A gradient flow for the p -elastic energy defined on closed planar curves
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Mathematische Annalen
A gradient flow for the p-elastic energy defined on closed planar curves Shinya Okabe1 · Paola Pozzi2 · Glen Wheeler3 Received: 22 December 2018 / Revised: 29 July 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We study the evolution of closed inextensible planar curves under a second order flow that decreases the p-elastic energy. A short time existence result for p ∈ (1, ∞) is obtained via a minimizing movements method. For p = 2, that is in the case of the classic elastic energy, long-time existence is retrieved. Mathematics Subject Classification 35K92 · 53A04 · 53C44
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Notation and useful preliminary estimates . . . . . . . . . . . . . . . . . 3 Short time existence via minimizing movements . . . . . . . . . . . . . 3.1 Discretization procedure . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Uniform bounds for approximating functions (1 < p < ∞) . 3.1.2 Regularity of approximating functions (1 < p < ∞) . . . . . 3.1.3 Discrete Lagrange multipliers (1 < p < ∞) . . . . . . . . . . 3.1.4 The case p = 2: control of the velocities . . . . . . . . . . . . 3.1.5 The case p = 2: control of the Lagrange multipliers . . . . . . 3.1.6 The case p > 2: control of the velocities for small initial data
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Communicated by Y. Giga.
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Glen Wheeler [email protected] Shinya Okabe [email protected] Paola Pozzi [email protected]
1
Mathematical Institute, Tohoku University, Sendai, Japan
2
Fakultät für Mathematik, Universität Duisburg-Essen, Duisburg, Germany
3
Institute for Mathematics and Its Applications, University of Wollongong, Wollongong, Australia
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S. Okabe et al. 3.1.7 The case p > 2: control of the Lagrange multipliers for small initial data 3.2 Convergence procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Convergence to a weak solution of (P) (1 < p < ∞) . . . . . . . . . . . 3.2.2 On regularity and uniqueness of weak solutions for (P) for p ≥ 2 . . . . 4 Long time existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1 Introduction Let γ : S 1 → R2 denote a smooth regular closed planar curve and let the p-elastic energy be defined by 1 E p (γ ) := p
γ
|κ(s)| p ds,
(1.1)
where κ and s denote the scalar curvature and the arc length parameter of γ , and 1 < p < ∞. In the following w
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