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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 [7], extending it to the Lagrangian framework.

We investigate the relation between energy minimizing maps valued into spheres having topological singularities at given points and optimal networks connecting them (e.g. Steiner trees, Gilbert-Steiner irrigation networks). We show the equivalence of the corresponding variational problems, interpreting in particular the branched optimal transport problem as a homological Plateau problem for rectifiable currents with values in a suitable normed group. This generalizes the pioneering work by Brezis, Coron and Lieb [10]., 19 pages, 2 figures

La presente investigación examina la situación hídrica del Departamento de Cerrillos (Provincia de Salta) entre 1857 y 1886, analizando los conflictos por el agua dados entre los regantes de los Departamentos de Cerrillos y Rosario de Lerma. Con el propósito de comprender la dinámica de las relaciones sociales y del poder que giraba en torno a ella, hemos avanzado en un mayor conocimiento sobre el problema de la irrigación, que se manifestó no sólo cómo una dificultad técnica a resolver sino, como un problema político relacionado con el accionar de la élite terrateniente local quien, frente a los procesos de la institucionalización del manejo del agua, pretendió conservar el dominio tradicional del agua con el argumento del derecho privado y apropiarse de la misma haciendo uso de los resortes del poder político. De allí, que el cúmulo de demandas, conflictos y juicios a favor de dar respuesta a la escasez del agua, a los intereses privados y a los procesos de institucionalización estatales (municipio de Cerrillos y de Rosario de Lerma), fueran llevados a cabo por los propietarios terratenientes. The present investigation examines the water situation of the Department of Cerrillos (Province of Salta) between 1857 and 1886, analyzing the conflicts over water given between the irrigators of the Departments of Cerrillos and Rosario de Lerma. In order to understand the dynamics of social relations and the power that revolved around it, we have advanced in a greater knowledge about the problem of irrigation, which manifested itself not only as a technical difficulty to be resolved but, as a political problem related to the actions of the local landowning elite who, faced with the processes of the institutionalization of water management, sought to preserve the traditional domain of water with the argument of private law and appropriate it by making use of the springs of political power. Hence, the accumulation of demands, conflicts and judgments in favor of responding to water shortages, private interests and state institutionalization processes (municipality of Cerrillos and Rosario de Lerma), were carried out by the landlord’s owners.

In this paper 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 $\Gamma$-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 [11]., Comment: 30 pages, 5 figures

The Gilbert-Steiner problem is a mass transportation problem, where the cost of the transportation depends on the network used to move the mass and it is proportional to a certain power of the "flow". In this paper, we introduce a new formulation of the problem, which turns it into the minimization of a convex functional in a class of currents with coefficients in a group. This framework allows us to define calibrations, which can be used to prove the optimality of concrete configurations. We apply this technique to prove the optimality of a certain irrigation network, having the topological property mentioned in the title.