Approximation of Length Minimization Problems Among Compact Connected Sets

Source abstract

In this paper we provide an approximation à la Ambrosio--Tortorelli of some classical minimization problems involving the length of one-dimensional sets. The minimization is performed under an additional connectedness constraint, in dimension 2. 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 such as a uniform lower bound for a sort of average directional Minkowski content of a family of compact connected sets.