Immiscible fluid clusters in \({\mathbb{R}}^2\) and \({\mathbb{R}}^3\) (Q1590922)

The author considers a system of immiscible fluids \(F_1\), \(F_2\), \(\dots\), \(F_{m}\) in \({\mathbb R}^{n}\) for \(n = 2,3.\) The ambient space is represented by \(F_0\). The energy of each interface is defined to be proportional to the surface area with constant of proportionality \(a_{ij}>0\). A non-degeneracy condition \(a_{ik}<a_{ij}+a_{jk}\) is imposed. By a nice argument which he summarizes in Section 1.1, the author shows that the energy-minimizing cluster in the planar case consists of arcs of circles meeting at isolated points. The final section of the paper considers the problem in \({\mathbb R}^3\) and uses a compactness argument to generalize Taylor's classification of soap-bubble cluster singularities [\textit{J. E. Taylor}, Ann. Math. (2) 103, 489-539 (1976; Zbl 0335.49032)] to clusters of immiscible fluids with interface energies near 1.