F Ur Mathematik in Den Naturwissenschaften Leipzig a Nonlocal Anisotropic Model for Phase Transitions Part Ii: Asymptotic Behaviour of Rescaled Energies a Nonlocal Anisotropic Model for Phase Transitions
F Ur Mathematik in Den Naturwissenschaften Leipzig a Nonlocal Anisotropic Model for Phase Transitions Part Ii: Asymptotic Behaviour of Rescaled Energies a Nonlocal Anisotropic Model for Phase Transitions
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F Ur Mathematik in Den Naturwissenschaften Leipzig 相变的非局部各向异性模型第二部分：重标度能量的渐近行为相变的非局部各向异性模型
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通讯作者：
G. Bellettini
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作者：
Giovanni Alberti;G. Bellettini
我们研究由缩放自由能 F " (u) := 1 4" Z J " (x 0 ? x) ? 描述的非局域相变模型的渐近行为“! 0” u(x 0) ? u(x) 2 dx 0 dx + 1 " Z W ? u(x) dx ；其中 u 是标量密度函数，W 是在 1 处消失的双阱势，J 是非负相互作用势， J " (h) := " ?N J(h=")。我们证明泛函 F " 在变分意义上收敛于各向异性表面能 F(u) := Z Su (u) ；其中 u 只允许取值 1，u 是相 fu = +1g 和 fu = ?1g 之间界面 Su 的法线，并且 是表面张力。本文总结了从 AB] 开始的分析。
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 J " (x 0 ? x) ? u(x 0) ? u(x) 2 dx 0 dx + 1 " Z W ? u(x) dx ; where u is a scalar density function, W is a double-well potential which vanishes at 1, J is a non-negative interaction potential and J " (h) := " ?N J(h="). We prove that the functionals F " converge in a variational sense to the anisotropic surface energy F(u) := Z Su (u) ; where u is allowed to take the values 1 only, u is the normal to the interface Su between the phases fu = +1g and fu = ?1g, and is the surface tension. This paper concludes the analisys started in AB].