Higher-order phase transitions with line-tension effect

Bernardo Galvão-Sousa

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 3, page 603-647
  • ISSN: 1292-8119

Abstract

top
The behavior of energy minimizers at the boundary of the domain is of great importance in the Van de Waals-Cahn-Hilliard theory for fluid-fluid phase transitions, since it describes the effect of the container walls on the configuration of the liquid. This problem, also known as the liquid-drop problem, was studied by Modica in [Ann. Inst. Henri Poincaré, Anal. non linéaire4 (1987) 487–512], and in a different form by Alberti et al. in [Arch. Rational Mech. Anal.144 (1998) 1–46] for a first-order perturbation model. This work shows that using a second-order perturbation Cahn-Hilliard-type model, the boundary layer is intrinsically connected with the transition layer in the interior of the domain. Precisely, considering the energieswhere u is a scalar density function and W and V are double-well potentials, the exact scaling law is identified in the critical regime, when .

How to cite

top

Galvão-Sousa, Bernardo. "Higher-order phase transitions with line-tension effect." ESAIM: Control, Optimisation and Calculus of Variations 17.3 (2011): 603-647. <http://eudml.org/doc/221903>.

@article{Galvão2011,
abstract = { The behavior of energy minimizers at the boundary of the domain is of great importance in the Van de Waals-Cahn-Hilliard theory for fluid-fluid phase transitions, since it describes the effect of the container walls on the configuration of the liquid. This problem, also known as the liquid-drop problem, was studied by Modica in [Ann. Inst. Henri Poincaré, Anal. non linéaire4 (1987) 487–512], and in a different form by Alberti et al. in [Arch. Rational Mech. Anal.144 (1998) 1–46] for a first-order perturbation model. This work shows that using a second-order perturbation Cahn-Hilliard-type model, the boundary layer is intrinsically connected with the transition layer in the interior of the domain. Precisely, considering the energies$\[ \{\cal F\}_\{\varepsilon\}(u) := \varepsilon^\{3\} \int_\{\Omega\} |D^\{2\}u|^\{2\} + \frac\{1\}\{\varepsilon\} \int_\{\Omega\} W (u) + \lambda_\{\varepsilon\} \int_\{\partial \Omega\} V(Tu), \]$where u is a scalar density function and W and V are double-well potentials, the exact scaling law is identified in the critical regime, when $\varepsilon\lambda_\{\varepsilon\}^\{\frac\{2\}\{3\}\} \sim 1$. },
author = {Galvão-Sousa, Bernardo},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Gamma limit; functions of bounded variations; functions of bounded variations on manifolds; phase transitions; limit},
language = {eng},
month = {8},
number = {3},
pages = {603-647},
publisher = {EDP Sciences},
title = {Higher-order phase transitions with line-tension effect},
url = {http://eudml.org/doc/221903},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Galvão-Sousa, Bernardo
TI - Higher-order phase transitions with line-tension effect
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2011/8//
PB - EDP Sciences
VL - 17
IS - 3
SP - 603
EP - 647
AB - The behavior of energy minimizers at the boundary of the domain is of great importance in the Van de Waals-Cahn-Hilliard theory for fluid-fluid phase transitions, since it describes the effect of the container walls on the configuration of the liquid. This problem, also known as the liquid-drop problem, was studied by Modica in [Ann. Inst. Henri Poincaré, Anal. non linéaire4 (1987) 487–512], and in a different form by Alberti et al. in [Arch. Rational Mech. Anal.144 (1998) 1–46] for a first-order perturbation model. This work shows that using a second-order perturbation Cahn-Hilliard-type model, the boundary layer is intrinsically connected with the transition layer in the interior of the domain. Precisely, considering the energies$\[ {\cal F}_{\varepsilon}(u) := \varepsilon^{3} \int_{\Omega} |D^{2}u|^{2} + \frac{1}{\varepsilon} \int_{\Omega} W (u) + \lambda_{\varepsilon} \int_{\partial \Omega} V(Tu), \]$where u is a scalar density function and W and V are double-well potentials, the exact scaling law is identified in the critical regime, when $\varepsilon\lambda_{\varepsilon}^{\frac{2}{3}} \sim 1$.
LA - eng
KW - Gamma limit; functions of bounded variations; functions of bounded variations on manifolds; phase transitions; limit
UR - http://eudml.org/doc/221903
ER -

References

top
  1. R. Adams, Sobolev Spaces. Academic Press (1975).  
  2. G. Alberti, G. Bouchitté and P. Seppecher, Phase transition with the line tension effect. Arch. Rational Mech. Anal.144 (1998) 1–46.  
  3. L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Clarendon Press, Oxford (2000).  
  4. J. Ball, A version of the fundamental theorem for Young measures, in PDEs and continuum models of phase transitions (Nice, 1988), Lecture Notes in Phys.344, Springer, Berlin (1989) 207–215.  
  5. R. Choksi and R. Kohn, Bounds on the micromagnetic energy of a uniaxial ferromagnet. Comm. Pure Appl. Math.51 (1998) 259–289.  
  6. R. Choksi, R. Kohn and F. Otto, Domain branching in uniaxial ferromagnets: a scaling law for the minimum energy. Comm. Math. Phys.201 (1999) 61–79.  
  7. R. Choksi, R. Kohn and F. Otto, Energy minimization and flux domain structure in the intermediate state of a type-I superconductor. J. Nonlinear Sci.14 (2004) 119–171.  
  8. R. Choksi, S. Conti, R. Kohn and F. Otto, Ground state energy scaling laws during the onset and destruction of the intermediate state in a type I superconductor. Comm. Pure Appl. Math.61 (2008) 595–626.  
  9. S. Conti, I. Fonseca and G. Leoni, A Γ-convergence result for the two-gradient theory of phase transitions. Comm. Pure Applied Math.55 (2002) 857–936.  
  10. G. Dal Maso, An Introduction to Γ-Convergence. Birkhäuser (1993).  
  11. E. DiBenedetto, Real Analysis. Birkhäuser (2002).  
  12. L. Evans and R. Gariepy, Measure Theory and fine Properties of Functions. CRC Press (1992).  
  13. I. Fonseca and G. Leoni, Modern methods in the calculus of variations: Lp spaces, Springer Monographs in Mathematics. Springer (2007).  
  14. I. Fonseca and C. Mantegazza, Second order singular perturbation models for phase transitions. SIAM J. Math. Anal.31 (2000) 1121–1143.  
  15. E. Gagliardo, Ulteriori prorietà di alcune classi di funzioni in più variabili. Ric. Mat.8 (1959) 24–51.  
  16. A. Garroni and G. Palatucci, A singular perturbation result with a fractional norm, in Variational problems in materials science, Progr. Nonlinear Differential Equations Appl.68, Birkhäuser, Basel (2006) 111–126.  
  17. E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhäuser (1984).  
  18. E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities. AMS/CIMS (1999).  
  19. M. Miranda, D. Pallara, F. Paronetto and M. Preunkert, Heat semigroup and functions of bounded variation on Riemannian manifolds. J. Reine Angew. Math.613 (2007) 99–119.  
  20. L. Modica, The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal.98 (1987) 123–142.  
  21. L. Modica, The gradient theory of phase transitions with boundary contact energy. Ann. Inst. Henri Poincaré, Anal. non linéaire4 (1987) 487–512.  
  22. L. Modica and S. Mortola, Un esempio de Γ--convergenza. Boll. Un. Mat. Ital. B14 (1977) 285–299.  
  23. S. Müller, Variational models for microstructure and phase transitions, in Calculus of variations and geometric evolution problems (Cetraro, 1996), Lecture Notes in Math.1713, Springer (1999) 85–210.  
  24. L. Nirenberg, An extended interpolation inequality. Ann. Sc. Normale Pisa - Scienze fisiche e matematiche20 (1966) 733–737.  
  25. E. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton University Press (1970).  
  26. L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear analysis and mechanics: Heriot-Watt SymposiumIV, Res. Notes in Math.39, Pitman, Boston (1979) 136–212.  
  27. W. Ziemer, Weakly differentiable functions – Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics120. Springer-Verlag, New York (1989).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.