Approximation of Length Minimization Problems Among Compact Connected Sets

Matthieu Bonnivard, Antoine Lemenant, Filippo Santambrogio
2015 SIAM Journal on Mathematical Analysis  
In this paper we provide an approximationà 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
more » ... lem relying on the p-compliance energy. The proof of convergence of the approximating functional, which is stated in terms of Γ-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.
doi:10.1137/14096061x fatcat:2ocovirsrjf4vce4nzunrniycu