Geodesic
On a phase field approximation of the planar Steiner problem: Existence, regularity, and asymptotic of minimizers
Matthieu Bonnivard1 orcid ; Antoine Lemenant2 orcid ; Vincent Millot3
1Université Denis Diderot – Paris 7, France
2Université Paris Diderot – Paris 7, France
3Université Denis Diderot - Paris 7, France
Interfaces and free boundaries, Tome 20 (2018) no. 1, pp. 69-106

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DOI

In this article, we consider and analyse a variant of a functional originally introduced in [9, 27] to approximate the (geometric) planar Steiner problem. This functional depends on a small parameter ε>0 and resembles the (scalar) Ginzburg–Landau functional from phase transitions. In a first part, we prove existence and regularity of minimizers for this functional. Then we provide a detailed analysis of their behavior as ε→0, showing in particular that sublevel sets Hausdorff converge to optimal Steiner sets. Applications to the average distance problem and optimal compliance are also discussed.
DOI : 10.4171/ifb/397
Classification : 49-XX
Mots-clés : Steiner problem, gamma-convergence, Ginzburg–Landau, Modica–Mortola, phase field approximation

Matthieu Bonnivard  1   ; Antoine Lemenant  2   ; Vincent Millot  3

1 Université Denis Diderot – Paris 7, France
2 Université Paris Diderot – Paris 7, France
3 Université Denis Diderot - Paris 7, France 
Matthieu Bonnivard; Antoine Lemenant; Vincent Millot. On a phase field approximation of the planar Steiner problem: Existence, regularity, and asymptotic of minimizers. Interfaces and free boundaries, Tome 20 (2018) no. 1, pp. 69-106. doi: 10.4171/ifb/397
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     author = {Matthieu Bonnivard and Antoine Lemenant and Vincent Millot},
     title = {On a phase field approximation of the planar {Steiner} problem: {Existence,} regularity, and asymptotic of minimizers},
     journal = {Interfaces and free boundaries},
     pages = {69--106},
     year = {2018},
     volume = {20},
     number = {1},
     doi = {10.4171/ifb/397},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/397/}
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