单阱Modica-Mortola泛函的奇异极限及其在Kobayashi-Warren-Carter能量中的应用
A finer singular limit of a single-well Modica--Mortola functional and its applications to the Kobayashi--Warren--Carter energy
摘要：
在图收敛条件下，给出了一维单阱Modica-Mortola泛函的Gamma极限的一个显式表示，这种图收敛条件比传统的$L^1$ -收敛或依测度收敛条件更精确.作为应用，给出了材料科学中常用的Kobayashi-Warren-Carter能量奇异极限的显式表示。在图收敛的条件下，得到了一些紧性定理.这样的公式，以及紧性，是有用的，以表征极限的极小Kobayashi-Warren-Carter能量。为了刻画图收敛条件下的Gamma极限，引入了一种新的思想，这种思想对一维问题特别有用。它是变量的参数通过其图形的弧长参数的变化，本文称之为弧长参数展开。
英文摘要：
An explicit representation of the Gamma limit of a single-well Modica--Mortola functional is given for one-dimensional space under the graph convergence which is finer than conventional $L^1$-convergence or convergence in measure. As an application, an explicit representation of a singular limit of the Kobayashi-Warren-Carter energy, which is popular in materials science, is given. Some compactness under the graph convergence is also established. Such formulas, as well as compactness, is useful to characterize the limit of minimizers the Kobayashi-Warren-Carter energy. To characterize the Gamma limit under the graph convergence, a new idea which is especially useful for one-dimensional problem is introduced. It is a change of parameter of the variable by arc-length parameter of its graph, which is called unfolding by the arc-length parameter in this paper.
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