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Variational approximation of functionals defined on 1-dimensional connected sets in ℝ n
Advances in Calculus of Variations ( IF 1.3 ) Pub Date : 2021-10-01 , DOI: 10.1515/acv-2019-0031
Mauro Bonafini 1 , Giandomenico Orlandi 2 , Édouard Oudet 3
Affiliation  

In this paper we consider the Euclidean Steiner tree problem and, more generally, (single sink) Gilbert–Steiner problems as prototypical examples of variational problems involving 1-dimensional connected sets in ℝ n {\mathbb{R}^{n}} . Following the analysis for the planar case presented in [M. Bonafini, G. Orlandi and E. Oudet, Variational approximation of functionals defined on 1-dimensional connected sets: The planar case, SIAM J. Math. Anal. 50 2018, 6, 6307–6332], we provide a variational approximation through Ginzburg–Landau type energies proving a Γ-convergence result for n ≥ 3 {n\geq 3} .

中文翻译:

在 ℝ n 中的一维连通集上定义的泛函的变分逼近

在本文中,我们将欧几里得斯坦纳树问题和更一般的(单汇)吉尔伯特-斯坦纳问题视为涉及 ℝ n {\mathbb{R}^{n}} 中的一维连通集的变分问题的原型示例。根据[M. Bonafini、G. Orlandi 和 E. Oudet,在一维连接集上定义的泛函变分逼近:平面案例,SIAM J. Math。肛门。50 2018, 6, 6307–6332],我们通过 Ginzburg-Landau 型能量提供了变分近似,证明了 n ≥ 3 {n\geq 3} 的 Γ 收敛结果。
更新日期:2021-10-01
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