The existence of an energy minimizing configuration for multiple solid objects floating in a bath of three liquids (Q2309451)

An \(\mathbb R^n\) domain filled by three immiscible fluids is considered; a finite number of rigid bodies can moving inside. The fluid-regions are Caccioppoli sets (of finite perimeter). Thus the corresponding perimeters can be estimated with respect to the total variation of the characteristic function of each set. An energy-functional \(F\) for this configuration is described, by using the theory of functions of bounded variation. \(F\) contains some terms depending on the surface tensions between the fluids and some terms terms due to gravity effects. The authors obtain a specific admissible configuration for which the energy-functional has a minimum value. To this end, a lower bound on \(F\) is obtained, a minimizing sequence is given and the lower semicontinuity of \(F\) with respect to this sequence is proved. A basic assumption is that the number of the contact points between the bodies is finite; a very interesting analysis given concerning this point. Important mathematical tools are some Poincaré-type inequalities for functions of bounded variation and specific strip domains near the contact points (given in Lemma 2.1). The obtained results are related with two important previous works: [\textit{J. Bemelmans} et al., J. Math. Fluid Mech. 14, No. 4, 751--770 (2012; Zbl 1288.76025); Ann. Mat. Pura Appl. (4) 193, No. 4, 1185--1200 (2014; Zbl 1299.76030)].