Minimal clusters of four planar regions with the same area
ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 3, pp. 1303-1331.

We prove that the optimal way to enclose and separate four planar regions with equal area using the less possible perimeter requires all regions to be connected. Moreover, the topology of such optimal clusters is uniquely determined.

DOI : 10.1051/cocv/2017066
Classification : 53A10, 49Q05, 52C99
Mots-clés : Minimal clusters, planar networks
Paolini, Emanuele 1 ; Tamagnini, Andrea 1

1 
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Paolini, Emanuele; Tamagnini, Andrea. Minimal clusters of four planar regions with the same area. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 3, pp. 1303-1331. doi : 10.1051/cocv/2017066. http://www.numdam.org/articles/10.1051/cocv/2017066/

[1] F.J. Almgren, Existence and regularity almost everywhere of solutions to elliptic variation problems viwh constraints. Mem. AMS 165 (1976) | MR | Zbl

[2] C. Bandle, Isoperimetric Inequalities and Applications. In Vol. 7 of Monographs and Studies in Mathematics. Pitman Publishing (1980) | MR | Zbl

[3] W. Blaschke, Kreis und Kugel. Leipzig (1916) | DOI | JFM

[4] M.N. Bleicher, Isoperimetric division into a finite number of cells in the plane, Stud. Sci. Math. Hungar 22 (1987) 123–137 | MR | Zbl

[5] M.N. Bleicher, Isoperimetric networks in the Euclidean plane. Studia Sci. Math. Hungar 31 (1996) 455–478 | MR | Zbl

[6] C. Cox, L. Harrison, M. Hutchings, S. Kim, J. Light, A. Mauer and M. Tilton, The shortest enclosure of three connected areas in ℝ2. Real Anal. Exchange 20 (1994/1995) 313–335 | DOI | MR | Zbl

[7] M. Engelstein, The least-perimeter partition of a sphere into four equal areas. Discrete Comput. Geom. 44 (2010) 645–653 | DOI | MR | Zbl

[8] J. Foisy, M. Alfaro, J. Brock, N. Hodges and J. Zimba, The standard soap bubble in ℝ2 uniquely minimizes perimeter. Pacific J. Math. 159 (1993) 47–59 | DOI | MR | Zbl

[9] T.C. Hales, The honeycomb conjecture. Discrete and Computational Geometry 25 (1999) 1–22 | DOI | MR | Zbl

[10] M. Hutchings, F. Morgan, M. Ritoré, A. Ros, Proof of the double bubble conjecture. Ann. Math. 155 (2002) 459–489 | DOI | MR | Zbl

[11] A. Kaewkhao and W. Wichiramala, Shortest enclosures for four regions of given areas. Thai J. Math. special issue 5 (2007) 127–146 | MR | Zbl

[12] F. Maggi, Sets of finite perimeter and geometric variational problems. In Vol. 135 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2012) | MR | Zbl

[13] A. Montesinos Amilibia Existence and uniqueness of standard bubble clusters of given volumes in ℝ2. Asian J. Math. 5 (2001) 25–32 | DOI | MR | Zbl

[14] F. Morgan, Soap bubbles in ℝ2 and in surfaces. Pacific J. Math. 165 (1994) 347–361 | DOI | MR | Zbl

[15] F. Morgan, The hexagonal honeycomb conjecture. Trans. AMS 351 (1999) 1753–1763 | DOI | MR | Zbl

[16] F. Morgan and J. M. Sullivan, Open problems in soap bubble geometry. Int. J. Math. 7 (1996) 833–842 | DOI | MR | Zbl

[17] F. Morgan and W. Wichiramala, The standard double bubble is the unique stable double bubble in ℝ2. Proc. Amer. Math. Soc. 130 (2002) 2745–2751 | DOI | MR | Zbl

[18] E. Paolini and A. Tamagnini, Sandwich cluster analysis. GitHub repository, https://github.com/paolini/sandwich (2017)

[19] J. Steiner, Gesammelte Mathematische Abhandlungen. Vol. II. Springer Verlag, Berlin (1890)

[20] A. Tamagnini, Planar Clusters. Ph.D. thesis, University of Florence, http://cvgmt.sns.it/paper/2967/ (2016)

[21] J.E. Taylor, The structure of singularities in soap-bubble-like and soapfilm-like minimal surfaces. Ann. Math. 103 (1976) 489–539 | DOI | MR | Zbl

[22] R. Vaughn, Planar soap bubbles. Ph.D. thesis, University of California, Davis (1998) | MR

[23] W. Wichiramala, The Planar triple bubble problem. Ph.D. thesis, University of Illinois, Urbana Champ (2002) | MR

[24] W. Wichiramala, Proof of the planar triple bubble conjecture. J. reine angew. Math. 567 (2004) 1–49 | DOI | MR | Zbl

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