The landscape function in irrigation problems with an extended source

This article focuses on the landscape function, which is an essential and now standard tool to study variational problems involving branched transport, as it appears as a first variation of the irrigation functional. The goal is to extend its definition and fundamental properties to the case of an extended source. This is achieved in the case of optimal networks having a finite root system, that is when all curves pass through a finite set, which is true for instance when the source and target measures have disjoint supports. We give a satisfying definition of the landscape function in that case, such that the first variation property holds and we prove its Hölder regularity under reasonable assumptions on the measures we want to connect, namely when they have a density bounded from below and are supported on separated open sets which are lower Ahlfors-regular