Math 2.0

Calculus of variations and optimal control; optimization Geometric measure theory and optimization on manifolds Theorem

Double bubble theorem

Builds on Caccioppoli setHausdorff measureIsoperimetric inequality

Statement

Let V1,V2>0V_1, V_2 > 0. Consider pairs (E1,E2)(E_1, E_2) of disjoint Caccioppoli sets in ℝ3\R^3 with volumes ∣E1∣=V1\abs{E_1} = V_1 and ∣E2∣=V2\abs{E_2} = V_2. For such a pair, let A(E1,E2)=ℋ2(∂∗E1∪∂∗E2)A(E_1, E_2) = \mathcal{H}^2(\partial^* E_1 \cup \partial^* E_2) be the total area of the interfaces. Here ∂∗\partial^* is the reduced boundary and ℋ2\mathcal{H}^2 is the 22-dimensional Hausdorff measure, so that a common wall is counted once. Then AA is least for the standard double bubble with volumes V1V_1 and V2V_2. Every minimizing pair is a standard double bubble, up to a rigid motion and up to sets of measure zero.

The standard double bubble is made of three spherical caps, or of two caps and a flat disc when V1=V2V_1 = V_2. The three pieces have a common boundary circle and meet along it at angles of 120∘120^\circ.

Remarks

Michael Hutchings, Frank Morgan, Manuel Ritoré and Antonio Ros proved the theorem, published in 2002. Existence and regularity of a minimizer come from geometric measure theory. Hutchings showed that a minimizer is a surface of revolution about a line, with strong limits on the number of components of its regions. The authors then excluded the remaining non-standard candidates by an instability argument based on rotating parts of the surface. The case of equal volumes had been proved earlier by Joel Hass and Roger Schlafly with computer help. The analogous result in the plane, with circular arcs in place of spherical caps, had been proved earlier by a group of undergraduates. Ben Reichardt later extended the theorem to all dimensions. For one region instead of two, the corresponding statement is the isoperimetric inequality: the ball encloses a given volume with least area.

Going deeper

For a student

The theorem is the simplest case of the problem of least-area clusters, and it shows how hard such problems are. The single-bubble case is classical, but the two-bubble case needed new ideas in the 1990s. The 120∘120^\circ angles agree with Plateau's laws, which every minimizer must satisfy. The analogous problem for three or more bubbles in ℝ3\R^3 is much harder.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

Show the path as a graph
Borel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptCaccioppoli set, concept, no evidence levelCaccioppoli setconceptCompact set, concept, no evidence levelCompact setconceptEuclidean space, object, no evidence levelEuclidean spaceobjectHausdorff measure, concept, no evidence levelHausdorff measureconceptOuter measure, concept, no evidence levelOuter measureconceptParametrised surface, concept, no evidence levelParametrised surfaceconceptLebesgue measure, object, no evidence levelLebesgue measureobjectSurface area, concept, no evidence levelSurface areaconceptBrunn–Minkowski theorem, statement, no evidence levelBrunn–Minkowski theor…statementIsoperimetric inequality, statement, no evidence levelIsoperimetric inequal…statementDouble bubble theorem, statement, no evidence levelDouble bubble theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Borel sigma-algebra concept
  2. Caccioppoli set concept
  3. Compact set concept
  4. Euclidean space object
  5. Hausdorff measure concept
  6. Outer measure concept
  7. Parametrised surface concept
  8. Lebesgue measure object
  9. Surface area concept
  10. Brunn–Minkowski theorem statement
  11. Isoperimetric inequality statement
  12. Double bubble theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Double_bubble_theorem
About this page
Tag6QG8QNHBF4
URIm2://6QG8QNHBF4
Kindstatement, theorem
WritingW0 stub
MSC49Q10, 49Q20, 53A10
StatusThis copy of the site has no build data. Run m2 status 6QG8QNHBF4.
AboutCaccioppoli set
UsesCaccioppoli set, Hausdorff measure, Isoperimetric inequality
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