Higher-order phase transitions with line-tension effect

Galvão-Sousa, Bernardo

ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 3, pp. 603-647.

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. 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

$${ℱ}_{\epsilon }\left(u\right):={\epsilon }^3{\int }_{\Omega }{\left|D^{2}u\right|}^2+\frac{1}{\epsilon }{\int }_{\Omega }W\left(u\right)+{\lambda }_{\epsilon }{\int }_{\partial \Omega }V\left(Tu\right),$$

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 $\epsilon {\lambda }_{\epsilon }^{\frac{2}{3}}\sim 1$.

Classification : 49Q20, 49J45, 58E50, 76M30
Mots clés : gamma limit, functions of bounded variations, functions of bounded variations on manifolds, phase transitions

[1] R. Adams, Sobolev Spaces. Academic Press (1975). | MR | Zbl

[2] G. Alberti, G. Bouchitté and P. Seppecher, Phase transition with the line tension effect. Arch. Rational Mech. Anal. 144 (1998) 1-46. | MR | Zbl

[3] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Clarendon Press, Oxford (2000). | MR | Zbl

[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. | MR | Zbl

[5] R. Choksi and R. Kohn, Bounds on the micromagnetic energy of a uniaxial ferromagnet. Comm. Pure Appl. Math. 51 (1998) 259-289. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[10] G. Dal Maso, An Introduction to Γ-Convergence. Birkhäuser (1993). | MR | Zbl

[11] E. Dibenedetto, Real Analysis. Birkhäuser (2002). | MR | Zbl

[12] L. Evans and R. Gariepy, Measure Theory and fine Properties of Functions. CRC Press (1992). | MR | Zbl

[13] I. Fonseca and G. Leoni, Modern methods in the calculus of variations: Lp spaces, Springer Monographs in Mathematics. Springer (2007). | MR | Zbl

[14] I. Fonseca and C. Mantegazza, Second order singular perturbation models for phase transitions. SIAM J. Math. Anal. 31 (2000) 1121-1143. | MR | Zbl

[15] E. Gagliardo, Ulteriori prorietà di alcune classi di funzioni in più variabili. Ric. Mat. 8 (1959) 24-51. | MR | Zbl

[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. | MR | Zbl

[17] E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhäuser (1984). | MR | Zbl

[18] E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities. AMS/CIMS (1999). | MR | Zbl

[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. | MR | Zbl

[20] L. Modica, The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98 (1987) 123-142. | MR | Zbl

[21] L. Modica, The gradient theory of phase transitions with boundary contact energy. Ann. Inst. Henri Poincaré, Anal. non linéaire 4 (1987) 487-512. | Numdam | MR | Zbl

[22] L. Modica and S. Mortola, Un esempio de Γ--convergenza. Boll. Un. Mat. Ital. B 14 (1977) 285-299. | MR | Zbl

[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. | MR | Zbl

[24] L. Nirenberg, An extended interpolation inequality. Ann. Sc. Normale Pisa - Scienze fisiche e matematiche 20 (1966) 733-737. | Numdam | MR | Zbl

[25] E. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton University Press (1970). | MR | Zbl

[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. | MR | Zbl

[27] W. Ziemer, Weakly differentiable functions - Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics 120. Springer-Verlag, New York (1989). | MR | Zbl