On the honeycomb conjecture for a class of minimal convex partitions

Bucur, Dorin; Fragalà, Ilaria; Velichkov, Bozhidar; Verzini, Gianmaria

doi:10.1090/tran/7326

On the honeycomb conjecture for a class of minimal convex partitions
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by Dorin Bucur, Ilaria Fragalà, Bozhidar Velichkov and Gianmaria Verzini PDF

Trans. Amer. Math. Soc. 370 (2018), 7149-7179 Request permission

Abstract:

We prove that the planar hexagonal honeycomb is asymptotically optimal for a large class of optimal partition problems, in which the cells are assumed to be convex, and the criterion is to minimize either the sum or the maximum among the energies of the cells, the cost being a shape functional which satisfies a few assumptions. They are: monotonicity under inclusions; homogeneity under dilations; a Faber-Krahn inequality for convex hexagons; a convexity-type inequality for the map which associates with every $n \in \mathbb {N}$ the minimizers of $F$ among convex $n$-gons with given area. In particular, our result allows us to obtain the honeycomb conjecture for the Cheeger constant and for the logarithmic capacity (still assuming the cells to be convex). Moreover, we show that, in order to get the conjecture also for the first Dirichlet eigenvalue of the Laplacian, it is sufficient to establish some facts about the behaviour of $\lambda _1$ among convex pentagons, hexagons, and heptagons with prescribed area.

References

Beniamin Bogosel and Bozhidar Velichkov, A multiphase shape optimization problem for eigenvalues: qualitative study and numerical results, SIAM J. Numer. Anal. 54 (2016), no. 1, 210–241. MR 3504604, DOI 10.1137/140976406
Virginie Bonnaillie-Noël, Bernard Helffer, and Gregory Vial, Numerical simulations for nodal domains and spectral minimal partitions, ESAIM Control Optim. Calc. Var. 16 (2010), no. 1, 221–246. MR 2598097, DOI 10.1051/cocv:2008074
Dorin Bucur, Giuseppe Buttazzo, and Antoine Henrot, Existence results for some optimal partition problems, Adv. Math. Sci. Appl. 8 (1998), no. 2, 571–579. MR 1657219
Dorin Bucur and Ilaria Fragalà, A Faber-Krahn inequality for the Cheeger constant of $N$-gons, J. Geom. Anal. 26 (2016), no. 1, 88–117. MR 3441504, DOI 10.1007/s12220-014-9539-5
L. A. Cafferelli and Fang Hua Lin, An optimal partition problem for eigenvalues, J. Sci. Comput. 31 (2007), no. 1-2, 5–18. MR 2304268, DOI 10.1007/s10915-006-9114-8
M. Caroccia, Cheeger $N$-clusters, Calc. Var. Partial Differential Equations 56 (2017), no. 2, Paper No. 30, 35. MR 3610172, DOI 10.1007/s00526-017-1109-9
M. Caroccia and R. Neumayer, A note on the stability of the Cheeger constant of $N$-gons, J. Convex Anal. 22 (2015), no. 4, 1207–1213. MR 3436708
Jeff Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Papers dedicated to Salomon Bochner, 1969) Princeton Univ. Press, Princeton, N. J., 1970, pp. 195–199. MR 0402831
Andrea Colesanti and Paola Cuoghi, The Brunn-Minkowski inequality for the $n$-dimensional logarithmic capacity of convex bodies, Potential Anal. 22 (2005), no. 3, 289–304. MR 2134723, DOI 10.1007/s11118-004-1326-7
M. Conti, S. Terracini, and G. Verzini, An optimal partition problem related to nonlinear eigenvalues, J. Funct. Anal. 198 (2003), no. 1, 160–196. MR 1962357, DOI 10.1016/S0022-1236(02)00105-2
Monica Conti, Susanna Terracini, and Gianmaria Verzini, On a class of optimal partition problems related to the Fučík spectrum and to the monotonicity formulae, Calc. Var. Partial Differential Equations 22 (2005), no. 1, 45–72. MR 2105968, DOI 10.1007/s00526-004-0266-9
Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
L. Fejes Tóth, Regular figures, A Pergamon Press Book, The Macmillan Company, New York, 1964. MR 0165423
T. C. Hales, The honeycomb conjecture, Discrete Comput. Geom. 25 (2001), no. 1, 1–22. MR 1797293, DOI 10.1007/s004540010071
Bernard Helffer, Domaines nodaux et partitions spectrales minimales (d’après B. Helffer, T. Hoffmann-Ostenhof et S. Terracini), Séminaire: Équations aux Dérivées Partielles. 2006–2007, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 2007, pp. Exp. No. VIII, 23 (French, with French summary). MR 2385195
B. Helffer, On spectral minimal partitions: a survey, Milan J. Math. 78 (2010), no. 2, 575–590. MR 2781853, DOI 10.1007/s00032-010-0129-0
B. Helffer, T. Hoffmann-Ostenhof, and S. Terracini, Nodal domains and spectral minimal partitions, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 1, 101–138. MR 2483815, DOI 10.1016/j.anihpc.2007.07.004
Antoine Henrot, Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006. MR 2251558, DOI 10.1007/3-7643-7706-2
R. Jones, Computing ultra-precise eigenvalues of the Laplacian within polygons, Arxiv Preprint, arXiv:1602.08636 (2016).
Bernd Kawohl and Thomas Lachand-Robert, Characterization of Cheeger sets for convex subsets of the plane, Pacific J. Math. 225 (2006), no. 1, 103–118. MR 2233727, DOI 10.2140/pjm.2006.225.103
N. S. Landkof, Foundations of modern potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy. MR 0350027, DOI 10.1007/978-3-642-65183-0
Gian Paolo Leonardi, An overview on the Cheeger problem, New trends in shape optimization, Internat. Ser. Numer. Math., vol. 166, Birkhäuser/Springer, Cham, 2015, pp. 117–139. MR 3467379, DOI 10.1007/978-3-319-17563-8_{6}
Frank Morgan, The hexagonal honeycomb conjecture, Trans. Amer. Math. Soc. 351 (1999), no. 5, 1753–1763. MR 1615934, DOI 10.1090/S0002-9947-99-02356-9
Frank Morgan and Roger Bolton, Hexagonal economic regions solve the location problem, Amer. Math. Monthly 109 (2002), no. 2, 165–172. MR 1903153, DOI 10.2307/2695328
Enea Parini, An introduction to the Cheeger problem, Surv. Math. Appl. 6 (2011), 9–21. MR 2832554
Miguel Ramos, Hugo Tavares, and Susanna Terracini, Extremality conditions and regularity of solutions to optimal partition problems involving Laplacian eigenvalues, Arch. Ration. Mech. Anal. 220 (2016), no. 1, 363–443. MR 3458166, DOI 10.1007/s00205-015-0934-2
Alexander Yu. Solynin and Victor A. Zalgaller, An isoperimetric inequality for logarithmic capacity of polygons, Ann. of Math. (2) 159 (2004), no. 1, 277–303. MR 2052355, DOI 10.4007/annals.2004.159.277

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On the honeycomb conjecture for a class of minimal convex partitions
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Abstract: