Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T06:53:23.077Z Has data issue: false hasContentIssue false
  • English
  • Français

Quantitative estimates for bending energies and applications to non-local variational problems

Part of: Existence theories Global differential geometry Magnetohydrodynamics and electrohydrodynamics

Published online by Cambridge University Press:  23 January 2019

Michael Goldman ,
Matteo Novaga  and
Matthias Röger
Michael Goldman
Affiliation:
Laboratoire Jacques-Louis Lions (CNRS, UMR 7598), Université Paris Diderot, F-75005Paris, France ([email protected])
Matteo Novaga
Affiliation:
Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127Pisa, Italy ([email protected])
Matthias Röger
Affiliation:
Technische Universität Dortmund, Fakultät für Mathematik, Vogelpothsweg 87, D-44227 Dortmund, Germany ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We discuss a variational model, given by a weighted sum of perimeter, bending and Riesz interaction energies, that could be considered as a toy model for charged elastic drops. The different contributions have competing preferences for strongly localized and maximally dispersed structures. We investigate the energy landscape in dependence of the size of the ‘charge’, that is, the weight of the Riesz interaction energy.

In the two-dimensional case, we first prove that for simply connected sets of small elastica energy, the elastica deficit controls the isoperimetric deficit. Building on this result, we show that for small charge the only minimizers of the full variational model are either balls or centred annuli. We complement these statements by a non-existence result for large charge. In three dimensions, we prove area and diameter bounds for configurations with small Willmore energy and show that balls are the unique minimizers of our variational model for sufficiently small charge.

Keywords

Geometric variational problemscompeting interactionsnon-local perimeter perturbationWillmore functionalbending energyglobal minimizers

MSC classification

Primary: 49J20: Optimal control problems involving partial differential equations
Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions 76W99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 1 , February 2020 , pp. 131 - 169
Copyright
Copyright © Royal Society of Edinburgh 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Acerbi, E., Fusco, N. and Morini, M.. Minimality via second variation for a nonlocal isoperimetric problem. Comm. Math. Phys. 322 (2013), 515–557.CrossRefGoogle Scholar
2Alberti, G., Choksi, R. and Otto, F.. Uniform energy distribution for an isoperimetric problem with long-range interactions. J. Amer. Math. Soc. 22 (2009), 569–605.CrossRefGoogle Scholar
3Bates, F. S. and Fredrickson, G. H.. Block copolymers-designer soft materials. Phys. Today 52 (1999), 32–38.CrossRefGoogle Scholar
4Bella, P., Goldman, M. and Zwicknagl, B.. Study of island formation in epitaxially strained films on unbounded domains. Arch. Ration. Mech. Anal. 218 (2015), 163–217.CrossRefGoogle Scholar
5Bonacini, M. and Cristoferi, R.. Local and global minimality results for a nonlocal isoperimetric problem on ℝn. SIAM J. Math. Anal. 46 (2014), 2310–2349.CrossRefGoogle Scholar
6Brasco, L., De Philippis, G. and Velichkov, B.. Faber-Krahn inequalities in sharp quantitative form. Duke Math. J. 164 (2015), 1777–1831.CrossRefGoogle Scholar
7Bucur, D. and Henrot, A.. A new isoperimetric inequality for the elasticae. J. Eur. Math. Soc. 19 (2017), 3355–3376.CrossRefGoogle Scholar
8Choksi, R. and Peletier, M. A.. Small volume fraction limit of the diblock copolymer problem. I: Sharp-interface functional. SIAM J. Math. Anal. 42 (2010), 1334–1370.CrossRefGoogle Scholar
9Choksi, R., Muratov, C. B. and Topaloglu, I. A.. An old problem resurfaces nonlocally: Gamow's liquid drops inspire today's research and applications. Notices Amer. Math. Soc. 64 64 (2017), 1275–1283.Google Scholar
10Cicalese, M. and Leonardi, G. P.. A selection principle for the sharp quantitative isoperimetric inequality. Arch. Ration. Mech. Anal. 206 (2012), 617–643.CrossRefGoogle Scholar
11Cicalese, M. and Spadaro, E.. Droplet minimizers of an isoperimetric problem with long-range interactions. Comm. Pure Appl. Math. 66 (2013), 1298–1333.CrossRefGoogle Scholar
12Cicalese, M., Leonardi, G. P. and Maggi, F.. Improved convergence theorems for bubble clusters I. The planar case. Indiana Univ. Math. J. 65 (2016), 1979–2050.Google Scholar
13De Lellis, C. and Müller, S.. Optimal rigidity estimates for nearly umbilical surfaces. J. Diff. Geom. 69 (2005), 75–110.CrossRefGoogle Scholar
14De Simone, A., Kohn, R. V., Müller, S. and Otto, F.. Recent analytical developments in micromagnetics. In The science of hysteresis. Physical modeling, micromagnetics, and magnetization dynamics (ed. Bertotti, G. and Mayergoyz, I. D.), pp. 269–381 (Amsterdam: Elsevier [u. a.], 2006).Google Scholar
15Ferone, V., Kawohl, B. and Nitsch, C.. The elastica problem under area constraint. Math. Annalen 365 (2016), 987–1015.CrossRefGoogle Scholar
16Figalli, A., Fusco, N., Maggi, F., Millot, V. and Morini, M.. Isoperimetry and stability properties of balls with respect to nonlocal energies. Comm. Math. Phys. 336 (2015), 441–507.CrossRefGoogle Scholar
17Frank, R. L., Killip, R. and Nam, P. T.. Nonexistence of large nuclei in the liquid drop model. Lett. Math. Phys. 106 (2016), 1033–1036.CrossRefGoogle Scholar
18Fuglede, B.. Stability in the isoperimetric problem for convex or nearly spherical domains in R n. Trans. Amer. Math. Soc. 314 (1989), 619–638.Google Scholar
19Fusco, N., Maggi, F. and Pratelli, A.. The sharp quantitative isoperimetric inequality. Ann. Math. (2) 168 (2008), 941–980.CrossRefGoogle Scholar
20Gamow, G.. Mass defect curve and nuclear constitution. Proc. R. Soc. London. Series A 126 (1930), 632–644.Google Scholar
21Goldman, M. and Ruffini, B.. Equilibrium shapes of charged liquid droplets and related problems: (mostly) a review. Geom. Flows 2 (2017), 94–104.Google Scholar
22Goldman, M. and Runa, E.. On the optimality of stripes in a variational model with non-local interactions. Preprint, 2016.Google Scholar
23Goldman, D., Muratov, C. B. and Serfaty, S.. The Γ-limit of the two-dimensional Ohta-Kawasaki energy. I. Droplet density. Arch. Ration. Mech. Anal. 210 (2013), 581–613.CrossRefGoogle Scholar
24Goldman, M., Novaga, M. and Ruffini, B.. Existence and stability for a non-local isoperimetric model of charged liquid drops. Arch. Ration. Mech. Anal. 217 (2015), 1–36.CrossRefGoogle Scholar
25Hutchinson, J. E.. Second fundamental form for varifolds and the existence of surfaces minimising curvature. Indiana Univ. Math. J. 35 (1986), 45–71.CrossRefGoogle Scholar
26Julin, V.. Isoperimetric problem with a Coulomb repulsive term. Indiana Univ. Math. J. 63 (2014), 77–89.CrossRefGoogle Scholar
27Julin, V.. Remark on a nonlocal isoperimetric problem. Nonlinear Anal. 154 (2017), 174–188.CrossRefGoogle Scholar
28Knüpfer, H. and Muratov, C. B.. On an isoperimetric problem with a competing nonlocal term I: The planar case. Comm. Pure Appl. Math. 66 (2013), 1129–1162.CrossRefGoogle Scholar
29Knüpfer, H. and Muratov, C. B.. On an isoperimetric problem with a competing nonlocal term II: The general case. Comm. Pure Appl. Math. 67 (2014), 1974–1994.Google Scholar
30Knüpfer, H., Kohn, R. V. and Otto, F.. Nucleation barriers for the cubic-to-tetragonal phase transformation. Comm. Pure Appl. Math. 66 (2013), 867–904.CrossRefGoogle Scholar
31Knüpfer, H., Muratov, C. B. and Novaga, M.. Low density phases in a uniformly charged liquid. Commun. Math. Phys. 345 (2016), 141–183.CrossRefGoogle Scholar
32Kohn, R. V. and Müller, S.. Surface energy and microstructure in coherent phase transitions. Comm. Pure Appl. Math. 47 (1994), 405–435.CrossRefGoogle Scholar
33Kuwert, E. and Schätzle, R.. Removability of point singularities of Willmore surfaces. Ann. Math. (2) 160 (2004), 315–357.CrossRefGoogle Scholar
34Kuwert, E. and Schätzle, R.. The Willmore functional. In Topics in modern regularity theory, volume 13 of CRM Series (ed. Mingione, G.), pp. 1–115 (Pisa: Ed. Norm., 2012).Google Scholar
35Lamm, T. and Schätzle, R. M.. Optimal rigidity estimates for nearly umbilical surfaces in arbitrary codimension. Geom. Funct. Anal. 24 (2014), 2029–2062.CrossRefGoogle Scholar
36Leoni, G.. Variational models for epitaxial growth. In Free discontinuity problems, volume 19 of CRM Series (ed. Fusco, N. and Pratelli, A.), pp. 69–152 (Pisa: Ed. Norm., 2016).CrossRefGoogle Scholar
37Lieb, E. H. and Loss, M.. Analysis. 2nd ed. Grad. Stud. Math. (Providence, RI: American Mathematical Society (AMS), 2001).Google Scholar
38Lu, J. and Otto, F.. Nonexistence of a minimizer for Thomas–Fermi–Dirac–von Weizsäcker model. Comm. Pure Appl. Math. 67 (2014), 1605–1617.CrossRefGoogle Scholar
39Marques, F. C. and Neves, A.. Min-max theory and the Willmore conjecture. Ann. Math. Second Series 179 (2014), 683–782.CrossRefGoogle Scholar
40Müller, S. and Röger, M.. Confined structures of least bending energy. J. Diff. Geom. 97 (2014), 109–139.CrossRefGoogle Scholar
41Muratov, C. B. and Novaga, M.. On well-posedness of variational models of charged drops. Proc. R. Soc. A 472 (2016), 20150808.CrossRefGoogle ScholarPubMed
42Muratov, C. B., Novaga, M. and Ruffini, B.. On equilibrium shapes of charged flat drops. Comm. Pure Appl. Math. 71 (2018), 1049–1073.CrossRefGoogle Scholar
43Ohta, T. and Kawasaki, K.. Equilibrium morphology of block copolymer melts. Macromolecules 19 (1986), 2621–2632.CrossRefGoogle Scholar
44Okamoto, M., Maruyama, T., Yabana, K. and Tatsumi, T.. Nuclear ‘pasta’ structures in low-density nuclear matter and properties of the neutron-star crust. Phys. Rev. C 88 (Aug 2013), 025801.CrossRefGoogle Scholar
45Rayleigh, L.. On the equilibrium of liquid conducting masses charged with electricity. Phil. Mag. 14 (1882), 184–186.CrossRefGoogle Scholar
46Ren, X. and Wei, J.. Spherical solutions to a nonlocal free boundary problem from diblock copolymer morphology. SIAM J. Math. Anal. 39 (2008), 1497–1535.CrossRefGoogle Scholar
47Röger, M. and Schätzle, R.. Control of the isoperimetric deficit by the Willmore deficit. Analysis 32 (2012), 1–7.CrossRefGoogle Scholar
48Schygulla, J.. Willmore minimizers with prescribed isoperimetric ratio. Arch. Ration. Mech. Anal. 203 (2012), 901–941.CrossRefGoogle Scholar
49Simon, L.. Lectures on geometric measure theory, volume 3 of Proceedings of the Centre for Mathematical Analysis (Canberra: Australian National University, 1983).Google Scholar
50Simon, L.. Existence of surfaces minimizing the Willmore functional. Comm. Anal. Geom. 1 (1993), 281–326.CrossRefGoogle Scholar
51Topping, P.. Mean curvature flow and geometric inequalities. J. Reine Angew. Math. 503 (1998), 47–61.CrossRefGoogle Scholar
52Willmore, T. J.. Note on embedded surfaces. Ann. Stiint. Univ. Al. I. Cuza, Iaşi, Sect. I a Mat. (N.S.) 11B (1965), 493–496.Google Scholar