Interior and Boundary Regularity

Abstract

In this chapter we first define connected components of an optimal traffic plan X (Section 8.1), and the sub-traffic plans obtained by cutting X at a point x∈S _X (Section 8.2). These definitions and the fact that connected components of an optimal traffic plan are themselves optimal traffic plans will permit to perform some surgery leading to the main regularity theorems. The first “interior” regularity theorem (Section 8.3) states that outside the supports of μ⁺ and μ⁻ an optimal traffic plan for the irrigation problem is a locally finite graph. This result is not proved in full generality, but under the restriction that either μ⁺ or μ⁻ is finite atomic, or their supports are disjoint. What happens inside the support of the measures? Several properties can be established. Section 8.4 proves that at each branching point the number of branches is uniformly bounded by a constant depending only on α and the dimension N. The terms interior and boundary regularity for irrigation networks were proposed by Xia [96], [95]. Some techniques are borrowed from these papers. Yet, most of the statements and proofs follow [13].