Abstract
Given thatα, β are two Lipschitz continuous functions of Ω to ℝ+ and thatf(x, u, p) is a continuous function of\(\bar \Omega \) × ℝ × ℝN to [0, + ∞[ such that, for everyx, f(x,·, 0) reaches its minimum value 0 at exactly two pointsα(x) andβ(x), we prove the convergence ofFε(u) = (1/ε)∫Ωf (x, u, εDu) dx when the perturbation parameterε goes to zero. A formula is given for the limit functional and a general minimal interface criterium is deduced for a wide class of two-phase transition models. Earlier results of [19], [21], and [22] are extended with new proofs.
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References
L. Ambrosio. In preparation and private communication, Pise (1988).
S. S. Antman. Nonuniqueness of equilibrium states for bars in tension, J. Math. Appl., 44 (1973), 333.
H. Attouch. Variational Convergence for Functions and Operators, Appl. Math. Series, Pitman, Boston, (1984).
S. Baldo. Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids, Ann. Inst. H. Poincaré, to appear.
G. Bouchitte and M. Valadier. Multifonctions s.c.i. et régularisée s.c.i. essentielle. Fonctions de mesures dans le cas sous-linéaire, Analyse Non Linéaire (Contributions en l'honneur de J. J. Moreau), Gauthier-Villars, Paris (1989).
J. W. Cahn and J. E. Hilliard. J. Chem. Phys., 31 (1959), 688–699.
J. Carr, M. E. Gurtin, and M. Slemrod. Structure phase transitions on a finite interval, Arch. Rational Mech. Anal., 86 (1984), 317–351.
P. Cascal and H. Gouin. Représentation des interfaces liquide-vapeur à l'aide des fluides thermocapillaires, Séminaire de Mécanique Théorique, Novembre 1987, Université de Paris VI.
P. Casal and H. Gouin. Sur les interfaces liquide-vapeur, J. Méc. Théor. Appl., 7 (6) (1988), 689–718.
G. Dal Maso. Integral representation onBV(Ω) of Γ-limit of variational integrals, Manuscripta Math., 30 (1980), 387–416.
E. De Giorgi. Su una teoria generale della misure (r − 1)-dimensionali in uno spazio ar dimensioni, Ann. Mat. Pura Appl. (4), 36 (1954), 191–213.
E. De Giorgi. Convergence problems for functionals and operators, Proc. Internat. Meeting on Recent Methods in Nonlinear Analysis, eds. De Giorgi, Magenes, Mosco Pitagora, Bologne (1979), pp. 131–188.
D. Gilbarg and N. S. Trudinger. Elliptic Differential Equations of Second Order, Springer-Verlag, Berlin (1977).
E. Giusti. Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Basel (1984).
C. Goffman and J. Serrin. Sublinear functions of measures and variational integrals, Duke Math. J., 31 (1964), 159–178.
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).
M. E. Gurtin. On phase transitions with bulk, interfacial and boundary energy, Arch. Rational Mech. Anal., 96 (1986), 242–264.
U. Massari and M. Miranda. Minimal Surfaces of Codimension One, Math. Studies 91, North Holland, Amsterdam (1984).
L. Modica. Gradient theory of phase transitions and minimal interface criterium, Arch. Rational Mech. Anal., 98 (1987), 123–142.
L. Modica and S. Mortola. Un esempio di Γ-convergenza, Boll. Un. Mat. Ital. B (5) 14 (1977), 285–299.
N. Owen. Nonconvex variational problems with general singular perturbations, Preprint Trans. A. M. S., to appear.
N. Owen and P. Sternberg. Nonconvex problems with anisotropic perturbations, Preprint (1988).
Y. G. Reschetniak. Weak convergence of completely additive vector measures on a set, Sibirsk. Mat. Zh., (1968), 1386–1394.
P. Sternberg. The effect of a singular perturbation on nonconvex variational problems, Ph.D. Thesis, New York University (1986).
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Communicated by R. Conti
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Bouchitte, G. Singular perturbations of variational problems arising from a two-phase transition model. Appl Math Optim 21, 289–314 (1990). https://doi.org/10.1007/BF01445167
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DOI: https://doi.org/10.1007/BF01445167