Phase field models for two-dimensional branched transportation problems
- 67 Downloads
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 Classification49J15 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.
- 6.Ferrari, L., Roßmanith, C., Wirth, B.: Phase field approximations of branched transportation problems (2018). https://arxiv.org/abs/1805.11399
- 9.Santambrogio, F.: Optimal Transport for Applied Mathematicians, Volume 87 of Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser, Boston (2015)Google Scholar