Calibration of Euclidean Steiner Trees by Currents

in collaboration with A. Massaccesi and B. Velichkov


In this work 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.

You can find illustrations of counter examples of calibration identification on the following link

Sub optimal trees

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Representation of an optimal calibration associated to three equilateral points

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Representation of an optimal calibration associated to the four vertices of a square

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Representation of an optimal calibration associated to the five vertices of a Pentagon

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Representation of an optimal calibration associated to the six vertices of an Hexagon

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Representation of an optimal calibration associated to a configuration made of six points

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Representation of an optimal calibration associated to a regular polygon of 7 vertices