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Second order asymptotic development for the anisotropic Cahn–Hilliard functional

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Abstract

The asymptotic behavior of an anisotropic Cahn–Hilliard functional with prescribed mass and Dirichlet boundary condition is studied when the parameter \(\varepsilon \) that determines the width of the transition layers tends to zero. The double-well potential is assumed to be even and equal to \(|s-1|^\beta \) near \(s=1\), with \(1<\beta <2\). The first order term in the asymptotic development by \(\Gamma \)-convergence is well-known, and is related to a suitable anisotropic perimeter of the interface. Here it is shown that, under these assumptions, the second order term is zero, which gives an estimate on the rate of convergence of the minimum values.

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References

  1. Alvino, A., Ferone, V., Trombetti, G., Lions, P.-L.: Convex symmetrization and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 14(2), 275–293 (1997)

  2. Anzellotti, G., Baldo, S.: Asymptotic development by \(\Gamma \)-convergence. Appl. Math. Optim. 27(2), 105–123 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Barroso, A.C., Fonseca, I.: Anisotropic singular perturbations—the vectorial case. Proc. Roy. Soc. Edinb. Sect. A 124(3), 527–571 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bellettini, G., Nayam, A., Novaga, M.: \(\Gamma \)-type estimates for the one-dimensional Allen–Cahn’s action (submitted)

  6. Bouchitté, G.: Singular perturbations of variational problems arising from a two-phase transition model. Appl. Math. Optim. 21(3), 289–314 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Braides, A.: Approximation of Free-Discontinuity Problems. Lecture Notes in Mathematics, vol. 1694. Springer, Berlin (1998)

  8. Braides, A.: \(\Gamma \) -convergence for beginners. Oxford Lecture Series in Mathematics and its Applications, vol. 22. Oxford University Press, Oxford (2002)

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Dal Maso, G.: An Introduction to \(\Gamma \)-Convergence, Progress in Nonlinear Differential Equations and their Applications, vol. 8. Birkhäuser, Basel (1993)

  11. Dal Maso, G., Musina, R.: An approach to the thin obstacle problem for variational functionals depending on vector valued functions. Commun. Partial Differ. Equ. 14(12), 1717–1743 (1989)

  12. Fonseca, I.: The Wulff theorem revisited. Proc. R. Soc. Lond. Ser. A 432(1884), 125–145 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fonseca, I., Müller, S.: A uniqueness proof for the Wulff theorem. Proc. R. Soc. Edinb. Sect. A 119(1–2), 125–136 (1991)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Gurtin, M.E.: Some results and conjectures in the gradient theory of phase transitions. IMA, preprint 156 (1985)

  16. Leoni, G.: A first course in Sobolev spaces. Graduate Studies in Mathematics, vol. 105. American Mathematical Society, Providence (2009)

  17. Leoni, G., Murray, R.: Second order \(\Gamma \)-limit for the Cahn–Hilliard functional (in preparation)

  18. 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 

  19. Modica, L., Mortola, S.: Un esempio di \(\Gamma ^{-}\)-convergenza. Boll. Un. Mat. Ital. B (5) 14(1), 285–299 (1977)

  20. 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 

  21. Morgan, F.: The cone over the Clifford torus in \(\mathbf{R}^{4}\) is \(\Phi \)-minimizing. Math. Ann. 289(2), 341–354 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  22. Owen, N.C., Sternberg, P.: Nonconvex variational problems with anisotropic perturbations. Nonlinear Anal. 16(7–8), 705–719 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rockafellar, R.T.: Convex analysis. Reprint of the 1970 original. Princeton University Press, Princeton, NJ, Princeton Landmarks in Mathematics. Princeton Paperbacks (1977)

  24. 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 

  25. Taylor, J.E.: Existence and structure of solutions to a class of nonelliptic variational problems. Symposia Mathematica, vol. XIV (Convegno di Teoria Geometrica dell’Integrazione e Varietà Minimali, INDAM, Roma, Maggio 1973), pp. 499–508. Academic Press, London (1974)

  26. Taylor, J.E.: Unique structure of solutions to a class of nonelliptic variational problems. (Proc. Sympos. Pure. Math., vol. XXVII, Stanford Univ., Stanford, Calif. 1973), Part 1, pp. 419–427. Amer. Math. Soc, Providence (1975)

  27. Taylor, J.E.: Crystalline variational problems. Bull. Am. Math. Soc. 84(4), 568–588 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wulff, G.: Zur Frage der Geschwindigkeit des Wachstums und der Auflösung der Kristallflächen. Z. Kristallogr. 34, 449–530 (1901)

    Google Scholar 

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Acknowledgments

The authors warmly thank the Center for Nonlinear Analysis (NSF Grant No. DMS-0635983), where part of this research was carried out. The research of I. Fonseca was partially funded by the National Science Foundation under Grant No. DMS-0905778 and that of G. Leoni under Grant No. DMS-1007989. I. Fonseca and G. Leoni also acknowledge support of the National Science Foundation under the PIRE Grant No. OISE-0967140. The research of G. Dal Maso was also supported by by the Italian Ministry of Education, University, and Research under the Project “Variational Problems with Multiple Scales” 2008 and by the European Research Council under Grant No. 290888 “Quasistatic and Dynamic Evolution Problems in Plasticity and Fracture”. The authors wish to thank Matteo Focardi for several discussions on the subject of this paper and Michael Goldman, who called their attention to reference [1]. The results of this paper led to a dramatic simplification in the proof of the \(\Gamma \)-limsup inequality and were crucial in the proof of the \(\Gamma \)-liminf inequality in the anisotropic case.

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Correspondence to Giovanni Leoni.

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Communicated by L. Ambrosio.

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Dal Maso, G., Fonseca, I. & Leoni, G. Second order asymptotic development for the anisotropic Cahn–Hilliard functional. Calc. Var. 54, 1119–1145 (2015). https://doi.org/10.1007/s00526-015-0819-0

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