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

HTML articles powered by AMS MathViewer

- by Giovanni Alberti, Rustum Choksi and Felix Otto PDF
- J. Amer. Math. Soc.
**22**(2009), 569-605 Request permission

## 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

- Giovanni Alberti and Stefan Müller,
*A new approach to variational problems with multiple scales*, Comm. Pure Appl. Math.**54**(2001), no. 7, 761–825. MR**1823420**, DOI 10.1002/cpa.1013
BF Bates, F.S.; Fredrickson, G.H.: Block copolymers - Designer soft materials. - H. Brézis,
*Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert*, North-Holland Mathematics Studies, No. 5, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973 (French). MR**0348562** - Xinfu Chen and Yoshihito Oshita,
*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**, DOI 10.1137/S0036141004441155 - Xinfu Chen and Yoshihito Oshita,
*An application of the modular function in nonlocal variational problems*, Arch. Ration. Mech. Anal.**186**(2007), no. 1, 109–132. MR**2338353**, DOI 10.1007/s00205-007-0050-z - R. Choksi,
*Scaling laws in microphase separation of diblock copolymers*, J. Nonlinear Sci.**11**(2001), no. 3, 223–236. MR**1852942**, DOI 10.1007/s00332-001-0456-y - Rustum Choksi, Robert V. Kohn, and Felix Otto,
*Domain branching in uniaxial ferromagnets: a scaling law for the minimum energy*, Comm. Math. Phys.**201**(1999), no. 1, 61–79. MR**1669433**, DOI 10.1007/s002200050549
CPW 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.
- Rustum Choksi and Xiaofeng Ren,
*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**, DOI 10.1023/A:1025722804873 - Rustum Choksi and Peter Sternberg,
*On the first and second variations of a nonlocal isoperimetric problem*, J. Reine Angew. Math.**611**(2007), 75–108. MR**2360604**, DOI 10.1515/CRELLE.2007.074 - Sergio Conti,
*Branched microstructures: scaling and asymptotic self-similarity*, Comm. Pure Appl. Math.**53**(2000), no. 11, 1448–1474. MR**1773416**, DOI 10.1002/1097-0312(200011)53:11<1448::AID-CPA6>3.0.CO;2-C - Giorgio Bertotti and Isaak D. Mayergoyz (eds.),
*The science of hysteresis. Vol. I*, Elsevier/Academic Press, Amsterdam, 2006. Mathematical modeling and applications. MR**2307929** - Enrico Giusti,
*Minimal surfaces and functions of bounded variation*, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. MR**775682**, DOI 10.1007/978-1-4684-9486-0 - Raymond E. Goldstein, David J. Muraki, and Dean M. Petrich,
*Interface proliferation and the growth of labyrinths in a reaction-diffusion system*, Phys. Rev. E (3)**53**(1996), no. 4, 3933–3957. MR**1388238**, DOI 10.1103/PhysRevE.53.3933 - Robert V. Kohn,
*Energy-driven pattern formation*, International Congress of Mathematicians. Vol. I, Eur. Math. Soc., Zürich, 2007, pp. 359–383. MR**2334197**, DOI 10.4171/022-1/15 - Elliott H. Lieb and Michael Loss,
*Analysis*, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 1997. MR**1415616**, DOI 10.2307/3621022 - Stefan Müller,
*Singular perturbations as a selection criterion for periodic minimizing sequences*, Calc. Var. Partial Differential Equations**1**(1993), no. 2, 169–204. MR**1261722**, DOI 10.1007/BF01191616 - C. B. Muratov,
*Theory of domain patterns in systems with long-range interactions of Coulomb type*, Phys. Rev. E (3)**66**(2002), no. 6, 066108, 25. MR**1953930**, DOI 10.1103/PhysRevE.66.066108 - Yasumasa Nishiura and Isamu Ohnishi,
*Some mathematical aspects of the micro-phase separation in diblock copolymers*, Phys. D**84**(1995), no. 1-2, 31–39. MR**1334695**, DOI 10.1016/0167-2789(95)00005-O - Isamu Ohnishi, Yasumasa Nishiura, Masaki Imai, and Yushu Matsushita,
*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**, DOI 10.1063/1.166410
OK Ohta, T.; Kawasaki, K.: Equilibrium morphology of block copolymer melts. - Xiaofeng Ren and Juncheng Wei,
*On energy minimizers of the diblock copolymer problem*, Interfaces Free Bound.**5**(2003), no. 2, 193–238. MR**1980472**, DOI 10.4171/IFB/78 - Xiaofeng Ren and Juncheng Wei,
*Wriggled lamellar solutions and their stability in the diblock copolymer problem*, SIAM J. Math. Anal.**37**(2005), no. 2, 455–489. MR**2176111**, DOI 10.1137/S0036141003433589 - Xiaofeng Ren and Juncheng Wei,
*Existence and stability of spherically layered solutions of the diblock copolymer equation*, SIAM J. Appl. Math.**66**(2006), no. 3, 1080–1099. MR**2216732**, DOI 10.1137/040618771
SA Seul, M.; Andelman, D.: Domain shapes and patterns: The phenomenology of modulated phases. - Florian Theil,
*A proof of crystallization in two dimensions*, Comm. Math. Phys.**262**(2006), no. 1, 209–236. MR**2200888**, DOI 10.1007/s00220-005-1458-7
TAHH Thomas, E.L; Anderson, D.M.; Henkee, C.S.; Hoffman, D.: Periodic area-minimizing surfaces in block copolymers. - Nung Kwan Yip,
*Structure of stable solutions of a one-dimensional variational problem*, ESAIM Control Optim. Calc. Var.**12**(2006), no. 4, 721–751. MR**2266815**, DOI 10.1051/cocv:2006019

*Physics Today*

**52**(1999), no. 2, 32–38.

*Macromolecules*

**19**(1986), no. 10, 2621–2632.

*Science*

**267**(1995), no. 5197, 476–483. TN 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.

*Nature*

**334**(1988), no. 6183, 598–601.

## Additional Information

**Giovanni Alberti**- Affiliation: Dipartimento di Matematica, Università di Pisa, largo Pontecorvo 5, 56127 Pisa, Italy
- Email: galberti1@dm.unipi.it
**Rustum Choksi**- Affiliation: Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC V5A 1S6 Canada
- MR Author ID: 604242
- Email: choksi@math.sfu.ca
**Felix Otto**- Affiliation: Institute for Applied Mathematics, Universität Bonn, Wegelerstr 10, D-53115 Bonn, Germany
- Email: otto@iam.uni-bonn.de
- Received by editor(s): October 16, 2007
- Published electronically: November 6, 2008
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**22**(2009), 569-605 - MSC (2000): Primary 49Q10, 49N60, 49S05, 35B10
- DOI: https://doi.org/10.1090/S0894-0347-08-00622-X
- MathSciNet review: 2476783