On the Well‐Posedness of Branched Transportation

We show in full generality the stability of optimal traffic paths in branched transport: namely we prove that any limit of optimal traffic paths is optimal as well. This solves an open problem in the field (cf. Open problem 1 in the book Optimal transportation networks, by Bernot, Caselles and Morel), which has been addressed up to now only under restrictive assumptions.

[1]  Salvatore Stuvard,et al.  A multi-material transport problem with arbitrary marginals , 2018, Calculus of Variations and Partial Differential Equations.

[2]  Antoine Lemenant,et al.  Approximation of Length Minimization Problems Among Compact Connected Sets , 2014, SIAM J. Math. Anal..

[3]  Harold R. Parks,et al.  Geometric Integration Theory , 2008 .

[4]  F. Santambrogio,et al.  A fractal shape optimization problem in branched transport , 2017, Journal de Mathématiques Pures et Appliquées.

[5]  A. Brancolini,et al.  Equivalent formulations for the branched transport and urban planning problems , 2015, 1509.06698.

[6]  Qinglan Xia.  OPTIMAL PATHS RELATED TO TRANSPORT PROBLEMS , 2003 .

[7]  Benedikt Wirth,et al.  Optimal Micropatterns in 2D Transport Networks and Their Relation to Image Inpainting , 2016, 1601.07402.

[8]  R. Hardt,et al.  Rectifiable and flat G chains in a metric space , 2012 .

[9]  Leon Simon,et al.  Lectures on Geometric Measure Theory , 1984 .

[10]  B. White.  The deformation theorem for flat chains , 1999 .

[11]  Antonio De Rosa,et al.  Improved stability of optimal traffic paths , 2017, Calculus of Variations and Partial Differential Equations.

[12]  Robert V. Kohn,et al.  Bounds on the Micromagnetic Energy of a Uniaxial Ferromagnet , 1998 .

[13]  A. Brancolini,et al.  General transport problems with branched minimizers as functionals of 1-currents with prescribed boundary , 2017, 1705.00162.

[14]  H. Fédérer.  Geometric Measure Theory , 1969 .

[15]  S. Solimini,et al.  Fractal regularity results on optimal irrigation patterns , 2014 .

[16]  Andrea Marchese,et al.  A Multimaterial Transport Problem and its Convex Relaxation via Rectifiable G-currents , 2017, SIAM J. Math. Anal..

[17]  Qinglan Xia.  Interior regularity of optimal transport paths , 2004 .

[18]  A. Chambolle,et al.  A phase-field approximation of the Steiner problem in dimension two , 2019, Advances in Calculus of Variations.

[19]  F. Otto,et al.  A branched transport limit of the Ginzburg-Landau functional , 2017, 1704.02764.

[20]  R. Hardt,et al.  Connecting rational homotopy type singularities , 2008 .

[21]  M. F,et al.  A variational model of irrigation patterns , 2003 .

[22]  Giuseppe M. Buttazzo,et al.  A Benamou-Brenier Approach to Branched Transport , 2010, SIAM J. Math. Anal..

[23]  E. Gilbert.  Minimum cost communication networks , 1967 .

[24]  Qinglan Xia.  Boundary regularity of optimal transport paths , 2010 .

[25]  Edouard Oudet,et al.  Numerical Calibration of Steiner trees , 2017, Applied Mathematics & Optimization.

[26]  Robert V. Kohn,et al.  Domain Branching in Uniaxial Ferromagnets: A Scaling Law for the Minimum Energy , 1999 .

[27]  F. Santambrogio,et al.  A Modica-Mortola Approximation for Branched Transport and Applications , 2011 .

[28]  Andrea Marchese,et al.  The Steiner tree problem revisited through rectifiable G-currents , 2014, 1408.2696.

[29]  Eugene Stepanov,et al.  Optimal transportation networks as flat chains , 2006 .

[30]  Robert Hardt,et al.  Size minimization and approximating problems , 2003 .

[31]  F. Otto,et al.  Ground state energy scaling laws during the onset and destruction of the intermediate state in a type I superconductor , 2008 .

[32]  J. Morel,et al.  The Regularity of Optimal Irrigation Patterns , 2010 .

[33]  Filippo Santambrogio,et al.  Path Functionals over Wasserstein Spaces , 2006 .

[34]  Édouard Oudet,et al.  Variational Approximation of Functionals Defined on 1-dimensional Connected Sets: The Planar Case , 2016, SIAM J. Math. Anal..

[35]  Antonio De Rosa,et al.  Stability for the mailing problem , 2018, Journal de Mathématiques Pures et Appliquées.

[36]  Antonio De Rosa,et al.  On the lower semicontinuous envelope of functionals defined on polyhedral chains , 2017, 1703.01938.

[37]  Andrea Marchese,et al.  An optimal irrigation network with infinitely many branching points , 2014, 1408.2406.

[38]  A. Chambolle,et al.  Strong approximation in h-mass of rectifiable currents under homological constraint , 2018, Advances in Calculus of Variations.

[39]  Jean-Michel Morel,et al.  The structure of branched transportation networks , 2008 .

[40]  Benedikt Wirth,et al.  Approximation of rectifiable 1-currents and weak-⁎ relaxation of the h-mass , 2018, Journal of Mathematical Analysis and Applications.

[41]  E. Stepanov,et al.  Decomposition of acyclic normal currents in a metric space , 2012, 1303.5664.

[42]  J. Morel,et al.  Optimal Transportation Networks: Models and Theory , 2008 .

[43]  S. Solimini,et al.  Elementary properties of optimal irrigation patterns , 2006 .

[44]  Paul Pegon.  On the Lagrangian branched transport model and the equivalence with its Eulerian formulation , 2015, 1709.01414.

[45]  A. Bressan,et al.  On the optimal shape of tree roots and branches , 2018, Mathematical Models and Methods in Applied Sciences.

[46]  Robert V. Kohn,et al.  Surface energy and microstructure in coherent phase transitions , 1994 .