Research article Special Issues

Polar tangential angles and free elasticae

  • Received: 13 April 2020 Accepted: 29 April 2020 Published: 21 August 2020
  • In this note we investigate the behavior of the polar tangential angle of a general plane curve, and in particular prove its monotonicity for certain curves of monotone curvature. As an application we give (non)existence results for an obstacle problem involving free elasticae.

    Citation: Tatsuya Miura. Polar tangential angles and free elasticae[J]. Mathematics in Engineering, 2021, 3(4): 1-12. doi: 10.3934/mine.2021034

    Related Papers:

  • In this note we investigate the behavior of the polar tangential angle of a general plane curve, and in particular prove its monotonicity for certain curves of monotone curvature. As an application we give (non)existence results for an obstacle problem involving free elasticae.


    加载中


    [1] Dall'Acqua A, Deckelnick K (2018) An obstacle problem for elastic graphs. SIAM J Math Anal 50: 119-137.
    [2] Dayrens F, Masnou S, Novaga M (2018) Existence, regularity and structure of confined elasticae. ESAIM Contr Optim Ca 24: 25-43.
    [3] Ghys E, Tabachnikov S, Timorin V (2013) Osculating curves: Around the Tait-Kneser theorem. Math Intell 35: 61-66.
    [4] Kneser A (1912) Bemerkungen über die anzahl der extreme der krümmung auf geschlossenen kurven und über verwandte fragen in einer nicht-euklidischen geometrie. Festschrift H Weber 170-180.
    [5] Linnér A (1993) Existence of free nonclosed Euler-Bernoulli elastica. Nonlinear Anal 21: 575-593.
    [6] Linnér A (1998) Curve-straightening and the Palais-Smale condition. T Am Math Soc 350: 3743-3765.
    [7] Linnér A (1998) Explicit elastic curves. Ann Global Anal Geom 16: 445-475.
    [8] Miura T (2016) Singular perturbation by bending for an adhesive obstacle problem. Calc Var Partial Dif 55: 19.
    [9] Miura T (2017) Overhanging of membranes and filaments adhering to periodic graph substrates. Phys D 355: 34-44.
    [10] Miura T (2020) Elastic curves and phase transitions. Math Ann 376: 1629-1674.
    [11] Müller M (2019) An obstacle problem for elastic curves: existence results. Interface Free Bound 21: 87-129.
    [12] Sachkov YL (2008) Maxwell strata in the Euler elastic problem. J Dyn Control Syst 14: 169-234.
    [13] Singer DA (2008) Lectures on elastic curves and rods, In: Curvature and Variational Modeling in Physics and Biophysics, AIP Conference Proceedings, 3-32.
    [14] Tait PG (1896) Note on the circles of curvature of a plane curve. P Edinburgh Math Soc 14: 403.
    [15] Watanabe K (2014) Planar p-elastic curves and related generalized complete elliptic integrals. Kodai Math J 37: 453-474.
    [16] Yoshizawa K, A remark on elastic graphs with the symmetric cone obstacle. preprint (personal communication with the author).
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3969) PDF downloads(646) Cited by(1)

Article outline

Figures and Tables

Figures(5)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog