Proof of Double Bubble Conjecture

Morgan, Frank

doi:10.1016/b978-0-12-804489-6.50014-8

Cited by 1 publication

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“…In 2D, due to the fact that the Green's function is a logarithmic term, the interaction was simply the product of the masses, hence it was equivalent to minimize the perimeter, subject to two mass constraints. It is well known that the double bubble is the unique such minimizer (see, e.g., [8,19] for the 2D case, and [12] for the 3D case, and also [7,17,18,23]). In the ternary 3D case, however, such simplification is not available, and the shape of the minimizers is unclear, even for small masses.…”

Section: Introductionmentioning

confidence: 99%

Uniform bound on the number of partitions for optimal configurations of the Ohta–Kawasaki energy in 3D

Wei²

2023

Can. Math. Bull.

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We study a 3D ternary system which combines an interface energy with a long-range interaction term. Several such systems were derived as a sharp-interface limit of the Nakazawa–Ohta density functional theory of triblock copolymers. Both the binary case in 2D and 3D, and the ternary case in 2D, are quite well understood, whereas very little is known about the ternary case in 3D. In particular, it is even unclear whether minimizers are made of finitely many components. In this paper, we provide a positive answer to this, by proving that the number of components in a minimizer is bounded from above by a computable quantity depending only on the total masses and the interaction coefficients. There are two key difficulties, namely, the impossibility to decouple the long-range interaction from the perimeter term, and the absence of a quantitative isoperimetric inequality with two mass constraints in 3D. Therefore, the actual shape of minimizers is unknown, even for small masses, making the construction of suitable competing configurations significantly more delicate.

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