Connecting Measures By Means Of Branched Transportation Networks At Finite Cost

摘要

We study the couples of finite Borel measures φ_0 and φ_1 with compact support in R~n which can be transported to each other at a finite W~α cost, where W~α(φ_0,φ_1):=inf{M~α(T): partial deriv T = φ_0-φ_1}, α ∈[0,1], the infimum is taken over real normal currents of finite mass and M~α(T) denotes the α-mass of T. Besides the class of α-irrigable measures (i.e., measures which can be transported to a Dirac measure with the appropriate total mass at a finite W~α cost), two other important classes of measures are studied, which are called in the paper purely α-nonirrigable and marginally α-nonirrigable and are in a certain sense complementary to each other. For instance, purely α-nonirrigable and Ahlfors-regular measures are, roughly speaking, those having sufficiently high dimension. One shows that for φ_0 to be transported to φ_1 at finite W~α cost their naturally defined purely α-nonirrigable parts have to coincide.