Max-flow min-cut theorem in an anisotropic network

Abstract

The aim is to give rigorous formulations of max-flow problems and min-cut problems an a bounded domain Ω in R n with Lipschitz boundary and to prove max-flow min-cut theorems. For example, one has to give a rigorous notion of a flow σ, the outer unit normal v, the inner product σ.v and the cut capacity of a cut. To do so, one basically follows Strang's idea. The space of essentially bounded vector fields with divergence in L n (Ω) and the space of functions of bounded variations will play important roles in this study. In fact, a static flow is represented by an element in the former space and a static cut is represented by a characteristic function which belongs to the latter space. One of the mathematical tools is a generalized Greens' formula, for functions of bounded variation and essentially bounded vector fields with divergence in L n (Ω). Furthermore one treats the general case when the network is anisotropic