The role of Cheeger sets in the steady flows of viscoplastic fluids in pipes: A survey.

Given a convex or a Jordan domain Ω , let Ω ′ be a subset of this domain, with P (Ω ′) denoting its perimeter and A (Ω ′) its area. If a subset Ωc exists such that h = P (Ω c) / A (Ω c) is a minimum, the subset Ωc is called the Cheeger set of Ω and h, the Cheeger constant of the given domain. If one considers the reciprocal of this minimum or the maximum ratio of the area of the subset to its perimeter, t ∗ = 1 / h. It follows from the work of Mosolov and Miasnikov that the minimum pressure gradient G to sustain the steady flow of a viscoplastic fluid in a pipe, with a cross section defined by Ω , is given by G > τ y / t ∗ , where τy is the constant yield stress of the fluid. In this survey, we summarize several results to determine the constant h when the given domain is self-Cheeger or a Cheeger-regular set that touches each boundary of a convex polygon and when the Cheeger-irregular set does not do so. The determination of the constant h for an arbitrary ellipse, a strip, and a region with no necks is also mentioned. [ABSTRACT FROM AUTHOR]