Transport distances and irrigation models

Some variational models have been recently introduced to the aim of modeling ramified structures, such as trees, rivers and so on. We introduce a general scheme in which the notion of transport distance is introduced starting from a general transport cost functional, through relaxation arguments. Then we apply this general framework to the irrigation cost, which is a particular cost functional depending on a parameter α ε]0,1[. We discuss the equivalence between this abstract approach and the above models.