Phase field models for two-dimensional branched transportation problems

  • Benedikt WirthEmail author


We analyse the following inverse problem. Given a nonconvex functional (from a specific, but quite general class) of normal, codimension-1 currents (which in two spatial dimensions can be interpreted as transportation networks), find the potential of a phase field energy which approximates the given functional. We prove existence of a solution as well as its characterization via a linear deconvolution problem. We also provide an explicit formula that allows to approximate the solution arbitrarily well in the supremum norm.

Mathematics Subject Classification

49J15 49K15 49N45 



The author thanks Filippo Santambrogio for various discussions on the topic. This work has been supported by by the Alfried Krupp Prize for Young University Teachers awarded by the Alfried Krupp von Bohlen und Halbach-Stiftung. The work was also supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy-EXC 2044, Mathematics Münster: Dynamics-Geometry-Structure.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Applied Mathematis Münster, University of MünsterMünsterGermany

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