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Advances in Calculus of Variations

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Regularity theory for tangent-point energies: The non-degenerate sub-critical case

Simon Blatt
  • Karlsruhe Institute of Technology, Institut für Analysis, Kaiserstrasse 89-93, 76131 Karlsruhe, Germany
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/ Philipp Reiter
  • Fakultät für Mathematik, University of Duisburg-Essen, Forsthausweg 2, 47057 Duisburg, Germany
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Published Online: 2014-03-25 | DOI: https://doi.org/10.1515/acv-2013-0020


In this article we introduce and investigate a new two-parameter family of knot energies 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 TP(p,q) in the sub-critical range p ∈ (q+2,2q+1) and see that those are all injective and regular curves in the Sobolev–Slobodeckiĭ space W(p-1)/q,q(/,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 TP(p,2) + λ length, p ∈ (4,5), λ > 0, are smooth – so especially all local minimizers are smooth.

Keywords: Knot energies; existence of minimizers; regularity theory

MSC: 49N60; 49J10

About the article

Received: 2013-09-12

Revised: 2014-02-06

Accepted: 2014-02-14

Published Online: 2014-03-25

Published in Print: 2015-04-01

Funding Source: Swiss National Science Foundation

Award identifier / Grant number: 200020_125127

Funding Source: Leverhulm trust

Funding Source: DFG Transregional Collaborative Research Centre

Award identifier / Grant number: SFB TR 71

Funding Source: Czech Ministry of Education

Award identifier / Grant number: ERC CZ LL1203

Citation Information: Advances in Calculus of Variations, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2013-0020. Export Citation

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