The standard double soap bubble in R2 uniquely minimizes perimeter
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In this article, it was shown that the double bubble is the least-perimeter way to enclose and separate two regions of prescribed areas in the plane, and the solution for three regions remains conjectural.Abstract:
Of course the circle is the least-perimeter way to enclose a region of prescribed area in the plane. This paper proves that a certain standard «double bubble» is the least-perimeter way to enclose and separate two regions of prescribed areas. The solution for three regions remains conjecturalread more
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References
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The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces
TL;DR: In this paper, a complete classification of the local structure of singularities in a wide class of two-dimensional surfaces in R3 collected under the adjective (M, i, a) minimal by Almgren [A3] was provided.
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Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints
TL;DR: In this article, the authors studied the structure of m-dimensional subsets of R which are well behaved with respect to deformations of R and also showed the existence of such sets as solutions to geometric variational problems satisfying various constraints.