Cheeger sets and the minimum pressure gradient problem for viscoplastic fluids

A Cheeger set of a domain, Omega, is a subset of this domain such that the ratio of its perimeter to its area, h, is a minimum, if such a subset exists, with the possibility that the subset may be the original domain as well. This value h is called the Cheeger constant for a given domain. If one considers the reciprocal of this minimum or, the maximum ratio of the area of the subset to its perimeter, M = 1/h, it follows from the work of Mosolov and Miasnikov that the minimum pressure gradient, G, to sustain a steady flow in a pipe with a cross-section defined by Omega is given by G > tau(y)/M, where tau(y) is the constant yield stress of a viscoplastic fluid. Using the results of Kawohl and Lachand-Robert, we derive the Cheeger constant for a square in two different ways. The application of their results to a convex polygon including a triangle, which leads to a different method to find the relevant Cheeger constant, and rotationally symmetric cross-sections are also described. Finally, a new method to determine the Cheeger constant for an ellipse is given.