Show Summary Details
More options …

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Balogh, Zoltán / Cardiliaguet, Pierre / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Gianazza, Ugo / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / Mingione, Giuseppe / Nystrom, Kaj / Riviére, Tristan / Schaetzle, Reiner / Shen, Zhongwei / Silvestre, Luis / Tonegawa, Yoshihiro / Touzi, Nizar / Wang, Guofang

IMPACT FACTOR 2018: 2.316

CiteScore 2018: 1.77

SCImago Journal Rank (SJR) 2018: 2.350
Source Normalized Impact per Paper (SNIP) 2018: 1.465

Mathematical Citation Quotient (MCQ) 2018: 1.44

Online
ISSN
1864-8266
See all formats and pricing
More options …
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 ${TP}^{\left(p,\phantom{\rule{0.166667em}{0ex}}q\right)}$ 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 ${TP}^{\left(p,\phantom{\rule{0.166667em}{0ex}}q\right)}$ 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 ${W}^{\left(p-1\right)/q,q}\left(ℝ/ℤ,{ℝ}^{n}\right)$. 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 ${TP}^{\left(p,2\right)}$ + λ length, p ∈ (4,5), λ > 0, are smooth – so especially all local minimizers are smooth.

MSC: 49N60; 49J10

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,

Export Citation