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Article Dans Une Revue SIAM Journal on Mathematical Analysis Année : 2015

Approximation of length minimization problems among compact connected sets

Matthieu Bonnivard
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Antoine Lemenant
Filippo Santambrogio
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Résumé

In this paper we provide an approximation à 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.
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Dates et versions

hal-00957105 , version 1 (08-03-2014)

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Matthieu Bonnivard, Antoine Lemenant, Filippo Santambrogio. Approximation of length minimization problems among compact connected sets. SIAM Journal on Mathematical Analysis, 2015. ⟨hal-00957105⟩
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