Variational formulae and estimates of O'Hara's knot energies

Shoya Kawakami, Takeyuki Nagasawa

英文摘要：

O'Hara's energies, introduced by Jun O'Hara, were proposed to answer the question of what is the canonical shape in a given knot type, and were configured so that the less the energy value of a knot is, the "better" its shape is. The existence and regularity of minimizers has been well studied. In this article, we calculate the first and second variational formulae of the $(α,p)$-O'Hara energies and show absolute integrability, uniform boundedness, and continuity properties. Although several authors have already considered the variational formulae of the $(α,1)$-O'Hara energies, their techniques do not seem to be applicable to the case $p>1$. We obtain the variational formulae in a novel manner by extracting a certain function from the energy density. All of the $(α,p)$-energies are made from this function, and by analyzing it, we obtain not only the variational formulae but also estimates in several function spaces.

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