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The gradient theory of phase transitions and the minimal interface criterion

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References

  1. N. D. Alikakos & K. C, Shaing. On the singular limit for a class of problems modelling phase transitions. To appear.

  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.

    Google Scholar 

  4. H. Attouch. Variational Convergence for Functions and Operators. Appl. Math. Series, Pitman Adv. Publ. Program, Boston, London, Melbourne, 1984.

  5. J. Carr, M. E. Gurtin & M. Slemrod. Structured phase transitions on a finite interval. Arch. Rational Mech. Anal., 86 (1984), 317–351.

    Google Scholar 

  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.

    Google Scholar 

  7. E. de Giorgi. Nuovi teoremi relativi alle misure (r — 1)-dimensionali in uno spazio a r dimensioni. Ricerche Mat., 4 (1955), 95–113.

    Google Scholar 

  8. E. de Giorgi. Sulla convergenza di alcune successioni di integrali del tipo dell'area. Rendiconti di Matematica. (4) 8 (1975), 277–294.

    Google Scholar 

  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.

    Google Scholar 

  10. H. Federer. Geometric Measure Theory. Springer-Verlag, Berlin, Heidelberg, New York, 1968.

    Google Scholar 

  11. W. H. Fleming & R. W. Rishel. An integral formula for total gradient variation. Arch. Math., 11 (1960), 218–222.

    Google Scholar 

  12. D. Gilbarg & N. S. Trudinger. Elliptic Partial Differential Equations of Second-Order. Springer-Verlag, Berlin, Heidelberg, New York, 1977.

    Google Scholar 

  13. E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhäuser Verlag, Basel, Boston, Stuttgart, 1984.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  19. U. Massari & M. Miranda. Minimal Surfaces of codimension one. North-Holland Math. Studies 91, North-Holland, Amsterdam, New York, Oxford, 1984.

    Google Scholar 

  20. L. Modica & S. Mortola. Un esempio di Γ-convergenza. Boll. Un. Mat. Ital., (5) 14-B (1977), 285–299.

    Google Scholar 

  21. L. Modica & S. Mortola. The Γ-convergence of some functional. Istituto Matematico “Leonida Tonelli”, Università di Pisa, preprint n. 77-7 (1977).

  22. A. Novick-Cohen & L. A. Segel. Nonlinear aspects of the Cahn-Hilliard equation. Physica, 10-D (1984), 278–298.

    Google Scholar 

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Modica, L. The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98, 123–142 (1987). https://doi.org/10.1007/BF00251230

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