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：

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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。