Variational approximation of functionals defined on 1-dimensional\n connected sets: the planar case

2016/10/12 by Mauro Bonafini, Bonafini, Mauro, Giandomenico Orlandi +3
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Composite Material Mechanics #FOS: Mathematics #Optimization and Control (math.OC) #Topology Optimization in Engineering

paper · pdf · doi:10.48550/arxiv.1610.03839

openalex publication_date 2016/10/12 · openalex created_date 2022/08/21 · openalex updated_date 2026/07/28

Abstract

In this paper we consider variational problems involving 1-dimensional\nconnected sets in the Euclidean plane, such as the classical Steiner tree\nproblem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal\npartition problems and provide a variational approximation through\nModica-Mortola type energies proving a \Γ-convergence result. We also\nintroduce a suitable convex relaxation and develop the corresponding numerical\nimplementations. The proposed methods are quite general and the results we\nobtain can be extended to n-dimensional Euclidean space or to more general\nmanifold ambients, as shown in the companion paper [11].\n

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