The $$ L^2 $$ L 2 -gradient of decomposed Möbius energies

Autor: Aya Ishizeki, Takeyuki Nagasawa
Rok vydání: 2016
Předmět:
Zdroj: Calculus of Variations and Partial Differential Equations. 55
ISSN: 1432-0835
0944-2669
DOI: 10.1007/s00526-016-0993-8
Popis: The Mobius energy is an example of a knot energy, so-called since it is invariant under Mobius transformations. It has been shown that it is defined on the fractional Sobolev space $$ H^{ \frac{3}{2} } \cap W^{1, \infty } $$ , where it can be decomposed into three parts, each of which retains the Mobius invariance. These results hold not only for knots in three-dimensional Euclidean space but also for closed curves embedded in n -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 in $$ L^2 $$ if the curve is in $$ H^3 $$ . Such a linear form is called the $$ L^2 $$ -gradient. In this paper, we confirm this expectation, and give an explicit expression of the $$ L^2 $$ -gradient for the decomposed Mobius energies including all lower-order terms.
Databáze: OpenAIRE
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