Planar m-Bubbles with m-1 Equal Highest Pressures

Banyat Sroysang, Wacharin Wichiramala

Abstract

The planar soap bubble problem asks for the least-perimeter way to enclose and separate open regions R1, R2, . . . , Rm of m given areas on the plane. In this work, we study properties for minimizing bubbles in case that the pressure of Rm is lower than the equal pressures of R1, R2, . . . and Rm−1. For m = 4, we show that a minimizing bubble with nonnegative pressures and without empty chambers has at most one internal component of the region R4.