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Second-Order Γ-limit for the Cahn–Hilliard Functional

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Abstract

The goal of this paper is to solve a long standing open problem, namely, the asymptotic development of order 2 by Γ-convergence of the mass-constrained Cahn–Hilliard functional.

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References

  1. Alberico A., Cianchi A.: Borderline sharp estimates for solutions to Neumann problems. Ann. Acad. Sci. Fenn. Math. 32(1), 27–53 (2007)

    MathSciNet  MATH  Google Scholar 

  2. Alikakos N., Bronsard L., Fusco G.: Slow motion in the gradient theory of phase transitions via energy and spectrum. Calc. Var. Partial Differ. Equ. 6(1), 39–66 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alikakos N., Fusco G.: Slow dynamics for the Cahn–Hilliard equation in higher space dimensions. I. Spectral estimates. Comm. Partial Differ. Equ. 19(9-10), 1397–1447 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alikakos N., Fusco G.: Slow dynamics for the Cahn-Hilliard equation in higher space dimensions: the motion of bubbles. Arch. Ration. Mech. Anal. 141(1), 1–61 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alikakos N.D., Bates P.W., Chen X.: Convergence of the cahn-hilliard equation to the hele-shaw model. Arch. Ration. Mech. Anal. 128(2), 165–205 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000

  7. Anzellotti G., Baldo S.: Asymptotic development by Γ-convergence. Appl. Math. Optim. 27(2), 105– (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. Anzellotti G., Baldo S., Orlandi G.: Γ-asymptotic developments, the Cahn-Hilliard functional, and curvatures. J. Math. Anal. Appl. 197(3), 908–924 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bavard C., Pansu P.: Sur le volume minimal de R 2. Ann. Sci. École Norm. Sup. 19(4), 479–490 (1986)

    MathSciNet  MATH  Google Scholar 

  10. Bellettini, G., hassem Nayam, A., Novaga, M.: Γ-type estimates for the one-dimensional allen-cahn’s action. Calc. Var. Geom. Meas. Theor. (2014). http://cvgmt.sns.it/paper/2209/

  11. Braides, A.: Local minimization, variational evolution and Γ-convergence, vol. 2094 of Lecture Notes in Mathematics. Springer, Cham, 2014

  12. Braides A., Truskinovsky L.: Asymptotic expansions by Γ-convergence. Contin. Mech. Thermodyn. 20(1), 21–62 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Bronsard L., Kohn R.: On the slowness of phase boundary motion in one space dimension. Commun. Pure Appl. Math. 43(8), 983–997 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bronsard L., Kohn R.: Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics. J. Differ. Equ. 90(2), 211–237 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Bronsard L., Stoth B.: Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg–Landau equation. SIAM J. Math. Anal. 28(4), 769–807 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cahn J., Hilliard J.: Free energy of a nonuniform system. i. interfacial free energy. J. Chem. Phys. 28(2), 258–267 (1958)

    Article  ADS  Google Scholar 

  17. Carr J., Gurtin M., Slemrod M.: Structured phase transitions on a finite interval. Arch. Ration. Mech. Anal. 86(4), 317–351 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  18. Carr J., Pego R.: Metastable patterns in solutions of ut= 2uxx- f (u). Commun. Pure Appl. Math. 42(5), 523–576 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chen X.: Generation and propagation of interfaces for reaction-diffusion equations. J. Differ. Equ. 96(1), 116–141 (1992)

    Article  ADS  MATH  Google Scholar 

  20. Cianchi A., Edmunds D.E., Gurka P.: On weighted poincaré inequalities. Math. Nachrichten 180(1), 15–41 (1996)

    MathSciNet  MATH  Google Scholar 

  21. Cianchi A., Esposito L., Fusco N., Trombetti C.: A quantitative Pólya-Szegö principle. J. Reine Angew. Math. 614, 153–189 (2008)

    MathSciNet  MATH  Google Scholar 

  22. Cianchi A., Fusco N.: Functions of bounded variation and rearrangements. Arch. Ration. Mech. Anal. 165(1), 1–40 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Cianchi A., Maz’ya V.: Neumann problems and isocapacitary inequalities. J. Math. Pure. Appl. (9) 89(1), 71–105 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Cianchi A., Pick L.: Optimal gaussian sobolev embeddings. J. Funct. Anal. 256(11), 3588–3642 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Crandall M., Tartar L.: Some relations between nonexpansive and order preserving mappings. Proc. Am. Math. Soc. 78(3), 385–390 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  26. Dal Maso, G.: An introduction to Γ-convergence. Progress in Nonlinear Differential Equations and their Applications, 8. Birkhäuser Boston, Inc., Boston, MA, 1993

  27. Dal Maso, G., Fonseca, I., Leoni, G.: Second order asymptotic development for the Cahn–Hilliard functional. To Appear (2013)

  28. Evans, L., Gariepy, R.: Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992

  29. Focardi M.: Γ-convergence: a tool to investigate physical phenomena across scales. Math. Methods Appl. Sci. 35(14), 1613–1658 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Fonseca I., Tartar L.: The gradient theory of phase transitions for systems with two potential wells. Proc. R. Soc. Edinb: Sect. Math. 111(1-2), 89–102 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  31. Fusco G., Hale J.: Slow-motion manifolds, dormant instability, and singular perturbations. J. Dyn. Differ. Equ. 1(1), 75–94 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  32. Garcke H.: Curvature driven interface evolution. Jahresber. Dtsch. Math. Ver. 115(2), 63–100 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Gonzalez E., Massari U., Tamanini I.: On the regularity of boundaries of sets minimizing perimeter with a volume constraint. Indian Univ. Math. J. 32(1), 25–37 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  34. Grant C.: Slow motion in one-dimensional Cahn–Morral systems. SIAM J. Math. Anal. 26(1), 21–34 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  35. Grüter M.: Boundary regularity for solutions of a partitioning problem. Arch. Ration. Mech. Anal. 97(3), 261–270 (1987)

    Article  MATH  Google Scholar 

  36. Gurtin M.: On a theory of phase transitions with interfacial energy. Arch. Ration. Mech. Anal. 87(3), 187–212 (1985)

    Article  MathSciNet  Google Scholar 

  37. Gurtin M., Matano H.: On the structure of equilibrium phase transitions within the gradient theory of fluids. Q. Appl. Math. 46(2), 301–317 (1988)

    MathSciNet  MATH  Google Scholar 

  38. Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex analysis and minimization algorithms. I, vol. 305 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1993. Fundamentals

  39. Kawohl, B.: Rearrangements and convexity of level sets in PDE, vol. 1150 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1985

  40. Kesavan, S.: Symmetrization & applications, vol. 3 of Series in Analysis. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006

  41. Kohn R., Sternberg P.: Local minimisers and singular perturbations. Proc. R. Soc. Edinburgh Sect. A 111(1-2), 69–84 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  42. Krantz S.G., Parks H.R.: Distance to c k hypersurfaces. J. Differ. Equ. 40(1), 116–120 (1981)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Kurata K., Shibata M.: On a one-dimensional variational problem related to the Cahn–Hilliard energy in a bent strip-like domain. Nonlinear Anal. Theor. Methods Appl. 47(2), 1059–1068 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  44. Leoni, G.: A first course in Sobolev spaces, vol. 105 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2009

  45. Leoni G.: A remark on the compactness for the Cahn-Hilliard functional. ESAIM Control Optim. Calc. Var. 20(2), 517–523 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  46. Luckhaus S., Modica L.: The Gibbs–Thompson relation within the gradient theory of phase transitions. Arch. Ration. Mech. Anal. 107(1), 71–83 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  47. Maggi, F.: Sets of finite perimeter and geometric variational problems, vol. 135 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2012. An introduction to geometric measure theory

  48. Maz’ya, V.: Sobolev spaces with applications to elliptic partial differential equations, augmented ed., vol. 342 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, 2011

  49. Modica L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98(2), 123–142 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  50. Modica L., Mortola S.: Un esempio di Γ-convergenza. Boll. Un. Mat. Ital. B (5) 14(1), 285–299 (1977)

    MathSciNet  MATH  Google Scholar 

  51. Murray, R., Rinaldi, M.: Slow motion for the nonlocal Allen–Cahn equation in n dimensions. In Preparation (2015)

  52. Ni W.M.: On the positive radial solutions of some semilinear elliptic equations on R n. Appl. Math. Optim. 9(4), 373–380 (1983)

    MathSciNet  MATH  Google Scholar 

  53. Niethammer B.: Existence and uniqueness of radially symmetric stationary points within the gradient theory of phase transitions. Eur. J. Appl. Math. 6(1), 45–67 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  54. Oleksiv I.Y., Pesin N.: Finiteness of Hausdorff measure of level sets of bounded subsets of Euclidean space. Math. Notes 37(3), 237–242 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  55. Otto F., Reznikoff M.: Slow motion of gradient flows. J. Differ. Equ. 237(2), 372–420 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. Pego R.: Front migration in the nonlinear Cahn-Hilliard equation. Proc. R. Soc. Lond. Ser. A 422(1863), 261–278 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  57. Ros A.: The isoperimetric problem. Glob. Theor. Min. Surf. 2, 175–209 (2001)

    MathSciNet  Google Scholar 

  58. Sandier E., Serfaty S.: Gamma-convergence of gradient flows with applications to ginzburg-landau. Commun. Pure Appl. Math. 57(12), 1627–1672 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  59. Spector D.: Simple proofs of some results of Reshetnyak. Proc. Am. Math. Soc. 139(5), 1681–1690 (2004)

    Article  MathSciNet  Google Scholar 

  60. Sternberg P.: The effect of a singular perturbation on nonconvex variational problems. Arch. Ration. Mech. Anal. 101(3), 209–260 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  61. Sternberg P., Zumbrun K.: Connectivity of phase boundaries in strictly convex domains. Arch. Ration. Mech. Anal. 141(4), 375–400 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  62. Sternberg P., Zumbrun K.: On the connectivity of boundaries of sets minimizing perimeter subject to a volume constraint. Commun. Anal. Geom. 7(1), 199–220 (1999)

    MathSciNet  MATH  Google Scholar 

  63. Talenti G.: Elliptic equations and rearrangements. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 3(4), 697–718 (1976)

    MathSciNet  MATH  Google Scholar 

  64. Ziemer, W.: Weakly differentiable functions, vol. 120 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation

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Correspondence to Giovanni Leoni.

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Communicated by I. Fonseca

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Leoni, G., Murray, R. Second-Order Γ-limit for the Cahn–Hilliard Functional. Arch Rational Mech Anal 219, 1383–1451 (2016). https://doi.org/10.1007/s00205-015-0924-4

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