Variational and convex approximations of 1-dimensional optimal networks and hyperbolic obstacle problems

Cycle 31th Oral Defence of the Phd Thesis
11 aprile 2019
April 11, 2019

Venue: Seminar Room “-1” – Department of Mathematics – Via Sommarive, 14 Povo - Trento
Time: 1.00 p.m.

  • Mauro Bonafini - PhD in Mathematics

Abstract:
Connected 1-dimensional structures play a crucial role in very different areas like discrete geometry (graphs, networks, spanning, and Steiner trees), structural mechanics (crack formation and propagation), and inverse problems (defects identification, contour segmentation), etc. The modeling of these structures is a key problem both from the theoretical and the numerical points of view. Most of the difficulties encountered in studying such 1-dimensional objects are related to the fact that they are not canonically associated to standard mathematical quantities. In this seminar we focus on the Gilbert--Steiner problem as a prototypical example of problems involving such 1-dimensional optimal networks. On one side, we provide to it a variational approximation in the spirit of Gamma-convergence by means of Modica--Mortola type energies in the planar case and by means of Ginzburg--Landau type energies for any arbitrary dimension. On the other side, we introduce a convex relaxation of the problem in a fairly general context (arbitrary dimension and even manifold ambients) and we show through calibration type arguments that minimizers of the relaxed energy are generally (but not always) convex combinations of minimizers of the Gilbert--Steiner problem. We then present an extensive numerical investigation of the relaxed framework in the two and three dimensional case and in the surface scenario.

In the last part of the talk we switch our focus on the obstacle problem for (possibly non-local) wave equations: we show existence of suitably defined weak solutions through a convex minimization approach based on a time discrete approximation scheme and we exploit this approach from a numerical point of view to simulate the corresponding dynamic.

Supervisor: Giandomenico Orlandi

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