Proof of the planar double bubble conjecture using metacalibration methods

Dorff, Rebecca; Lawlor, Gary; Sampson, Donald; Wilson, Brandon

doi:10.2140/involve.2009.2.611

Cited by 5 publications

(6 citation statements)

References 5 publications

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“…The first proof of this result for n = 2 was given in [10] exploiting the analysis carried out in [22]. A second proof appeared in [8]. The case n = 3 was established first in [13] for equal volumes and then in [14] with no restrictions.…”

mentioning

confidence: 98%

Symmetric double bubbles in the Grushin plane

Franceschi

Stefani

2019

ESAIM: COCV

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We address the double bubble problem for the anisotropic Grushin perimeter P α , α ≥ 0, and the Lebesgue measure in R 2 , in the case of two equal volumes. We assume that the contact interface between the bubbles lies on either the vertical or the horizontal axis. We first prove existence of minimizers via the direct method by symmetrization arguments and then characterize them in terms of the given area by first variation techniques. Even though no regularity theory is available in this setting, we prove that angles at which minimal boundaries intersect satisfy the standard 120-degree rule up to a suitable change of coordinates. While for α = 0 the Grushin perimeter reduces to the Euclidean one and both minimizers coincide with the symmetric double bubble found in [10], for α = 1 vertical interface minimizers have Grushin perimeter strictly greater than horizontal interface minimizers. As the latter ones are obtained by translating and dilating the Grushin isoperimetric set found in [19], we conjecture that they solve the double bubble problem with no assumptions on the contact interface.

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confidence: 98%

Symmetric double bubbles in the Grushin plane

Franceschi

Stefani

2019

ESAIM: COCV

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“…• On R n -Long believed to be true, but appearing explicitly as a conjecture in an undergraduate thesis by J. Foisy in 1991 [16], the Euclidean double-bubble problem in R n was considered in the 1990's by various authors. In the Euclidean plane R 2 , the conjecture was confirmed in [17] (see also [39,14,9]). In R 3 , the equal volumes case v 1 = v 2 was settled in [22,23], and the structure of general double-bubbles was further studied in [25].…”

Section: Previously Known and Related Resultsmentioning

confidence: 78%

“…Below we list some of the previously known results regarding the above three conjectures, and refer to the excellent book by F. Morgan [38,Chapters 13,14,18,19] for additional information.…”

Section: Previously Known and Related Resultsmentioning

confidence: 99%

The Structure of Isoperimetric Bubbles on $\mathbb{R}^n$ and $\mathbb{S}^n$

Milman¹,

Neeman²

2022

Preprint

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The multi-bubble isoperimetric conjecture in n-dimensional Euclidean and spherical spaces from the 1990's asserts that standard bubbles uniquely minimize total perimeter among all q − 1 bubbles enclosing prescribed volume, for any q ≤ n + 2. The double-bubble conjecture on R 3 was confirmed in 2000 by Hutchings-Morgan-Ritoré-Ros, and is nowadays fully resolved for all n ≥ 2. The double-bubble conjecture on S 2 and triple-bubble conjecture on R 2 have also been resolved, but all other cases are in general open. We confirm the conjecture on R n and on S n for all q ≤ min(5, n + 1), namely: the double-bubble conjectures for n ≥ 2, the triple-bubble conjectures for n ≥ 3 and the quadruple-bubble conjectures for n ≥ 4. In fact, we show that for all q ≤ n + 1, a minimizing cluster necessarily has spherical interfaces, and after stereographic projection to S n , its cells are obtained as the Voronoi cells of q affine-functions, or equivalently, as the intersection with S n of convex polyhedra in R n+1 . Moreover, the cells (including the unbounded one) are necessarily connected and intersect a common hyperplane of symmetry, resolving a conjecture of Heppes. We also show for all q ≤ n + 1 that a minimizer with non-empty interfaces between all pairs of cells is necessarily a standard bubble. The proof makes crucial use of considering R n and S n in tandem and of Möbius geometry and conformal Killing fields; it does not rely on establishing a PDI for the isoperimetric profile as in the Gaussian setting, which seems out of reach in the present one.

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“…It is well known that the double bubble is the unique such minimizer (see e.g. [8,18] for the 2D case, and [12] for the 3D case, and also [22,7,16,17]). In the ternary 3D case, however, such simplification is not available, and the shape of the minimizers is unclear, even for small masses.…”

Section: Setting Up the Problemmentioning

confidence: 99%

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

Lu¹,

Wei²

2022

Preprint

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We study a 3D ternary system derived as a sharp-interface limit of the Nakazawa-Ohta density functional theory of triblock copolymers, which combines an interface energy with a long range interaction term. Although both the binary case in 2D and 3D, and the ternary case in 2D, are quite well studied, 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 some quantity depending only on the total masses and the interaction coefficients. One key difficulty is that the 3D structure prevents us from uncoupling the Coulomb-like long range interaction from the perimeter term, hence the actual shape of minimizers is unknown, not even for small masses. This is due to the lack of a quantitative isoperimetric inequality with two mass constraints in 3D, and it makes the construction of competitors significantly more delicate.

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Proof of the planar double bubble conjecture using metacalibration methods

Cited by 5 publications

References 5 publications

Symmetric double bubbles in the Grushin plane

Symmetric double bubbles in the Grushin plane

The Structure of Isoperimetric Bubbles on $\mathbb{R}^n$ and $\mathbb{S}^n$

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

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