A nonlocal anisotropic model for phase transitions. Part II: Asymptotic behaviour of rescaled energies

Giovanni Alberti and Giovanni Bellettini

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Submission date: 22. Apr. 1997
Pages: 28
published in: European journal of applied mathematics, 9 (1998) 3, p. 261-284 
DOI number (of the published article): 10.1017/S0956792598003453
Bibtex
MSC-Numbers: 49J45, 49N45, 82B24
Keywords and phrases: phrase transitions, singular perturbations, gamma-convergence, nonlocal integral functionals
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Abstract:
In this paper we study the asymptotic behaviour in the thermodynamic limit of a non local model for phase separation. More precisely we consider the free energy obtained by replacing the usual gradient term in the Cahn-Hilliard model with an interaction energy associated to a positive non-isotropic interaction potential, and then we show that the limit in the sense of Gamma-convergence of suitable rescalings of this energy leads to the classical (non-isotropic) model for surface tension. These functionals appear in statistical mechanics as free energies of Ising systems with Kac potential.