Variational Approximation of Functionals Defined on 1-dimensional Connected Sets: The Planar Case

Abstract : In this paper we consider variational problems involving 1-dimensional connected sets in the Euclidean plane, such as the classical Steiner tree problem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal partition problems and provide a variational approximation through Modica-Mortola type energies proving a full Γ-convergence result. We also introduce a suitable convex relaxation and develop the corresponding numerical implementations. The proposed methods are quite general and the results we obtain can be extended to n-dimensional Euclidean space or to more general manifold ambients, as shown in the companion paper [11].
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Mauro Bonafini, Giandomenico Orlandi, Edouard Oudet. Variational Approximation of Functionals Defined on 1-dimensional Connected Sets: The Planar Case. SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2018, 50 (6), pp.6307-6332. ⟨10.1137/17M1159452⟩. ⟨hal-02009439⟩

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