偏微分方程式セミナー(2015/7/24): A decomposition of the Möbius energy and consequences, 石関 彩 氏

開催日時
2015年   7月 24日 17時 00分 ~ 2015年   7月 24日 18時 00分
場所
北海道大学理学部3号館3-309室
講演者
石関 彩 氏(埼玉大学)
 
We consider the Möbius energy for closed curves in \( \mathbb{R}^n \), so-called since it is invariant under Möbius transformations. Since the energy was introduced for finding the canonical configuration of knots, the energy density contains negative powers of the intrinsic and extrinsic distance between any two points on the curve, and this causes significant difficulty with the analysis.

We can decompose the energy into three parts, each of which is Möbius invariant. The first part characterizes the proper domain of the energy; the second one plays the role of canceling the singularity of the density; and the third one gives us information about the minimal value of the energy.

The decomposition gives us easy-to-analyze components. For example, although the first and second variational formulae of the original energy had already been derived in the sense of Cauchy's principal value without our decomposition, we can more easily derive simpler expressions and hence obtain certain applicable estimates for the variational formulae in fractional Sobolev spaces (as well as other spaces) using our decomposition.

Furthermore, the Möbius invariance of each component gives information concerning the minimizers of the energies.

This is a joint work with Prof. Takeyuki Nagasawa (Saitama University).

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