Approximation of Length Minimization Problems Among Compact Connected Sets

Abstract

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 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.

Strong approximation inh-mass of rectifiable currents under homological constraint

“…This strategy is implemented in, e.g. [18,16,14,3,4,5]. The asymptotic equivalence between the approximate variational problem and (1.1) follows from the (expected) Γ-convergence of the family {M ε h } towards M h as ε ↓ 0.…”

“…In fact the result stated in [8] assumes that T is compactly supported but the general case can be recovered easily. 3 Setting P = P 1 + P ′ 1 , U = U 1 + U ′ 1 and V = V 1 + V ′ 1 , we have T = P + U + ∂V with the estimates…”

Phase field approximations of branched transportation problems

“…For instance, in [24] the authors propose an approximation to the branched transport problem based on the Modica-Mortola functional in which the phase field is replaced by a vector-valued function satisfying a divergence constraint. Similarly, in [5] Santambrogio et al study a variational approximation to the Steiner minimal tree problem in which the connectedness constraint of the graph is enforced trough the introduction of a geodesic distance depending on the phase field. Our phase field approximations can be viewed as a generalization of recent work by two of the authors [11,12], in which essentially (1) for τ (m) = αm + β with α, β > 0 is approximated by an Ambrosio-Tortorelli-type functional defined as [σ, ϕ] = ∞ otherwise.…”

A Phase-field Approximation of the Perimeter under a Connectedness Constraint

“…To incorporate a connectedness constraint, we follow an idea developed by two of the authors for a problem of surfaces in a three-dimensional ambient space [8] based on a similar model for the two-dimensional Steiner problem and related questions [4]. Due to its novelty, we include a heuristic motivation here.…”