The gradient theory of phase transitions and the minimal interface criterion. (English) Zbl 0616.76004

Some conjectures concerning the Van der Waals-Cahn-Hilliard theory of phase transitions are proved. The main result is the following: to recover, in the frame of this theory, the physical reasonable criterion that the interface has minimal area one has to model interfacial energy by the dependence on the density gradient.
Reviewer: V.A.Zagrebnov


76A02 Foundations of fluid mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics
49S05 Variational principles of physics
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[1] N. D. Alikakos & K. C, Shaing. On the singular limit for a class of problems modelling phase transitions. To appear. · Zbl 0649.34055
[2] F. Almgren & M. E. Gurtin. To appear.
[3] G. Anzellotti & M. Giaquinta. Funzioni BV e tracce. Rend. Sem. Mat. Univ.Padova, 60 (1978), 1–22. · Zbl 0432.46031
[4] H. Attouch. Variational Convergence for Functions and Operators. Appl. Math. Series, Pitman Adv. Publ. Program, Boston, London, Melbourne, 1984. · Zbl 0561.49012
[5] J. Carr, M. E. Gurtin & M. Slemrod. Structured phase transitions on a finite interval. Arch. Rational Mech. Anal., 86 (1984), 317–351. · Zbl 0564.76075 · doi:10.1007/BF00280031
[6] E. de Giorgi. Su una teoria generale della misura (r – 1)-dimensionale in uno spazio a r dimensioni. Ann. Mat. Pura Appl., (4) 36 (1954), 191–213. · Zbl 0055.28504 · doi:10.1007/BF02412838
[7] E. de Giorgi. Nuovi teoremi relativi alle misure (r – 1)-dimensionali in uno spazio a r dimensioni. Ricerche Mat., 4 (1955), 95–113. · Zbl 0066.29903
[8] E. de Giorgi. Sulla convergenza di alcune successioni di integrali del tipo dell’area. Rendiconti di Matematica. (4) 8 (1975), 277–294. · Zbl 0316.35036
[9] E. de Giorgi & T. Franzoni. Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei, Rend. Cl. Sc. Mat. Fis. Natur., (8) 58 (1975), 842–850. · Zbl 0339.49005
[10] H. Federer. Geometric Measure Theory. Springer-Verlag, Berlin, Heidelberg, New York, 1968. · Zbl 0176.00801
[11] W. H. Fleming & R. W. Rishel. An integral formula for total gradient variation. Arch. Math., 11 (1960), 218–222. · Zbl 0094.26301 · doi:10.1007/BF01236935
[12] D. Gilbarg & N. S. Trudinger. Elliptic Partial Differential Equations of Second-Order. Springer-Verlag, Berlin, Heidelberg, New York, 1977. · Zbl 0361.35003
[13] E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhäuser Verlag, Basel, Boston, Stuttgart, 1984. · Zbl 0545.49018
[14] E. Gonzalez, U. Massari & I. Tamanini. On the regularity of boundaries of sets minimizing perimeter with a volume constraint. Indiana Univ. Math. J., 32 (1983), 25–37. · Zbl 0486.49024 · doi:10.1512/iumj.1983.32.32003
[15] M. E. Gurtin. Some results and conjectures in the gradient theory of phase transitions. Institute for Mathematics and Its Applications, University of Minnesota, preprint n. 156 (1985).
[16] M. E. Gurtin. On phase transitions with bulk, interfacial, and boundary energy. Arch. Rational Mech. Anal., 96 (1986), 243–264. · doi:10.1007/BF00251908
[17] M. E. Gurtin & H. Matano. On the structure of equilibrium phase transitions within the gradient theory of fluids. To appear.
[18] M. Marcus & V. J. Mizel. Nemitsky Operators on Sobolev Spaces. Arch. Rational Mech. Anal., 51 (1973), 347–370. · Zbl 0266.46029 · doi:10.1007/BF00263040
[19] U. Massari & M. Miranda. Minimal Surfaces of codimension one. North-Holland Math. Studies 91, North-Holland, Amsterdam, New York, Oxford, 1984. · Zbl 0565.49030
[20] L. Modica & S. Mortola. Un esempio di {\(\Gamma\)}–convergenza. Boll. Un. Mat. Ital., (5) 14-B (1977), 285–299.
[21] L. Modica & S. Mortola. The {\(\Gamma\)}-convergence of some functional. Istituto Matematico ”Leonida Tonelli”, Università di Pisa, preprint n. 77-7 (1977). · Zbl 0356.49008
[22] A. Novick-Cohen & L. A. Segel. Nonlinear aspects of the Cahn-Hilliard equation. Physica, 10-D (1984), 278–298.
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