Large mass rigidity for a liquid drop model in $2$D with kernels of finite moments

Benoit Merlet; Marc Pegon

doi:10.5802/jep.178

Large mass rigidity for a liquid drop model in

2

D with kernels of finite moments
[Rigidité pour un modèle de goutte liquide en 2D avec des noyaux de moments finis et dans le régime des grandes masses]

Benoit Merlet ¹ ; Marc Pegon ¹

¹ Univ. Lille, CNRS, Inria, UMR 8524 - Laboratoire Paul Painlevé F-59000 Lille, France

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 63-100.

Résumé
Abstract

Motivé par l’étude du modèle de goutte liquide de Gamow dans le régime des grandes masses, nous considérons un problème isopérimétrique dans lequel le périmètre classique $P (E)$ est remplacé par $P (E) - γ P_{ε} (E)$ , où $0 < γ < 1$ et $P_{ε}$ est une énergie non locale telle que $P_{ε} (E) \to P (E)$ lorsque $ε$ tend vers zéro. Nous montrons que pour $ε$ assez petit les minimiseurs à aire fixée sont les disques. Pour cela, nous établissons d’abord qu’en dimension $2$ , les minimiseurs sont convexes dès que $ε$ est suffisamment petit. Ceci implique que le bord d’un minimiseur est une petite perturbation Lipschitz d’un cercle. Puis, par un argument à la Fuglede, nous prouvons (en dimension arbitraire $n \geq 2$ ) que si un minimiseur à volume fixé est une perturbation d’une boule au sens précédent, alors c’est une boule. Ce problème isopérimétrique est équivalent à une généralisation du modèle de goutte liquide pour le noyau atomique introduit par Gamow lorsque le potentiel répulsif non local est donné par un noyau suffisamment intégrable. Dans cette formulation, notre résultat principal indique que si le premier moment du noyau est inférieur à un seuil explicite, il existe une masse critique $m_{0}$ telle que les minimiseurs de masse prescrite $m > m_{0}$ sont les disques. Ceci contraste fortement avec le cas classique des noyaux de Riesz, où le problème n’admet pas de minimiseur au-delà d’une masse critique.

Motivated by Gamow’s liquid drop model in the large mass regime, we consider an isoperimetric problem in which the standard perimeter $P (E)$ is replaced by $P (E) - γ P_{ε} (E)$ , with $0 < γ < 1$ and $P_{ε}$ a nonlocal energy such that $P_{ε} (E) \to P (E)$ as $ε$ vanishes. We prove that unit area minimizers are disks for $ε > 0$ small enough. More precisely, we first show that in dimension $2$ , minimizers are necessarily convex, provided that $ε$ is small enough. In turn, this implies that minimizers have nearly circular boundaries, that is, their boundary is a small Lipschitz perturbation of the circle. Then, using a Fuglede-type argument, we prove that (in arbitrary dimension $n \geq 2$ ) the unit ball in $ℝ^{n}$ is the unique unit-volume minimizer of the problem among centered nearly spherical sets. As a consequence, up to translations, the unit disk is the unique minimizer. This isoperimetric problem is equivalent to a generalization of the liquid drop model for the atomic nucleus introduced by Gamow, where the nonlocal repulsive potential is given by a radial, sufficiently integrable kernel. In that formulation, our main result states that if the first moment of the kernel is smaller than an explicit threshold, there exists a critical mass $m_{0}$ such that for any $m > m_{0}$ , the disk is the unique minimizer of area $m$ up to translations. This is in sharp contrast with the usual case of Riesz kernels, where the problem does not admit minimizers above a critical mass.

Reçu le : 2021-06-05
Accepté le : 2021-11-01
Publié le : 2021-11-10

DOI : 10.5802/jep.178

Classification : 28A75, 49Q05, 49Q10, 49Q20, 52A10
Keywords: Geometric variational problems, nonlocal isoperimetric problems, nonlocal perimeters, regularity, liquid drop model
Mot clés : Problèmes variationnels géométriques, problèmes isopérimétriques non locaux, périmètres non locaux, régularité, modèle de goutte liquide

Affiliations des auteurs :

Benoit Merlet ¹ ; Marc Pegon ¹

¹ Univ. Lille, CNRS, Inria, UMR 8524 - Laboratoire Paul Painlevé F-59000 Lille, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Large mass rigidity for a liquid drop model in $2${D} with kernels of finite moments},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {63--100},
     publisher = {\'Ecole polytechnique},
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     year = {2022},
     doi = {10.5802/jep.178},
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Benoit Merlet; Marc Pegon. Large mass rigidity for a liquid drop model in $2$D with kernels of finite moments. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 63-100. doi : 10.5802/jep.178. https://jep.centre-mersenne.org/articles/10.5802/jep.178/

Bibliographie
Cité par

[1] S. Alama, L. Bronsard, R. Choksi & I. Topaloglu - “Ground-states for the liquid drop and TFDW models with long-range attraction”, J. Math. Phys. 58 (2017) no. 10, article ID 103503, 11 pages | DOI | MR | Zbl

[2] S. Alama, L. Bronsard, R. Choksi & I. Topaloglu - “Droplet breakup in the liquid drop model with background potential”, Commun. Contemp. Math. 21 (2019) no. 03, article ID 1850022, 23 pages | DOI | MR | Zbl

[3] S. Alama, L. Bronsard, I. Topaloglu & A. Zuniga - “A nonlocal isoperimetric problem with density perimeter”, Calc. Var. Partial Differential Equations 60 (2021) no. 1, article ID 1, 27 pages | DOI | MR | Zbl

[4] L. Ambrosio, V. Caselles, S. Masnou & J.-M. Morel - “Connected components of sets of finite perimeter and applications to image processing”, J. Eur. Math. Soc. (JEMS) 3 (2021) no. 1, p. 39-92 | DOI | MR | Zbl

[5] L. Ambrosio, N. Fusco & D. Pallara - Functions of bounded variation and free discontinuity problems, Oxford math. monographs, Clarendon Press, Oxford; New York, 2000 | Zbl

[6] J. Berendsen & V. Pagliari - “On the asymptotic behaviour of nonlocal perimeters”, ESAIM Control Optim. Calc. Var. 25 (2019), article ID 48, 27 pages | DOI | MR | Zbl

[7] M. Bonacini & R. Cristoferi - “Local and global minimality results for a nonlocal isoperimetric problem on $ℝ^{N}$ ”, SIAM J. Math. Anal. 46 (2014) no. 4, p. 2310-2349 | DOI | MR | Zbl

[8] J. Bourgain, H. Brezis & P. Mironescu - “Another look at Sobolev spaces”, in Optimal control and partial differential equations (J. L. Menaldi, E. Rofman & A. Sulem, eds.), IOS, Amsterdam, 2001, p. 439-455 | Zbl

[9] D. Carazzato, N. Fusco & A. Pratelli - “Minimality of balls in the small volume regime for a general Gamow type functional”, 2020 | arXiv

[10] A. Cesaroni & M. Novaga - “The isoperimetric problem for nonlocal perimeters”, Discrete Contin. Dynam. Syst. Ser. S 11 (2018) no. 3, p. 425-440 | DOI | MR | Zbl

[11] R. Choksi, C. B. Muratov & I. Topaloglu - “An old problem resurfaces nonlocally: Gamow’s liquid drops inspire today’s research and applications”, Notices Amer. Math. Soc. 64 (2017) no. 11, p. 1275-1283 | DOI | MR | Zbl

[12] J. Dávila - “On an open question about functions of bounded variation”, Calc. Var. Partial Differential Equations 15 (2002) no. 4, p. 519-527 | DOI | MR | Zbl

[13] L. C. Evans & R. F. Gariepy - Measure theory and fine properties of functions, Textbooks in math., CRC Press, Taylor & Francis Group, Boca Raton, 2015 | Zbl

[14] H. Federer - Geometric measure theory, Classics in Math., Springer, Berlin, Heidelberg, 1996 | DOI | Zbl

[15] A. Figalli, N. Fusco, F. Maggi, V. Millot & M. Morini - “Isoperimetry and stability properties of balls with respect to nonlocal energies”, Comm. Math. Phys. 336 (2015) no. 1, p. 441-507 | DOI | MR | Zbl

[16] R. L. Frank, R. Killip & P. T. Nam - “Nonexistence of large nuclei in the liquid drop model”, Lett. Math. Phys. 106 (2016) no. 8, p. 1033-1036 | DOI | MR | Zbl

[17] R. L. Frank & E. H. Lieb - “A compactness lemma and Its application to the existence of minimizers for the liquid drop model”, SIAM J. Math. Anal. 47 (2015) no. 6, p. 4436-4450 | DOI | MR | Zbl

[18] R. L. Frank & P. T. Nam - “Existence and nonexistence in the liquid drop model”, Calc. Var. Partial Differential Equations 60 (2021) no. 6, article ID 223, 12 pages | DOI | MR | Zbl

[19] B. Fuglede - “Stability in the isoperimetric problem for convex or nearly spherical domains in $ℝ^{n}$ ”, Trans. Amer. Math. Soc. 314 (1989) no. 2, p. 619-638 | MR | Zbl

[20] N. Fusco - “The quantitative isoperimetric inequality and related topics”, Bull. Sci. Math. 5 (2015-10) no. 3, p. 517-607 | DOI | MR | Zbl

[21] F. Générau & E. Oudet - “Large volume minimizers of a nonlocal isoperimetric problem: theoretical and numerical approaches”, SIAM J. Math. Anal. 50 (2018) no. 3, p. 3427-3450 | DOI | MR | Zbl

[22] L. Grafakos - Classical Fourier analysis, Graduate Texts in Math., vol. 249, Springer, New York, NY, 2014 | DOI | MR | Zbl

[23] V. Julin - “Isoperimetric problem with a Coulomb repulsive term”, Indiana Univ. Math. J. 63 (2014) no. 1, p. 77-89 | DOI | MR | Zbl

[24] H. Knüpfer & C. B. Muratov - “On an isoperimetric problem with a competing nonlocal term I: the planar case”, Comm. Pure Appl. Math. 66 (2013) no. 7, p. 1129-1162 | DOI | MR | Zbl

[25] H. Knüpfer & C. B. Muratov - “On an isoperimetric problem with a competing nonlocal term II: the general case”, Comm. Pure Appl. Math. 67 (2014) no. 12, p. 1974-1994 | DOI | MR | Zbl

[26] H. Knüpfer, C. B. Muratov & M. Novaga - “Low density phases in a uniformly charged liquid”, Comm. Math. Phys. 345 (2016) no. 1, p. 141-183 | DOI | MR | Zbl

[27] E. H. Lieb & M. Loss - Analysis, Graduate studies in math., vol. 14, American Mathematical Society, Providence, RI, 2001 | Zbl

[28] J. Lu & F. Otto - “Nonexistence of a minimizer for Thomas-Fermi-Dirac-von Weizsäcker model”, Comm. Pure Appl. Math. 67 (2014) no. 10, p. 1605-1617 | DOI | Zbl

[29] F. Maggi - Sets of finite perimeter and geometric variational problems. An introduction to geometric measure theory, Cambridge Studies in Advanced Math., Cambridge University Press, Cambridge, 2012 | DOI | Zbl

[30] A. Mellet & Y. Wu - “An isoperimetric problem with a competing nonlocal singular term”, Calc. Var. Partial Differential Equations 60 (2021) no. 3, article ID 106, 40 pages | DOI | MR | Zbl

[31] F. Morgan - Geometric measure theory: a beginner’s guide, Elsevier/AP, Amsterdam, 2016 | Zbl

[32] C. B. Muratov & T. M. Simon - “A nonlocal isoperimetric problem with dipolar repulsion”, Comm. Math. Phys. 372 (2019) no. 3, p. 1059-1115 | DOI | MR

[33] M. Novaga & A. Pratelli - “Minimisers of a general Riesz-type problem”, Nonlinear Anal. 209 (2021), article ID 112346, 27 pages | DOI | MR | Zbl

[34] R. Osserman - “A strong form of the isoperimetric inequality in $ℝ^{n}$ ”, Complex Variables Theory Appl. 9 (1987) no. 2-3, p. 241-249 | DOI | MR | Zbl

[35] M. Pegon - “Large mass minimizers for isoperimetric problems with integrable nonlocal potentials”, Nonlinear Anal. 211 (2021), article ID 112395, 48 pages | DOI | MR | Zbl

[36] S. Rigot - Ensembles quasi-minimaux avec contrainte de volume et rectifiabilité uniforme, Mém. Soc. Math. France (N.S.), vol. 82, Société Mathématique de France, Paris, 2000 | DOI | Numdam | Zbl

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