The L2-gradient of decomposed Mobius energies
The L2-gradient of decomposed Mobius energies
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分解莫比乌斯能量的 L2 梯度
DOI：
10.1007/s00526-016-0993-8
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
Takeyuki Nagasawa
中科院分区：
文献类型：
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作者：
Aya Ishizeki;Takeyuki Nagasawa
Möbius能量是结能的一个例子，之所以叫结能，是因为它在Möbius变换下是不变的。证明了它是在分数Sobolev空间上定义的，在分数Sobolev空间中，它可以分解为三部分，每一部分都保持Möbius不变性。这些结果不仅适用于三维欧几里得空间中的结点，也适用于嵌入在二维欧几里得空间中的封闭曲线。已经得到两种曲线的分解能量的变分公式及其变化都在同一分数Sobolev空间中。分部积分的形式意味着公式可以扩展到变分的线性形式，如果曲线是。这样的线性形式叫做梯度。在本文中，我们证实了这一期望，并给出了包含所有低阶项的分解Möbius能量的梯度的显式表达式。
The Möbius energy is an example of a knot energy, so-called since it is invariant under Möbius transformations. It has been shown that it is defined on the fractional Sobolev space, where it can be decomposed into three parts, each of which retains the Möbius invariance. These results hold not only for knots in three-dimensional Euclidean space but also for closed curves embedded inn-dimensional Euclidean space. It has already been obtained that the variational formulae of the decomposed energies for both curves and their variations are in the same fractional Sobolev space. A formal integration by parts implies that the formulae extend to linear forms for variationals inif the curve is in. Such a linear form is called the-gradient. In this paper, we confirm this expectation, and give an explicit expression of the-gradient for the decomposed Möbius energies including all lower-order terms.