Glasnik Matematicki, Vol. 59, No. 1 (2024), 125-145. \( \)

SOME RESULTS IN ASYMPTOTIC ANALYSIS OF FINITE-ENERGY SEQUENCES OF ONE-DIMENSIONAL CAHN–HILLIARD FUNCTIONAL WITH NON-STANDARD TWO-WELL POTENTIAL

Andrija Raguž

Department of Economics and Mathematics, Zagreb School of Economics and Management, Filipa Vukasovića 1, 10 000 Zagreb, Croatia
e-mail:araguz@zsem.hr

Abstract.   In this paper we extend the consideration of G. Leoni pertaining to the finite-energy sequences of the one-dimensional Cahn-Hilliard functional \[ I^{\varepsilon}_0(u)=\int_{0}^{1}\Big({\varepsilon}^2 u'^2(s)+W(u(s))\Big)ds, \] where \(u\in {\rm H}^{1}(0,1)\) and where \(W\) is a two-well potential with symmetrically placed wells endowed with a non-standard integrability condition. We introduce several new classes of finite-energy sequences, we recover their underlying geometric properties as \(\varepsilon\longrightarrow 0\), and we prove the related compactness result.

2020 Mathematics Subject Classification.   34E15, 49J45

Key words and phrases.   Asymptotic analysis, singular perturbation, Young measures, Cahn-Hilliard functional, compactness

Full text (PDF) (access from subscribing institutions only)

https://doi.org/10.3336/gm.59.1.06

References:

  1. N. Ansini, A. Braides and V. Chiadò Piat, Gradient theory of phase transitions in composite media, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), 265–296.
    CrossRef    MathSciNet

  2. G. Alberti, Variational models for phase transitions, an approach via \(\Gamma\)-convergence, in: Calculus of variations and partial differential equations, Springer-Verlag, Berlin, 2000, 95–114.
    CrossRef    MathSciNet

  3. G. Alberti and S. Müller, A new approach to variational problems with multiple scales, Comm. Pure Appl. Math. 54 (2001), 761–825.
    CrossRef    MathSciNet

  4. S. Baldo, Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids, Ann. Inst. H. Poincaré Non Linéare 7 (1990), 67–90.
    CrossRef    MathSciNet

  5. J. M. Ball, A version of the fundamental theorem for Young measures, in: PDEs and continuum models of phase transitions, Springer, Berlin 1989, 207–215.
    CrossRef    MathSciNet

  6. B. Benešová and M. Kružík, Weak lower semicontinuity of integral functionals and application, SIAM Rev. 59 (2017), 703–766.
    CrossRef    MathSciNet

  7. A. Braides, \(\Gamma\)-convergence for beginners, Oxford University Press, Oxford, 2002.
    CrossRef    MathSciNet

  8. A. Bradies, Chapter 2 A handbook of \(\Gamma\)-convergence, in: Handbook of Differential Equations: Stationary Partial Differential Equations, Elsevier, Amsterdam, 2006, 101–213.
    CrossRef

  9. H. Brezis, Analyse fonctionnelle. Théorie et applications, Masson, Paris, 1983.
    MathSciNet

  10. J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys. 28 (1958), 258–267.
    CrossRef

  11. S. Conti, I. Fonseca and G. Leoni, A \(\Gamma\)-convergence result for the two-gradient theory of phase transitions, Comm. Pure Appl. Math. 55 (2002), 857–936.
    CrossRef    MathSciNet

  12. L. C. Evans, R. F. Gariepy, Measure theory and fine properties of functions, CRC Press, Boca Raton, 1992.
    CrossRef    MathSciNet

  13. G. B. Folland, Real analysis, John Wiley and Sons, Inc., New York, 1984.
    MathSciNet

  14. I. Fonseca and G. Leoni, Modern methods in the calculus of variations: \({\rm L}^p\)-spaces, Springer, New York, 2007.
    CrossRef    MathSciNet

  15. I. Fonseca and L. Tartar, The gradient theory of phase transitions for systems with two potential wells, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), 89–102.
    CrossRef    MathSciNet

  16. D. J. H. Garling, A course in mathematical analysis, Vol. I. Foundations and elementary real analysis, Cambridge University Press, Cambridge, 2013.
    CrossRef    MathSciNet

  17. M. Giaquinta and G. Modica, Mathematical analysis. Foundations and advanced techniques for functions of several variables, Birkhauser, Boston, 2012.
    CrossRef

  18. M. E. Gurtin, Some results and conjectures in the gradient theory of phase transitions, in: Metastability and incompletely posed problems, Proc. Workshop, Springer, New York, 1987, 135–146.
    CrossRef    MathSciNet

  19. R. V. Kohn and P. Sternberg, Local minimisers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), 69–84.
    CrossRef    MathSciNet

  20. G. Leoni, A remark on compactness for the Cahn-Hilliard functional, ESAIM COCV 20(2) (2014), 517–523.

  21. G. Leoni, Gamma convergence and applications to phase transitions, CNA Lecture Notes, CMU, in preparation.

  22. G. Leoni, A first course in Sobolev spaces, American Mathematical Society, Providence, 2009.
    CrossRef    MathSciNet

  23. L. Modica and S. Mortola, Un esempio di \(\Gamma\)-convergenca, Bull. Un. Mat. Ital. (5) 14 (1977), 285–299.
    MathSciNet

  24. S. Müller, Singular perturbations as a selection criterion for minimizing sequences, Calc. Var. Partial Differential Equations 1 (1993), 169–204.
    CrossRef    MathSciNet

  25. S. Müller, Variational models for microstructure and phase transitions, Lecture notes, Max Planck Institut für Mathematik in den Naturwissenschaften, 1998.

  26. S. Müller, A sharp version of Zhang's theorem on truncating sequences of gradients, Trans. Amer. Math. Soc. 351 (1999), 4585–4597.
    CrossRef    MathSciNet

  27. N. Hugenbühler, A refinement of Ball's theorem on Young measures, New York J. Math. 3 (1997), 48–53.
    MathSciNet    Link

  28. N. C. Owen, Nonconvex variational problems with general singular perturbations, Tran. Amer. Math. Soc. 310 (1988), 393–404.
    CrossRef    MathSciNet

  29. A. Raguž, Relaxation of Ginzburg-Landau functional with \(1\)-Lipschitz penalizing term in one dimension by Young measures on micropatterns, Asymptot. Anal. 41 (2005), 331–361.
    MathSciNet

  30. A. Raguž, A note on calculation of asymptotic energy for Ginzburg-Landau functional with externally imposed lower-order oscillatory term in one dimension, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 10, bis (2007), 1125–1142.
    MathSciNet

  31. A. Raguž, A result in asymptotic analysis for the functional of Ginzburg-Landau type with externally imposed multiple small scales in one dimension, Glas. Mat. Ser. III 44(64) (2009), 401–421.
    CrossRef    MathSciNet

  32. A. Raguž, Relaxation of Ginzburg-Landau functional perturbed by continuous nonlinear lower-order term in one dimension, Anal. Appl. (Singap.) 13 (2015), 101–123.
    CrossRef    MathSciNet

  33. A. Raguž, On approximation of finite-energy sequences of Müller's functional with non-standard 2-well potential, Proc. Appl. Math. Mech. 17 (2017), 713–714.

  34. P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal. 101 (1988), 209–260.
    CrossRef    MathSciNet

  35. W. P. Ziemer, Weakly differentiable functions, Springer-Verlag, New York, 1989.
    CrossRef    MathSciNet
Glasnik Matematicki Home Page