A nonlocal anisotropic model for phase transitionsPart II: Asymptotic behaviour of rescaled energies
Giovanni Alberti, Giovanni Bellettini · 1997
We study the asymptotic behaviour as " ! 0, of the nonlocal models for phase transition described by the scaled free energy F " (u) := 1 4" Z \\Omega \\Theta\\Omega J " (x 0 \\Gamma x) \\Gamma u(x 0 ) \\Gamma u(x) \\Delta 2 dx 0 dx + 1 " Z \\Omega W \\Gamma u(x) \\Delta dx ; where u is a scalar density function, W is a double-well potential which vanishes at \\Sigma1, J is a non-negative interaction potential and J " (h) := " \\GammaN J(h="). We prove that the functionals F " converge in a variational sense to the anisotropic surface energy F (u) := Z Su oe( u ) ; where u is allowed to take the values \\Sigma1 only, u is the normal to the interface Su between the phases fu = +1g and fu = \\Gamma1g, and oe is the surface tension. This paper concludes the analisys started in [AB].