Approximation of length minimization problems among compact connected sets
Abstract
In this paper we provide an approximation \`a la Ambrosio-Tortorelli of some classical minimization problems involving the length of an unknown one-dimensional set, with an additional connectedness constraint, in dimension two. We introduce a term of new type relying on a weighted geodesic distance that forces the minimizers to be connected at the limit. We apply this approach to approximate the so-called Steiner Problem, but also the average distance problem, and finally a problem relying on the p-compliance energy. The proof of convergence of the approximating functional, which is stated in terms of Gamma-convergence relies on technical tools from geometric measure theory, as for instance a uniform lower bound for a sort of average directional Minkowski content of a family of compact connected sets.
Cite
@article{arxiv.1403.3004,
title = {Approximation of length minimization problems among compact connected sets},
author = {Matthieu Bonnivard and Antoine Lemenant and Filippo Santambrogio},
journal= {arXiv preprint arXiv:1403.3004},
year = {2014}
}