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
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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