含表面能项的材料微结构模型中的局部极小元和低能路径
Local minimizers and low energy paths in a model of material microstructure with a surface energy term
摘要：
一个家庭的积分泛函F，在一个简化的方式，模型材料的微观结构占据一个二维域D，并考虑到表面能和一个可变的井深进行了研究。结果表明，存在一个临界阱深，给出了临界阱深与表面能密度和畴尺寸的关系，在此临界阱深之下，状态u=0是F类中典型f的全局极小.还证明了u=0是f的严格局部极小元，在这个意义下，如果非零的v是可容许的，则它的L2范数或D的子集的测度，其中|v_{y}|超过1是足够小的（在能量泛函f中出现的参数方面给出了定量界限），则f（v）> f（0）。给出了u=0和全局最小值之间的低能量路径（在足够大的井深的情况下），使得引入小区域的成本，其中|v_{y}| 1（类似于奥氏体中马氏体的形核）到域D中的量可以任意小。
英文摘要：
A family of integral functionals F which, in a simplified way, model material microstructure occupying a two-dimensional domain D and which take account of surface energy and a variable well-depth is studied. It is shown that there is a critical well-depth, whose scaling with the surface energy density and domain dimensions is given, below which the state u=0 is the global minimizer of a typical f in the class F. It is also shown that u=0 is a strict local minimizer of f in the sense that if a non-zero v is admissible and either its L2 norm or the meaure of the subset of D where |v_{y}| exceeds 1 is sufficiently small (with quantitative bounds given in terms of the parameters appearing in the energy functional f) then f(v) > f(0). Low energy paths between u=0 and the global minimizer (in the case of a sufficiently large well-depth) are given such that the cost of introducing small regions where |v_{y}| is larger 1 (analogous to nucleation of martensite in austenite) into the domain D can be made arbitrarily small.
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