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- W2964175079 abstract "Abstract In this article we introduce and investigate a new two-parameter family of knot energies <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msup> <m:mo form=prefix>TP</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mspace width=0.166667em /> <m:mi>q</m:mi> <m:mo>)</m:mo> </m:mrow> </m:msup> </m:math> ${operatorname{TP}^{(p,,q)}}$ that contains the tangent-point energies. These energies are obtained by decoupling the exponents in the numerator and denominator of the integrand in the original definition of the tangent-point energies. We will first characterize the curves of finite energy <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msup> <m:mo form=prefix>TP</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mspace width=0.166667em /> <m:mi>q</m:mi> <m:mo>)</m:mo> </m:mrow> </m:msup> </m:math> ${operatorname{TP}^{(p,,q)}}$ in the sub-critical range p ∈ ( q +2,2 q +1) and see that those are all injective and regular curves in the Sobolev–Slobodeckiĭ space <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msup> <m:mi>W</m:mi> <m:mstyle scriptlevel=1 displaystyle=false> <m:mrow> <m:mo>(</m:mo> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> <m:mo>)</m:mo> <m:mo>/</m:mo> <m:mi>q</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> </m:mstyle> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mi>ℝ</m:mi> <m:mo>/</m:mo> <m:mi>ℤ</m:mi> <m:mo>,</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> ${W^{scriptstyle (p-1)/q,q}(mathbb {R}/mathbb {Z},mathbb {R}^n)}$ . We derive a formula for the first variation that turns out to be a non-degenerate elliptic operator for the special case q = 2: a fact that seems not to be the case for the original tangent-point energies. This observation allows us to prove that stationary points of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msup> <m:mo form=prefix>TP</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mn>2</m:mn> <m:mo>)</m:mo> </m:mrow> </m:msup> </m:math> $operatorname{TP}^{(p,2)}$ + λ length, p ∈ (4,5), λ > 0, are smooth – so especially all local minimizers are smooth." @default.
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- W2964175079 date "2014-03-25" @default.
- W2964175079 modified "2023-09-26" @default.
- W2964175079 title "Regularity theory for tangent-point energies: The non-degenerate sub-critical case" @default.
- W2964175079 doi "https://doi.org/10.1515/acv-2013-0020" @default.
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