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Deterministic equivalent for the Allen-Cahn energy of a scaling law in the Ising model

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Abstract

For the Allen-Cahn functional we study the following problem: for which prescribed amount m of volume is there the appearence of a droplet of one phase inside the other? Under a suitable assumption on the domain we show that the breaking of symmetry occurs at the same value of m as for the limit of the sharp interface energy. We also prove that there exists a threshold for m of order \(\varepsilon^\frac{n}{n+1}\) so that either there is the appearence of the droplet or there is no breaking of symmetry.

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References

  1. Berard, P., Meyer, D.: Inégalités isopérimétriques et applications. Ann. Sci. Ecole Norm. Sup. 15(4), 513–541 (1982)

    Google Scholar 

  2. Biskup, M., Chayes, L., Kotecky, R.: Critical region for droplet formation in the two-dimensional Ising model. Comm. Math. Phys., to appear

  3. Braides, A., Truskinovsky, L.: Γ-convergence of parametrized functionals. Preprint (2005)

  4. Carlen, E., Carvahlo, C., Esposito, R., Lebowitz, J.L., Marra, R.: Droplet minimizers for the Cahn–Hilliard free energy functional. Preprint (2005)

  5. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton (1992)

    MATH  Google Scholar 

  6. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, Grundleheren der Mathematischen Wissenschaften, vol. 224. Spinger-Verlag, Berlin New York (1977)

    Google Scholar 

  7. Gurtin, M., Matano, H.: On the structure of equilibrium phase transitions within the gradient theory of fluids. Quart. Appl. Math., 46, 301–317 (1988)

    MATH  MathSciNet  Google Scholar 

  8. Ioffe, D., Shlosman, S., Velenik, Y.: 2D models of statistical physics with continuous symmetry: the case of singular interactions. Comm. Math. Phys. 226 433–454 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kawohl, B.: Rearrangements and Convexity of Level Sets in PDE. Lecture Notes in Mathematics, Springer-Verlag, Berlin (1985)

  10. Modica, L., Mortola, S.: Un esempio di Γ-convergenza, Boll. Un. Mat. Ital. A, 14(5) 285–299 (1977)

    MATH  MathSciNet  Google Scholar 

  11. Niethammer, B.: Existence and uniqueness of radially symmetric stationary points within the gradient theory of phase transitions. Eur. J. Appl. Math. 6 45–67 (1995)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133, Roma, Italy

    G. Bellettini

  2. Dipartimento di Matematica, Università di Pisa, largo Pontecorvo 5, 56216, Pisa, Italy

    M. S. Gelli

  3. Universitat Leipzig, Fakultät für Mathematik und Informatik, Mathematisches Institut, Augustusplatz 10/11, 04109, Leipzig, Germany

    S. Luckhaus

  4. Dipartimento di Matematica, Università di Pisa, largo Pontecorvo 5, 56216, Pisa, Italy

    M. Novaga

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  1. G. Bellettini
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  2. M. S. Gelli
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  3. S. Luckhaus
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  4. M. Novaga
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Correspondence to G. Bellettini.

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Bellettini, G., Gelli, M.S., Luckhaus, S. et al. Deterministic equivalent for the Allen-Cahn energy of a scaling law in the Ising model. Calc. Var. 26, 429–445 (2006). https://doi.org/10.1007/s00526-006-0012-6

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  • DOI: https://doi.org/10.1007/s00526-006-0012-6

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