Approximation of Length Minimization Problems Among Compact Connected Sets
Approximation of Length Minimization Problems Among Compact Connected Sets
复制标题
紧连通集长度最小化问题的逼近
DOI：
10.1137/14096061x
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
F. Santambrogio
中科院分区：
文献类型：
--
作者：
M. Bonnivard;A. Lemenant;F. Santambrogio
本文利用Ambrosio-Tortorelli方法，给出了一些经典最小化问题的近似解，这些最小化问题涉及未知的一维集合的长度，并带有附加的连通性约束。我们引入了一种新的依赖于加权测地线距离的术语，它迫使最小值在极限处连接。我们将这种方法应用于近似所谓的斯坦纳问题，以及平均距离问题，最后是一个依赖于p-柔度能量的问题。近似泛函的收敛性的证明，是用伽玛收敛来表述的，它依赖于几何测度理论中的技术工具，例如紧连集族的一类平均方向Minkowski内容的一致下界。
In this paper we provide an approximation \`a 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.