Abstract
This paper discusses the asymptotic behavior as ɛ → 0+ of the chemical potentials λɛ associated with solutions of variational problems within the Van der Waals-Cahn-Hilliard theory of phase transitions in a fluid with free energy, per unit volume, given by ɛ2¦▽ϱ¦2+ W(ϱ), where ϱ is the density. The main result is that λɛ is asymptotically equal to ɛEλ/d+o(ɛ), with E the interfacial energy, per unit surface area, of the interface between phases, λ the (constant) sum of principal curvatures of the interface, and d the density jump across the interface. This result is in agreement with a formula conjectured by M. Gurtin and corresponds to the Gibbs-Thompson relation for surface tension, proved by G. Caginalp within the context of the phase field model of free boundaries arising from phase transitions.
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References
N. D. Alikakos & P. W. Bates, On the singular limit in a phase field model of phase transitions. Ann. Inst. H. Poincaré, Analyse Non-linéaire, (1988).
H. Attouch, Variational Convergence for Functions and Operators. Pitman, London, 1984.
G. Caginalp, An Analysis of a Phase Field Model of a Free Boundary. Arch. Rational Mech. Anal. 92 (1986) 205–245.
G. Caginalp & P. Fife, Elliptic Problems Involving Phase Boundaries Satisfying a Curvature Condition. IMA Journal of Appl. Math. 38 (1987) 195–217.
E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhäuser Verlag, Basel, 1984.
E. Gonzalez, U. Massari & I. Tamanini, On the regularities of boundaries of sets minimizing perimeter with a volume constraint. Indiana Univ. Math. J. 32 (1983) 25–37.
M. E. Gurtin, Some Results and Conjectures in the Gradient Theory of Phase Transitions. Inst. for Math. and Its Applications, University of Minnesota, Preprint # 156, 1985.
M. E. Gurtin & H. Matano, On the Structure of Equilibrium Phase Transitions within the Gradient Theory of Fluids. Quarterly of Appl. Math. 46 (1988) 301–317.
U. Massari & M. Miranda, Minimal Surfaces of Codimension One. Mathematics Studies 91, North-Holland, Amsterdam, 1984.
L. Modica, The Gradient Theory of Phase Transitions and the Minimal Interface Criterion. Arch. Rational Mech. Anal. 98 (1986) 123–142.
L. Modica, Gradient Theory of Phase Transitions with Boundary Contact Energy. Ann. Inst. H. Poincaré, Analyse Nonlinéaire, 4 (1987) 487–512.
Yu. G. Reshetnyak, Weak Convergence of Completely Additive Vector Functions on a Set. Siberian Math. J. 9 (1968) 1039–1045; translated from Sibirskii Mathematicheskii Zhurnal 9 (1968) 1386–1394.
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Communicated by M. E. Gurtin
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Luckhaus, S., Modica, L. The Gibbs-Thompson relation within the gradient theory of phase transitions. Arch. Rational Mech. Anal. 107, 71–83 (1989). https://doi.org/10.1007/BF00251427
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DOI: https://doi.org/10.1007/BF00251427