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Volumn 8, Issue 3, 2006, Pages 371-392

Periodic phase separation: The periodic Cahn-Hilliard and isoperimetric problems

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EID: 33750002269     PISSN: 14639963     EISSN: None     Source Type: Journal    
DOI: 10.4171/IFB/148     Document Type: Article
Times cited : (37)

References (60)
  • 1
    • 78751619232 scopus 로고    scopus 로고
    • Morphology of surfaces in mesoscopic polymers, surfactants, electrons, or reaction-diffusion systems: Methods, simulations, and measurements
    • AKSIMENTIEV, A., FIALKOWSKI, M., & HOLYST, R. Morphology of surfaces in mesoscopic polymers, surfactants, electrons, or reaction-diffusion systems: methods, simulations, and measurements. Adv. Chem. Phys. 121 (2002), 141-239.
    • (2002) Adv. Chem. Phys. , vol.121 , pp. 141-239
    • Aksimentiev, A.1    Fialkowski, M.2    Holyst, R.3
  • 3
    • 0012520952 scopus 로고    scopus 로고
    • A new approach to variational problems with multiple scales
    • Zbl 1021.49012 MR 1823420
    • ALBERTI, G., & MÜLLER, S. A new approach to variational problems with multiple scales. Comm. Pure Appl. Math. 54 (2001), 761-825. Zbl 1021.49012 MR 1823420
    • (2001) Comm. Pure Appl. Math. , vol.54 , pp. 761-825
    • Alberti, G.1    Müller, S.2
  • 5
    • 0023829056 scopus 로고
    • Minimal surfaces and structures: From inorganic and metal crystals to cell membranes and biopolymers
    • ANDERSSON, S., HYDE, S. T., LARSSON, K., & LIDIN, S. Minimal surfaces and structures: from inorganic and metal crystals to cell membranes and biopolymers. Chem. Rev. 88 (1988), 221-242.
    • (1988) Chem. Rev. , vol.88 , pp. 221-242
    • Andersson, S.1    Hyde, S.T.2    Larsson, K.3    Lidin, S.4
  • 6
    • 0348046829 scopus 로고    scopus 로고
    • Block copolymers - Designer soft materials
    • BATES, F. S., & FREDRICKSON, G. H. Block copolymers - designer soft materials. Physics Today 52 (1999), no. 2, 32-38.
    • (1999) Physics Today , vol.52 , Issue.2 , pp. 32-38
    • Bates, F.S.1    Fredrickson, G.H.2
  • 7
    • 17144426836 scopus 로고    scopus 로고
    • Γ-convergence for beginners
    • Oxford Univ. Press Zbl pre01865939 MR 1968440
    • BRAIDES, A. Γ-convergence for Beginners. Oxford Lecture Ser. Math. Appl. 22, Oxford Univ. Press (2002). Zbl pre01865939 MR 1968440
    • (2002) Oxford Lecture Ser. Math. Appl. , vol.22
    • Braides, A.1
  • 8
    • 11744273384 scopus 로고
    • The surface evolver
    • Software available at
    • BRAKKE, K. The surface evolver. Experiment. Math. 1 (1992), 141-165. Software available at: http://www.susqu.edu/facstaff/b/brakke/evolver/evolver. html.
    • (1992) Experiment. Math. , vol.1 , pp. 141-165
    • Brakke, K.1
  • 9
    • 33746012315 scopus 로고
    • Free energy of a nonuniform system I. Interfacial Free Energy
    • CAHN, J. W., & HILLIARD, J. E. Free energy of a nonuniform system I. Interfacial Free Energy. J. Chem. Phys. 28 (1958), 258-267.
    • (1958) J. Chem. Phys. , vol.28 , pp. 258-267
    • Cahn, J.W.1    Hilliard, J.E.2
  • 10
    • 0000410819 scopus 로고    scopus 로고
    • Scaling laws in microphase separation of diblock copolymers
    • Zbl 1023.82015 MR 1852942
    • CHOKSI, R. Scaling laws in microphase separation of diblock copolymers. J. Nonlinear Sci. 11 (2001), 223-236. Zbl 1023.82015 MR 1852942
    • (2001) J. Nonlinear Sci. , vol.11 , pp. 223-236
    • Choksi, R.1
  • 11
    • 0346801867 scopus 로고    scopus 로고
    • On a derivation of a density functional theory for microphase separation of diblock copolymers
    • Zbl 1034.82037 MR 2012976
    • CHOKSI, R., & REN, X. On a derivation of a density functional theory for microphase separation of diblock copolymers. J. Statist. Phys. 113 (2003), 151-176. Zbl 1034.82037 MR 2012976
    • (2003) J. Statist. Phys. , vol.113 , pp. 151-176
    • Choksi, R.1    Ren, X.2
  • 12
    • 36248952676 scopus 로고    scopus 로고
    • On the first and second variations of a nonlocal isoperimetric problem
    • to appear
    • CHOKSI, R., & STERNBERG, P. On the first and second variations of a nonlocal isoperimetric problem. J. Reine Angew. Math., to appear.
    • J. Reine Angew. Math.
    • Choksi, R.1    Sternberg, P.2
  • 15
    • 3142523730 scopus 로고    scopus 로고
    • Reaction-limited sintering in nearly saturated environments
    • DAVIDOVITCH, B., ERTAS, D., & HALSEY, T. C. Reaction-limited sintering in nearly saturated environments. Philos. Mag. 84 (2004), 1937.
    • (2004) Philos. Mag. , vol.84 , pp. 1937
    • Davidovitch, B.1    Ertas, D.2    Halsey, T.C.3
  • 16
    • 0003418232 scopus 로고
    • Springer, New York Zbl 0874.49001 MR 0257325
    • FEDERER, H. Geometric Measure Theory. Springer, New York (1969). Zbl 0874.49001 MR 0257325
    • (1969) Geometric Measure Theory
    • Federer, H.1
  • 17
    • 54749104300 scopus 로고    scopus 로고
    • Models for phase separation and their mathematics
    • Zbl 0957.35062 MR 1772733
    • FIFE, P. Models for phase separation and their mathematics. Electron. J. Differential Equations 2000, no. 48, 1-26. Zbl 0957.35062 MR 1772733
    • Electron. J. Differential Equations , vol.2000 , Issue.48 , pp. 1-26
    • Fife, P.1
  • 18
    • 0003174115 scopus 로고    scopus 로고
    • Perturbation of doubly periodic solution branches with applications to the Cahn-Hilliard equation
    • Zbl 0890.35043 MR 1429160
    • FIFE, P., KIELHÖFER, H., MAIER-PAAPE, S., & WANNER, T. Perturbation of doubly periodic solution branches with applications to the Cahn-Hilliard equation. Phys. D 100 (1997), 257-278. Zbl 0890.35043 MR 1429160
    • (1997) Phys. D , vol.100 , pp. 257-278
    • Fife, P.1    Kielhöfer, H.2    Maier-Paape, S.3    Wanner, T.4
  • 19
    • 0028767777 scopus 로고
    • Complex phase behavior of polyisoprene-polystyrene diblock copolymers near the order-disorder transition
    • FÖRSTER S., KHANDPUR, K., ZHAO, J., BATES, F., HAMLEY, I., RYAN, J., & BRAS, W. Complex phase behavior of polyisoprene-polystyrene diblock copolymers near the order-disorder transition. Macromolecules 27 (1994), 6922-6935.
    • (1994) Macromolecules , vol.27 , pp. 6922-6935
    • Förster, S.1    Khandpur, K.2    Zhao, J.3    Bates, F.4    Hamley, I.5    Ryan, J.6    Bras, W.7
  • 21
    • 0002246356 scopus 로고
    • On the regularity of boundaries of sets minimizing perimeter with a volume constraint
    • Zbl 0486.49024 MR 0684753
    • GONZALEZ, E., MASSARI, U., & TAMANINI, I. On the regularity of boundaries of sets minimizing perimeter with a volume constraint. Indiana Univ. Math. J. 32 (1983), 25-37. Zbl 0486.49024 MR 0684753
    • (1983) Indiana Univ. Math. J. , vol.32 , pp. 25-37
    • Gonzalez, E.1    Massari, U.2    Tamanini, I.3
  • 22
    • 0001220422 scopus 로고    scopus 로고
    • Stable constant mean curvature surfaces minimize area
    • Zbl 0878.49026 MR 1432843
    • GROSSE-BRAUCKMANN, K. Stable constant mean curvature surfaces minimize area. Pacific J. Math. 175 (1997), 527-534. Zbl 0878.49026 MR 1432843
    • (1997) Pacific J. Math. , vol.175 , pp. 527-534
    • Grosse-Brauckmann, K.1
  • 23
    • 0031537321 scopus 로고    scopus 로고
    • Gyroids of constant mean curvature
    • Zbl 0898.53010 MR 1464580
    • GROSSE-BRAUCKMANN, K. Gyroids of constant mean curvature. Experiment. Math. 6 (1997), 33-50. Zbl 0898.53010 MR 1464580
    • (1997) Experiment. Math. , vol.6 , pp. 33-50
    • Grosse-Brauckmann, K.1
  • 25
  • 26
    • 0021853508 scopus 로고
    • On a theory of phase transitions with interfacial energy
    • MR 0768066
    • GURTIN, M. On a theory of phase transitions with interfacial energy. Arch. Ration. Mech. Anal. 87 (1985), no. 3, 187-212. MR 0768066
    • (1985) Arch. Ration. Mech. Anal. , vol.87 , Issue.3 , pp. 187-212
    • Gurtin, M.1
  • 27
    • 0001031874 scopus 로고
    • Gitterperiodische Punktmengen und Isoperimetrie
    • Zbl 0248.52012 MR 0324550
    • HADWIGER, H. Gitterperiodische Punktmengen und Isoperimetrie. Monatsh. Math. 76 (1972), 410-418. Zbl 0248.52012 MR 0324550
    • (1972) Monatsh. Math. , vol.76 , pp. 410-418
    • Hadwiger, H.1
  • 28
    • 0028461343 scopus 로고
    • The gyroid: A new equilibrium morphology in weakly segregated diblock copolymers
    • HAJDUK, D. A. ET AL. The gyroid: a new equilibrium morphology in weakly segregated diblock copolymers. Macromolecules 27 (1994), 4063-4075.
    • (1994) Macromolecules , vol.27 , pp. 4063-4075
    • Hajduk, D.A.1
  • 29
    • 2142834706 scopus 로고    scopus 로고
    • The periodic isoperimetric problem
    • Zbl 1046.52002 MR 2031051
    • HAUSWIRTH, L., PEREZ, J., ROMON, P., & ROS, A. The periodic isoperimetric problem. Trans. Amer. Math. Soc. 356 (2004), 2025-2047. Zbl 1046.52002 MR 2031051
    • (2004) Trans. Amer. Math. Soc. , vol.356 , pp. 2025-2047
    • Hauswirth, L.1    Perez, J.2    Romon, P.3    Ros, A.4
  • 30
    • 0029854260 scopus 로고    scopus 로고
    • A new turn for Archimedes
    • HOFFMAN, D. A new turn for Archimedes. Nature 384 (1996), 28-29.
    • (1996) Nature , vol.384 , pp. 28-29
    • Hoffman, D.1
  • 31
    • 0033449295 scopus 로고    scopus 로고
    • The isoperimetric problem on surfaces
    • Zbl 1003.52011 MR 1699261
    • HOWARDS, H., HUTCHINGS, M., & MORGAN, F. The isoperimetric problem on surfaces. Amer. Math. Monthly 106 (1999), 430-439. Zbl 1003.52011 MR 1699261
    • (1999) Amer. Math. Monthly , vol.106 , pp. 430-439
    • Howards, H.1    Hutchings, M.2    Morgan, F.3
  • 32
    • 0010598327 scopus 로고    scopus 로고
    • Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory
    • Zbl 1070.49026 MR 1803974
    • HUTCHINSON, J., & TONEGAWA, Y. Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory. Calc. Var. Partial Differential Equations 10 (2000), 49-84. Zbl 1070.49026 MR 1803974
    • (2000) Calc. Var. Partial Differential Equations , vol.10 , pp. 49-84
    • Hutchinson, J.1    Tonegawa, Y.2
  • 33
    • 0001162145 scopus 로고
    • The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions
    • Zbl 0687.53010 MR 1003093
    • KARCHER, H. The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions. Manuscripta Math. 64 (1989), 291-357. Zbl 0687.53010 MR 1003093
    • (1989) Manuscripta Math. , vol.64 , pp. 291-357
    • Karcher, H.1
  • 34
    • 27844439674 scopus 로고    scopus 로고
    • Transl. Math. Monogr. 221, Amer. Math. Soc. Zbl 1042.53001 MR 2013507
    • KENMOTSU, K. Surfaces with Constant Mean Curvature. Transl. Math. Monogr. 221, Amer. Math. Soc. (2003). Zbl 1042.53001 MR 2013507
    • (2003) Surfaces with Constant Mean Curvature
    • Kenmotsu, K.1
  • 35
    • 84975988204 scopus 로고
    • Local minimisers and singular perturbations
    • Zbl 0676.49011 MR 0985990
    • KOHN, R. V., & STERNBERG, P. Local minimisers and singular perturbations. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), 69-94. Zbl 0676.49011 MR 0985990
    • (1989) Proc. Roy. Soc. Edinburgh Sect. A , vol.111 , pp. 69-94
    • Kohn, R.V.1    Sternberg, P.2
  • 36
    • 34250109116 scopus 로고
    • The gradient theory of phase transitions and the minimal interface criterion
    • Zbl 0616.76004 MR 0866718
    • MODICA, L. The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98 (1987), 123-142. Zbl 0616.76004 MR 0866718
    • (1987) Arch. Ration. Mech. Anal. , vol.98 , pp. 123-142
    • Modica, L.1
  • 37
    • 0000359194 scopus 로고
    • --convergenza
    • Zbl 0356.49008 MR 0445362
    • -- convergenza. Boll. Un. Mat. Ital. (5) 14 (1977), 285-299. Zbl 0356.49008 MR 0445362
    • (1977) Boll. Un. Mat. Ital. , vol.14 , Issue.5 , pp. 285-299
    • Modica, L.1    Mortola, S.2
  • 38
    • 0000390068 scopus 로고
    • Singular perturbations as a selection criterion for periodic minimizing sequences
    • Zbl 0821.49015 MR 1261722
    • MÜLLER, S. Singular perturbations as a selection criterion for periodic minimizing sequences. Calc. Var. 1 (1993), 169-204. Zbl 0821.49015 MR 1261722
    • (1993) Calc. Var. , vol.1 , pp. 169-204
    • Müller, S.1
  • 39
    • 41349101280 scopus 로고    scopus 로고
    • Theory of domain patterns in systems with long-range interactions of Coulomb type
    • MR 1953930
    • MURATOV, C. B. Theory of domain patterns in systems with long-range interactions of Coulomb type. Phys. Rev. E 66 (2002), 066108. MR 1953930
    • (2002) Phys. Rev. E , vol.66 , pp. 066108
    • Muratov, C.B.1
  • 40
    • 0002733111 scopus 로고
    • Some mathematical aspects of the micro-phase separation in diblock copolymers
    • MR 1334695
    • NISHIURA, Y., & OHNISHI, I. Some mathematical aspects of the micro-phase separation in diblock copolymers. Phys. D 84 (1995), 31-39. MR 1334695
    • (1995) Phys. D , vol.84 , pp. 31-39
    • Nishiura, Y.1    Ohnishi, I.2
  • 42
    • 0000482598 scopus 로고
    • Equilibrium morphology of block copolymer melts
    • OHTA, T., & KAWASAKI, K. Equilibrium morphology of block copolymer melts. Macromolecules 19 (1986), 2621-2632.
    • (1986) Macromolecules , vol.19 , pp. 2621-2632
    • Ohta, T.1    Kawasaki, K.2
  • 43
    • 0034347328 scopus 로고    scopus 로고
    • On the multiplicity of solutions of two nonlocal variational problems
    • Zbl 0973.49007 MR 1752422
    • REN, X., & WEI, J. On the multiplicity of solutions of two nonlocal variational problems. SIAM J. Math. Anal. 31 (2000), 909-924. Zbl 0973.49007 MR 1752422
    • (2000) SIAM J. Math. Anal. , vol.31 , pp. 909-924
    • Ren, X.1    Wei, J.2
  • 44
    • 84978800888 scopus 로고    scopus 로고
    • On energy minimizers of the diblock copolymer problem
    • Zbl 1031.49035 MR 1980472
    • REN, X., & WEI, J. On energy minimizers of the diblock copolymer problem. Interfaces Free Bound. 5 (2003), 193-238. Zbl 1031.49035 MR 1980472
    • (2003) Interfaces Free Bound. , vol.5 , pp. 193-238
    • Ren, X.1    Wei, J.2
  • 45
    • 33644939600 scopus 로고    scopus 로고
    • Wriggled lamellar solutions and their stability in the diblock copolymer problem
    • MR 2176111
    • REN, X., & WEI, J. Wriggled lamellar solutions and their stability in the diblock copolymer problem. SIAM J. Math. Anal. 37 (2005), 455-489. MR 2176111
    • (2005) SIAM J. Math. Anal. , vol.37 , pp. 455-489
    • Ren, X.1    Wei, J.2
  • 46
    • 22544461938 scopus 로고    scopus 로고
    • Stability of spot and ring solutions of the diblock copolymer equation
    • Zbl 1064.82040 MR 2098120
    • REN, X., & WEI, J. Stability of spot and ring solutions of the diblock copolymer equation. J. Math. Phys. 45 (2004), 4106-4133. Zbl 1064.82040 MR 2098120
    • (2004) J. Math. Phys. , vol.45 , pp. 4106-4133
    • Ren, X.1    Wei, J.2
  • 47
    • 0036813973 scopus 로고    scopus 로고
    • Concentrically layered energy equilibria of the diblock copolymer problem
    • Zbl 1010.82041 MR 1939157
    • REN, X., & WEI, J. Concentrically layered energy equilibria of the diblock copolymer problem. Europ. J. Appl. Math. 13 (2002), 479-496. Zbl 1010.82041 MR 1939157
    • (2002) Europ. J. Appl. Math. , vol.13 , pp. 479-496
    • Ren, X.1    Wei, J.2
  • 48
    • 51249163412 scopus 로고
    • Stable constant mean curvature tori and the isoperimetric problem in three space forms
    • Zbl 0760.53037 MR 1161286
    • RITORE M., & ROS, A. Stable constant mean curvature tori and the isoperimetric problem in three space forms. Comment. Math. Helvet. 67 (1992), 293-305. Zbl 0760.53037 MR 1161286
    • (1992) Comment. Math. Helvet. , vol.67 , pp. 293-305
    • Ritore, M.1    Ros, A.2
  • 49
    • 8644240623 scopus 로고    scopus 로고
    • Proceedings from the Clay Mathematics Institute Summer School, MSRI, Berkeley, CA. Available at
    • ROS, A. The isoperimetric problem. Proceedings from the Clay Mathematics Institute Summer School, MSRI, Berkeley, CA. Available at www.ugr.es/~aros.
    • The Isoperimetric Problem
    • Ros, A.1
  • 50
    • 0000049028 scopus 로고
    • Schwarz' P and D surfaces are stable
    • Zbl 0747.53010 MR 1245555
    • ROSS, M. Schwarz' P and D surfaces are stable. Differential Geom. Appl. 2 (1992), 179-195. Zbl 0747.53010 MR 1245555
    • (1992) Differential Geom. Appl. , vol.2 , pp. 179-195
    • Ross, M.1
  • 51
    • 33750006083 scopus 로고    scopus 로고
    • The Scientific Graphics Project
    • The Scientific Graphics Project. http://www.msri.org/publications/sgp/ SGP/indexc.html
  • 52
    • 0004096780 scopus 로고
    • Proc. Centre Math. Anal. Austral. Nat. Univ. 3 Zbl 0546.49019 MR 0756417
    • SIMON, L. Lectures on Geometric Measure Theory. Proc. Centre Math. Anal. Austral. Nat. Univ. 3 (1983). Zbl 0546.49019 MR 0756417
    • (1983) Lectures on Geometric Measure Theory
    • Simon, L.1
  • 53
    • 34250095380 scopus 로고
    • The effect of a singular perturbation on nonconvex variational problems
    • Zbl 0647.49021 MR 0930124
    • STERNBERG, P. The effect of a singular perturbation on nonconvex variational problems. Arch. Ration. Mech. Anal. 101 (1988), 209-260. Zbl 0647.49021 MR 0930124
    • (1988) Arch. Ration. Mech. Anal. , vol.101 , pp. 209-260
    • Sternberg, P.1
  • 54
    • 0032370858 scopus 로고    scopus 로고
    • Connectivity of phase boundaries in strictly convex domains
    • Zbl 0911.49025 MR 1620498
    • STERNBERG, P., & ZUMBRUN, K. Connectivity of phase boundaries in strictly convex domains. Arch. Ration. Mech. Anal. 141 (1998), 375-400. Zbl 0911.49025 MR 1620498
    • (1998) Arch. Ration. Mech. Anal. , vol.141 , pp. 375-400
    • Sternberg, P.1    Zumbrun, K.2
  • 55
    • 0036335523 scopus 로고    scopus 로고
    • Double gyroid morphology in a gradient system with nonlocal effects
    • TERAMOTO, T., & NISHIURA, Y. Double gyroid morphology in a gradient system with nonlocal effects. J. Phys. Soc. Japan 71 (2002), 1611-1614.
    • (2002) J. Phys. Soc. Japan , vol.71 , pp. 1611-1614
    • Teramoto, T.1    Nishiura, Y.2
  • 56
    • 0024057538 scopus 로고
    • Periodic area-minimizing surfaces in block copolymers
    • THOMAS, E., ANDERSON, D. M., HENKEE, C. S., & HOFFMAN, D. Periodic area-minimizing surfaces in block copolymers. Nature 334 (1988), 598-601.
    • (1988) Nature , vol.334 , pp. 598-601
    • Thomas, E.1    Anderson, D.M.2    Henkee, C.S.3    Hoffman, D.4
  • 57
    • 0037119540 scopus 로고    scopus 로고
    • Phase field model with a variable chemical potential
    • Zbl 1013.35070 MR 1926927
    • TONEGAWA, Y. Phase field model with a variable chemical potential. Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), 993-1019. Zbl 1013.35070 MR 1926927
    • (2002) Proc. Roy. Soc. Edinburgh Sect. A , vol.132 , pp. 993-1019
    • Tonegawa, Y.1
  • 58
    • 33744932727 scopus 로고    scopus 로고
    • A diffused interface whose chemical potential lies in a Sobolev space
    • MR 2185866
    • TONEGAWA, Y. A diffused interface whose chemical potential lies in a Sobolev space. Ann. Scuola Norm. Sup. Pisa 4 (2005), 487-510. MR 2185866
    • (2005) Ann. Scuola Norm. Sup. Pisa , vol.4 , pp. 487-510
    • Tonegawa, Y.1
  • 59
    • 0041008860 scopus 로고
    • A strong minimax property of nondegenerate submanifolds
    • Zbl 0808.49037 MR 1305283
    • WHITE, B. A strong minimax property of nondegenerate submanifolds. J. Reine Angew. Math. 457 (1994), 203-218. Zbl 0808.49037 MR 1305283
    • (1994) J. Reine Angew. Math. , vol.457 , pp. 203-218
    • White, B.1


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