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Approximation of length minimization problems among compact connected sets

Abstract : 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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Contributor : Antoine Lemenant <>
Submitted on : Saturday, March 8, 2014 - 8:13:26 PM
Last modification on : Monday, January 18, 2021 - 9:18:02 PM
Long-term archiving on: : Sunday, June 8, 2014 - 10:42:24 AM

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  • HAL Id : hal-00957105, version 1
  • ARXIV : 1403.3004

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

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