discussion sessions can be organized if students are interested.

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