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Double Bubbles for Immiscible Fluids in ℝn

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Abstract

We use a new approach that we call unification to prove that standard weighted double bubbles in n-dimensional Euclidean space minimize immiscible fluid surface energy, that is, surface area weighted by constants. The result is new for weighted area, and also gives the simplest known proof to date of the (unit weight) double bubble theorem (Hass et al., Electron. Res. Announc. Am. Math. Soc., 1(3):98–102, 1995; Hutchings et al., Ann. Math., 155(2):459–489, 2002; Reichardt, J. Geom. Anal., 18(1):172–191, 2008).

As part of the proof, we introduce a striking new symmetry argument for showing that a minimizer must be a surface of revolution.

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Acknowledgements

The author gained valuable insight and momentum at the Workshop on isoperimetric problems, space-filling, and soap bubble geometry in Edinburgh, Scotland, in March of 2012, and would like to express gratitude to the hosting International Centre for Mathematical Sciences and to the workshop organizers.

The author would also like to thank the referee for many helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Brigham Young University, Provo, UT, USA

    Gary R. Lawlor

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  1. Gary R. Lawlor
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Correspondence to Gary R. Lawlor.

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Communicated by Steven G. Krantz.

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Lawlor, G.R. Double Bubbles for Immiscible Fluids in ℝn . J Geom Anal 24, 190–204 (2014). https://doi.org/10.1007/s12220-012-9333-1

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  • DOI: https://doi.org/10.1007/s12220-012-9333-1

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