Planar soap bubble problem with some conditions on pressures

Abstract

In this thesis, we solve the planar soap bubble problem with equal pressures for four, five and six regions, and find properties of solutions for the planar soap bubble problem in the case that all but one pressures are equal. We obtain the following results: (1)for four, five and six regions, every minimizing bubble with equal pressures and without empty chambers is standard; (2)in case of four regions, for a minimizing bubble without empty chambers and with three equal lowest pressures, if the regions of the highest pressures is connected, then the bubble is standard; (3)in case of four regions, for a minimizing bubble without empty chambers and with three equal highest pressures, the regions of the lowest pressures must have at least one external component