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Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Astala, Kari / Colding, Tobias / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Finster, Felix / Gianazza, Ugo / Gursky, Matthew / Hardt, Robert / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / McCann, Robert / Mingione, Giuseppe / Nystrom, Kaj / Pacard, Frank / Preiss, David / Riviére, Tristan / Schaetzle, Reiner / Silvestre, Luis

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IMPACT FACTOR 2016: 1.182

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1864-8266
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Volume 8, Issue 2

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

Simon Blatt / Philipp Reiter
Published Online: 2014-03-25 | DOI: https://doi.org/10.1515/acv-2013-0020

Abstract

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, Volume 8, Issue 2, Pages 93–116, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2013-0020.

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