Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Uniform energy distribution for an isoperimetric problem with long-range interactions

Authors: Giovanni Alberti, Rustum Choksi and Felix Otto
Journal: J. Amer. Math. Soc. 22 (2009), 569-605
MSC (2000): Primary 49Q10, 49N60, 49S05, 35B10
Published electronically: November 6, 2008
MathSciNet review: 2476783
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study minimizers of a nonlocal variational problem. The problem is a mathematical paradigm for the ubiquitous phenomenon of energy-driven pattern formation induced by competing short- and long-range interactions. The short-range interaction is attractive and comes from an interfacial energy, and the long-range interaction is repulsive and comes from a nonlocal energy contribution. In particular, the problem is the sharp interface version of a problem used to model microphase separation of diblock copolymers. A natural conjecture is that in all space dimensions, minimizers are essentially periodic on an intrinsic scale. However, proving any periodicity result turns out to be a formidable task in dimensions larger than one.

In this paper, we address a weaker statement concerning the distribution of energy for minimizers. We prove in any space dimension that each component of the energy (interfacial and nonlocal) of any minimizer is uniformly distributed on cubes which are sufficiently large with respect to the intrinsic length scale. Moreover, we also prove an $ L^\infty$ bound on the optimal potential associated with the long-range interactions. This bound allows for an interesting interpretation: Note that the average volume fraction of the optimal pattern in a subsystem of size $ R$ fluctuates around the system average $ m$. The bound on the potential yields a rate of decay of these fluctuations as $ R$ tends to $ +\infty$. This rate of decay is stronger than the one for a random checkerboard pattern. In this sense, the optimal pattern has less large-scale variations of the average volume fraction than a pattern with a finite correlation length.

References [Enhancements On Off] (What's this?)

  • 1. Alberti, G.; Müller, S.: A new approach to variational problems with multiple scales. Comm. Pure Appl. Math. 54 (2001), no. 7, 761-825. MR 1823420 (2002f:49016)
  • 2. Bates, F.S.; Fredrickson, G.H.: Block copolymers - Designer soft materials. Physics Today 52 (1999), no. 2, 32-38.
  • 3. Brézis, H.: Opèrateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Mathematics Studies 5. North-Holland Publishing Co., Amsterdam, 1973. MR 0348562 (50:1060)
  • 4. Chen, X.; Oshita, Y.: Periodicity and uniqueness of global minimizers of an energy functional containing a long-range interaction. SIAM J. Math. Anal. 37 (2005), no. 4, 1299-1332. MR 2192296 (2006k:49027)
  • 5. Chen, X.; Oshita, Y.: An application of the modular function in nonlocal variational problems. Arch. Ration. Mech. Anal. 186 (2007), no. 1, 109-132. MR 2338353
  • 6. Choksi, R.: Scaling laws in microphase separation of diblock copolymers. J. Nonlinear Sci. 11 (2001), no. 3, 223-236. MR 1852942 (2003h:82091)
  • 7. Choksi, R.; Kohn, R.V.; Otto, F.: Domain branching in uniaxial ferromagnets: a scaling law for the minimum energy. Comm. Math. Phys. 201 (1999), no. 1, 61-79. MR 1669433 (2000c:49060)
  • 8. Choksi, R.; Peletier, M.A.; Williams, J.F.: On the phase diagram for microphase separation of diblock copolymers: an approach via a nonlocal Cahn-Hilliard functional. Submitted.
  • 9. Choksi, R.; Ren, X.: On the derivation of a density functional theory for microphase separation of diblock copolymers. J. Statist. Phys. 113 (2003), no. 1-2, 151-176. MR 2012976 (2004k:82065)
  • 10. Choksi, R.; Sternberg, P.: On the first and second variations of a nonlocal isoperimetric problem. J. Reine Angew. Math. 611 (2007), 75-108. MR 2360604
  • 11. Conti, S.: Branched microstructures: scaling and asymptotic self-similarity. Comm. Pure Appl. Math. 53 (2000), no. 11, 1448-1474. MR 1773416 (2001j:74032)
  • 12. DeSimone, A.; Kohn, R.V.; Müller, S.; Otto, F.: Recent analytical developments in micromagnetics. In The science of hysteresis. Vol. II. Physical modeling, micromagnetics, and magnetization dynamics (G. Bertotti and I.D. Mayergoyz. eds.), pp. 269-381. Elsevier/Academic Press, Amsterdam, 2006. MR 2307930
  • 13. Giusti, E.: Minimal surfaces and functions of bounded variation. Monographs in Mathematics, 80. Birkhäuser Verlag, Basel, 1984. MR 775682 (87a:58041)
  • 14. Goldstein, R.E.; Muraki, D.J.; Petrich, D.M.: Interface proliferation and the growth of labyrinths in a reaction-diffusion system. Phys. Rev. E (3) 53 (1996), no. 4, part B, 3933-3957. MR 1388238 (97b:35100)
  • 15. Kohn, R.V.: Energy-driven pattern formation. In International Congress of Mathematicians. Proceedings of the Congress held in Madrid, August 22-30, 2006. (M. Sanz-Solž et al., eds.), vol. I, pp. 359-383. European Mathematical Society (EMS), Zürich, 2007. MR 2334197
  • 16. Lieb, E.; Loss, M.: Analysis. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 1997. MR 1415616 (98b:00004)
  • 17. Müller, S.: Singular perturbations as a selection criterion for periodic minimizing sequences. Calc. Var. Partial Differential Equations 1 (1993), no. 2, 169-204. MR 1261722 (95k:49030)
  • 18. Muratov, C.B.: Theory of domain patterns in systems with long-range interactions of Coulomb type. Phys. Rev. E (3) 66 (2002), no. 6, 066108. MR 1953930 (2003k:82028)
  • 19. Nishiura, Y.; Ohnishi, I.: Some mathematical aspects of the micro-phase separation in diblock copolymers. Physica D 84 (1995), no. 1-2, 31-39. MR 1334695 (96g:35196)
  • 20. Ohnishi, I.; Nishiura, Y.; Imai, M.; Matsushita, Y.: Analytical solutions describing the phase separation driven by a free energy functional containing a long-range interaction term. Chaos 9 (1999), no. 2, 329-341. MR 1697656 (2000d:35244)
  • 21. Ohta, T.; Kawasaki, K.: Equilibrium morphology of block copolymer melts. Macromolecules 19 (1986), no. 10, 2621-2632.
  • 22. Ren, X.; Wei, J.: On energy minimizers of the diblock copolymer problem. Interfaces Free Bound. 5 (2003), no. 2, 193-238. MR 1980472 (2004i:82077)
  • 23. Ren, X.; Wei, J.: Wriggled lamellar solutions and their stability in the diblock copolymer problem. SIAM J. Math. Anal. 37 (2005), no. 2, 455-489. MR 2176111 (2006m:35112)
  • 24. Ren, X.; Wei, J.: Existence and stability of spherically layered solutions of the diblock copolymer equation. SIAM J. Appl. Math. 66 (2006), no. 3, 1080-1099. MR 2216732 (2007a:34088)
  • 25. Seul, M.; Andelman, D.: Domain shapes and patterns: The phenomenology of modulated phases. Science 267 (1995), no. 5197, 476-483.
  • 26. Teramoto, T.; Nishiura, Y.: Double gyroid morphology in a gradient system with nonlocal effects. Journal of the Physical Society of Japan 71 (2002), no. 7, 1611-1614.
  • 27. Theil, F.: A proof of crystallization in two dimensions. Comm. Math. Phys. 262 (2006), no. 1, 209-236. MR 2200888 (2007f:82018)
  • 28. Thomas, E.L; Anderson, D.M.; Henkee, C.S.; Hoffman, D.: Periodic area-minimizing surfaces in block copolymers. Nature 334 (1988), no. 6183, 598-601.
  • 29. Yip, N.K.: Structure of stable solutions of a one-dimensional variational problem. ESAIM Control Optim. Calc. Var. 12 (2006), no. 4, 721-751. MR 2266815 (2007g:49027)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 49Q10, 49N60, 49S05, 35B10

Retrieve articles in all journals with MSC (2000): 49Q10, 49N60, 49S05, 35B10

Additional Information

Giovanni Alberti
Affiliation: Dipartimento di Matematica, Università di Pisa, largo Pontecorvo 5, 56127 Pisa, Italy

Rustum Choksi
Affiliation: Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC V5A 1S6 Canada

Felix Otto
Affiliation: Institute for Applied Mathematics, Universität Bonn, Wegelerstr 10, D-53115 Bonn, Germany

Received by editor(s): October 16, 2007
Published electronically: November 6, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society