两个最小化分数阶矩的斜率函数及其在失配位错中的应用
Two slope functions minimizing fractional seminorms and applications to misfit dislocations
摘要：
我们考虑周期分段仿射函数，定义在实直线上，具有两个给定的斜率和规定的长度范围的区域的斜率是负的。我们证明了，在这样一个类中，极小的$s$ -分数Gagliardo密度，与$0<s<1$，实际上是周期的最小可能的周期所确定的规定的斜率和长度尺度。然后，我们确定的渐近行为的能量密度之间的比率的斜率是常数的两个区间的长度为零。我们的结果，$s=\frac 1 2$，在半相干直界面的失配位错的van der Merwe理论有相关的应用。我们考虑两种弹性材料具有不同的弹性系数和铸造平行晶格具有不同的间距。作为我们的分析的副产品，我们证明了最佳位错配置的周期性，我们提供了尖锐的渐近能量密度在半相干极限的两个晶格间距之间的比率趋于一。
英文摘要：
We consider periodic piecewise affine functions, defined on the real line, with two given slopes and prescribed length scale of the regions where the slope is negative. We prove that, in such a class, the minimizers of $s$-fractional Gagliardo seminorm densities, with $0<s<1$, are in fact periodic with the minimal possible period determined by the prescribed slopes and length scale. Then, we determine the asymptotic behavior of the energy density as the ratio between the length of the two intervals where the slope is constant vanishes. Our results, for $s=\frac 1 2$, have relevant applications to the van der Merwe theory of misfit dislocations at semi-coherent straight interfaces. We consider two elastic materials having different elastic coefficients and casting parallel lattices having different spacing. As a byproduct of our analysis, we prove the periodicity of optimal dislocation configurations and we provide the sharp asymptotic energy density in the semi-coherent limit as the ratio between the two lattice spacings tends to one.
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