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 linéaire 4 (1987) 487–512], and in a different form by Alberti et al. in [Arch. Rational Mech. Anal.u is a scalar density function and W and Vare double-well potentials, the exact scaling law is identified in the critical regime, when ε λ ε 2 3 ∼ 1 .

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/272912>.

@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 linéaire 4 (1987) 487–512], and in a different form by Alberti et al. in [Arch. Rational Mech. Anal.u is a scalar density function and W and Vare 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},
number = {3},
pages = {603-647},
publisher = {EDP-Sciences},
title = {Higher-order phase transitions with line-tension effect},
url = {http://eudml.org/doc/272912},
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
PY - 2011
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 linéaire 4 (1987) 487–512], and in a different form by Alberti et al. in [Arch. Rational Mech. Anal.u is a scalar density function and W and Vare 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/272912
ER -

References

top
  1. [1] R. Adams, Sobolev Spaces. Academic Press (1975). Zbl1098.46001MR450957
  2. [2] G. Alberti, G. Bouchitté and P. Seppecher, Phase transition with the line tension effect. Arch. Rational Mech. Anal.144 (1998) 1–46. Zbl0915.76093MR1657316
  3. [3] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Clarendon Press, Oxford (2000). Zbl0957.49001MR1857292
  4. [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. Zbl0991.49500MR1036070
  5. [5] R. Choksi and R. Kohn, Bounds on the micromagnetic energy of a uniaxial ferromagnet. Comm. Pure Appl. Math.51 (1998) 259–289. Zbl0909.49004MR1488515
  6. [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. Zbl1023.82011MR1669433
  7. [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. Zbl1136.82044MR2041429
  8. [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. Zbl1142.82031MR2388657
  9. [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. Zbl1029.49040MR1894158
  10. [10] G. Dal Maso, An Introduction to Γ-Convergence. Birkhäuser (1993). Zbl0816.49001MR1201152
  11. [11] E. DiBenedetto, Real Analysis. Birkhäuser (2002). Zbl1012.26001MR1897317
  12. [12] L. Evans and R. Gariepy, Measure Theory and fine Properties of Functions. CRC Press (1992). Zbl0804.28001MR1158660
  13. [13] I. Fonseca and G. Leoni, Modern methods in the calculus of variations: Lp spaces, Springer Monographs in Mathematics. Springer (2007). Zbl1153.49001MR2341508
  14. [14] I. Fonseca and C. Mantegazza, Second order singular perturbation models for phase transitions. SIAM J. Math. Anal.31 (2000) 1121–1143. Zbl0958.49007MR1759200
  15. [15] E. Gagliardo, Ulteriori prorietà di alcune classi di funzioni in più variabili. Ric. Mat.8 (1959) 24–51. Zbl0199.44701MR109295
  16. [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. Zbl1107.82019MR2223366
  17. [17] E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhäuser (1984). Zbl0545.49018MR775682
  18. [18] E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities. AMS/CIMS (1999). Zbl0981.58006MR1688256
  19. [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. Zbl1141.58014MR2377131
  20. [20] L. Modica, The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal.98 (1987) 123–142. Zbl0616.76004MR866718
  21. [21] L. Modica, The gradient theory of phase transitions with boundary contact energy. Ann. Inst. Henri Poincaré, Anal. non linéaire 4 (1987) 487–512. Zbl0642.49009MR921549
  22. [22] L. Modica and S. Mortola, Un esempio de Γ--convergenza. Boll. Un. Mat. Ital. B14 (1977) 285–299. Zbl0356.49008MR445362
  23. [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. Zbl0968.74050MR1731640
  24. [24] L. Nirenberg, An extended interpolation inequality. Ann. Sc. Normale Pisa - Scienze fisiche e matematiche20 (1966) 733–737. Zbl0163.29905MR208360
  25. [25] E. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton University Press (1970). Zbl0207.13501MR290095
  26. [26] L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear analysis and mechanics: Heriot-Watt Symposium IV, Res. Notes in Math. 39, Pitman, Boston (1979) 136–212. Zbl0437.35004MR584398
  27. [27] W. Ziemer, Weakly differentiable functions – Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics 120. Springer-Verlag, New York (1989). Zbl0692.46022MR1014685

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.