Asymptotic expansions by Γ-convergence
Asymptotic expansions by Γ-convergence
复制标题
通过 Γ 收敛进行渐近展开
DOI：
10.1007/s00161-008-0072-2
复制
发表时间：
2008
影响因子：
2.6
通讯作者：
L. Truskinovsky
中科院分区：
文献类型：
--
作者：
Andrea Braides;L. Truskinovsky
我们的出发点是一个参数化的家庭的功能（一个“理论”），我们有兴趣在近似的全球最小值的能量时，这些参数之一为零。我们的目标是开发一套越来越准确的渐近变分模型，允许处理的情况下，这个参数是“小”，但有限的。由于在“理论”中Γ-收敛可能是非一致的，我们提出了一个寻找一致逼近的问题。为了实现这一目标，我们提出了一种方法的基础上纠正的奇异点在参数空间中使用爆破参数，然后渐近匹配这些点周围的近似与正规近似远离他们。我们说明的主要思想与物理意义的例子，涵盖了广泛的主题，从均匀化和降维断裂和相变。特别是，我们给予相当大的关注，从离散到连续的过渡时，内部和外部的尺度没有很好地分离的问题，和一个必须处理所谓的“大小”或“规模”的影响。
Our starting point is a parameterized family of functionals (a ‘theory’) for which we are interested in approximating the global minima of the energy when one of these parameters goes to zero. The goal is to develop a set of increasingly accurate asymptotic variational models allowing one to deal with the cases when this parameter is ‘small’ but finite. Since Γ-convergence may be non-uniform within the ‘theory’, we pose a problem of finding a uniform approximation. To achieve this goal we propose a method based on rectifying the singular points in the parameter space by using a blow-up argument and then asymptotically matching the approximations around such points with the regular approximation away from them. We illustrate the main ideas with physically meaningful examples covering a broad set of subjects from homogenization and dimension reduction to fracture and phase transitions. In particular, we give considerable attention to the problem of transition from discrete to continuum when the internal and external scales are not well separated, and one has to deal with the so-called ‘size’ or ‘scale’ effects.