Although a bubble inside a bubble as pictured below is unstable—the inner bubble pops out when it touches the outer bubble—students from the Williams SMALL undergraduate research project have found that a bubble inside a bubble can be stable—in fact, energy minimizing—if the space is weighted by a log-convex density, such as e^r^2.
They provided complete proofs in 1D and numerical evidence in higher dimensions. Their paper, “Double Bubbles on the Line with Log-Convex Density,” was accepted in May by Analysis and Geometry in Metric Spaces and has appeared online here. The authors, members of Frank Morgan’s 2016 and 2017 Geometry Groups, are Eliot Bongiovanni, Alejandro Diaz, Leonardo Di Giosia, Jay Habib, Arjun Kakkar, Lea Kenigsberg, Dustin Ping, Dylanger Pittman, Nat Sothanaphan, and Weitao Zhu.
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