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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Global analysis of the generalised Helfrich flow of closed curves immersed in $\mathbb {R}^n$
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by Glen Wheeler PDF
Trans. Amer. Math. Soc. 367 (2015), 2263-2300 Request permission

Abstract:

In this paper we consider the evolution of regular closed elastic curves $\gamma$ immersed in $\mathbb {R}^n$. Equipping the ambient Euclidean space with a vector field $\mathbb {c}:\mathbb {R}^n\rightarrow \mathbb {R}^n$ and a function $f:\mathbb {R}^n\rightarrow \mathbb {R}$, we assume the energy of $\gamma$ is smallest when the curvature $\vec {\kappa }$ of $\gamma$ is parallel to $\vec {c}_0 = (\mathbb {c}\circ \gamma ) + (f\circ \gamma )\tau$, where $\tau$ is the unit vector field spanning the tangent bundle of $\gamma$. This leads us to consider a generalisation of the Helfrich functional $\mathcal {H}^{\vec {c}_0}_{\lambda }$, defined as the sum of the integral of $|\vec {\kappa }-\vec {c}_0|^2$ and $\lambda$-weighted length. We primarily consider the case where $f:\mathbb {R}^n\rightarrow \mathbb {R}$ is uniformly bounded in $C^\infty (\mathbb {R}^n)$ and $\mathbb {c}:\mathbb {R}^n\rightarrow \mathbb {R}^n$ is an affine transformation. Our first theorem is that the steepest descent $L^2$-gradient flow of $\mathcal {H}^{\vec {c}_0}_{\lambda }$ with smooth initial data exists for all time and subconverges to a smooth solution of the Euler-Lagrange equation for a limiting functional $\mathcal {H}^{\vec {c}_\infty }_{\lambda }$. We additionally perform some asymptotic analysis. In the broad class of gradient flows for which we obtain global existence and subconvergence, there exist many examples for which full convergence of the flow does not hold. This may manifest in its simplest form as solutions translating or spiralling off to infinity. We prove that if either $\mathbb {c}$ and $f$ are constant, the derivative of $\mathbb {c}$ is invertible and non-vanishing, or $(f,\gamma _0)$ satisfy a ‘properness’ condition, then one obtains full convergence of the flow and uniqueness of the limit. This last result strengthens a well-known theorem of Kuwert, Schätzle and Dziuk on the elastic flow of closed curves in $\mathbb {R}^n$ where $f$ is constant and $\mathbb {c}$ vanishes.
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Additional Information
  • Glen Wheeler
  • Affiliation: Otto-von-Guericke-Universität, Postfach 4120, D-39016 Magdeburg, Germany
  • Address at time of publication: University of Wollongong, Institute for Mathematics and its Applications, School of Mathematics and Applied Statistics, Faculty of Engineering and Information Sciences, Northfields Avenue, Wollongong 2522, New South Wales, Australia
  • MR Author ID: 833897
  • Email: wheeler@ovgu.de, glenw@uow.edu.au
  • Received by editor(s): June 22, 2012
  • Published electronically: December 5, 2014
  • Additional Notes: Financial support from the Alexander-von-Humboldt Stiftung is gratefully acknowledged
  • © Copyright 2014 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 367 (2015), 2263-2300
  • MSC (2010): Primary 53C44, 58J35
  • DOI: https://doi.org/10.1090/S0002-9947-2014-06592-6
  • MathSciNet review: 3301865