Stability of optimal traffic plans in the irrigation problem

<p style='text-indent:20px;'>We prove the stability of optimal traffic plans in branched transport. In particular, we show that any limit of optimal traffic plans is optimal as well. This result goes beyond the Eulerian stability proved in [<xref ref-type="bibr" rid="b7">7</xref>], extending it to the Lagrangian framework.</p>

[1]  Branched transport and fractal structures , 2017 .

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

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

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

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

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

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

[8]  Pertti Mattila,et al.  Geometry of sets and measures in Euclidean spaces , 1995 .

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

[10]  Ronald F. Gariepy,et al.  Measure Theory and Fine Properties of Functions, Revised Edition , 1865 .

[11]  L. Evans Measure theory and fine properties of functions , 1992 .

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

[13]  J. Morel,et al.  A variational model of irrigation patterns , 2003 .

[14]  Maria Colombo,et al.  On the Well‐Posedness of Branched Transportation , 2019, Communications on Pure and Applied Mathematics.

[15]  F. Santambrogio A Dacorogna-Moser approach to flow decomposition and minimal flow problems , 2013, 1310.2738.

[16]  L. Ambrosio,et al.  Functions of Bounded Variation and Free Discontinuity Problems , 2000 .

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

[18]  S. Solimini,et al.  Synchronic and Asynchronic Descriptions of Irrigation Problems , 2013 .

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