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

Alberti, Giovanni¹ ; Choksi, Rustum² ; Otto, Felix³

¹ Dipartimento di Matematica, Università di Pisa, largo Pontecorvo 5, 56127 Pisa, Italy
² Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC V5A 1S6 Canada
³ Institute for Applied Mathematics, Universität Bonn, Wegelerstr 10, D-53115 Bonn, Germany

Journal of the American Mathematical Society, Tome 22 (2009) no. 2, pp. 569-605

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-08-00622-X

Bibliographie

[1] Alberti, Giovanni, Mã¼Ller, Stefan A new approach to variational problems with multiple scales Comm. Pure Appl. Math. 2001 761 825

[2] Brã©Zis, H. Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert 1973

[3] Chen, Xinfu, Oshita, Yoshihito Periodicity and uniqueness of global minimizers of an energy functional containing a long-range interaction SIAM J. Math. Anal. 2005 1299 1332

[4] Chen, Xinfu, Oshita, Yoshihito An application of the modular function in nonlocal variational problems Arch. Ration. Mech. Anal. 2007 109 132

[5] Choksi, R. Scaling laws in microphase separation of diblock copolymers J. Nonlinear Sci. 2001 223 236

[6] Choksi, Rustum, Kohn, Robert V., Otto, Felix Domain branching in uniaxial ferromagnets: a scaling law for the minimum energy Comm. Math. Phys. 1999 61 79

[7] Choksi, Rustum, Ren, Xiaofeng On the derivation of a density functional theory for microphase separation of diblock copolymers J. Statist. Phys. 2003 151 176

[8] Choksi, Rustum, Sternberg, Peter On the first and second variations of a nonlocal isoperimetric problem J. Reine Angew. Math. 2007 75 108

[9] Conti, Sergio Branched microstructures: scaling and asymptotic self-similarity Comm. Pure Appl. Math. 2000 1448 1474

[10] The science of hysteresis. Vol. I 2006

[11] Giusti, Enrico Minimal surfaces and functions of bounded variation 1984

[12] Goldstein, Raymond E., Muraki, David J., Petrich, Dean M. Interface proliferation and the growth of labyrinths in a reaction-diffusion system Phys. Rev. E (3) 1996 3933 3957

[13] Kohn, Robert V. Energy-driven pattern formation 2007 359 383

[14] Lieb, Elliott H., Loss, Michael Analysis 1997

[15] Mã¼Ller, Stefan Singular perturbations as a selection criterion for periodic minimizing sequences Calc. Var. Partial Differential Equations 1993 169 204

[16] Muratov, C. B. Theory of domain patterns in systems with long-range interactions of Coulomb type Phys. Rev. E (3) 2002

[17] Nishiura, Yasumasa, Ohnishi, Isamu Some mathematical aspects of the micro-phase separation in diblock copolymers Phys. D 1995 31 39

[18] Ohnishi, Isamu, Nishiura, Yasumasa, Imai, Masaki, Matsushita, Yushu Analytical solutions describing the phase separation driven by a free energy functional containing a long-range interaction term Chaos 1999 329 341

[19] Ren, Xiaofeng, Wei, Juncheng On energy minimizers of the diblock copolymer problem Interfaces Free Bound. 2003 193 238

[20] Ren, Xiaofeng, Wei, Juncheng Wriggled lamellar solutions and their stability in the diblock copolymer problem SIAM J. Math. Anal. 2005 455 489

[21] Ren, Xiaofeng, Wei, Juncheng Existence and stability of spherically layered solutions of the diblock copolymer equation SIAM J. Appl. Math. 2006 1080 1099

[22] Theil, Florian A proof of crystallization in two dimensions Comm. Math. Phys. 2006 209 236

[23] Yip, Nung Kwan Structure of stable solutions of a one-dimensional variational problem ESAIM Control Optim. Calc. Var. 2006 721 751