Some Network Flows Optimizing Ramified Transport
Abstract: Suppose A and B are finite sums of atoms in R^n with the same total weights and T is an oriented finite mass network going from A to B. Thus, Boundary(T) = B-A. Various segments of T may have different multiplicities and the mass M(T) can be found by integrating the multiplicity function theta(T) over the network. Q. Xia (2003) used, for 0 < a < 1, a different mass M_a(T) obtained by integrating theta(T)^a over the network T. Here M_a minimization favors higher multiplicity segments (See the text [Bernot-Caselles-Morel]). C.Downes recently constructed an M_alpha decreasing network flow T_t in analogy to the (ordinary mass) M decreasing flows of rectifiable currents of X. Cheng (1993) or Almgren-Taylor-Wang (1993) . In research with C.Downes and J.Wu, we consider time-parameterized versions of such networks, which give some models for optimal transport “routings or schedules”. Some higher order functionals lead to networks with C^1 junctures, like train tracks. We will discuss briefly existence and regularity of minimizers and flows.
Biographical Information: Bob began his academic career at MIT where he was an undergraduate, after which he obtained his PhD with Federer at Brown University. Since then he has trained many students at University of Minnesota and Rice University where he has been since the 1980’s. In addition to his well known accomplishments in geometric measure theory, he is also well known for his anecdotes and the fact he has to duck his head when entering a room.