Short time existence for the elastic flow of clamped curves

Abstract

Mathematische NachrichtenVolume 291, Issue 13 p. 2115-2116 ERRATUMFree Access Short time existence for the elastic flow of clamped curves This article corrects the following: Short time existence for the elastic flow of clamped curves Adrian Spener, Volume 290Issue 13Mathematische Nachrichten pages: 2052-2077 First Published online: January 20, 2017 First published: 17 July 2018 https://doi.org/10.1002/mana.201700368Citations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Published in Math. Nachr. 290 (2017), no. 13, 2052–2077 Adrian Spener* Universität Ulm, Helmholtzstraße 18, 89081 Ulm, Germany *Email: [email protected] In this erratum we want to correct a mistake in the statement and the proof of [3, Theorem 1.1]. In this theorem, the equivalence of the elastic flow equation to a quasilinear equation is shown using an ODE. Unfortunately, the terms contained on the right hand side of the flow equation failed to have enough regularity for the argument in the proof of [3, Theorem 1.1] to hold. The corrected statement of this theorem reads as follows. Theorem. (Corrected version of [[3], Theorem 1.1])Let be an immersion satisfying [3, Assumption 1.3] and let . If the elastic flow [3, (1.2)] has a unique (up to reparametrisation) solution in for some and , then there exists a unique solution in for some (possibly different) to the problem [3, (1.3)].Conversely, if is a unique solution to [3, (1.3)] for some and , then there exists a unique solution for some (possibly different) to the problem [3, (1.2)]. Proof.We start with the second part of the statement and since the corresponding ODE is easier. As in [3, (2.6)] we introduce the function where and u is analytic in the interior of its domain. Under the assumptions of the original paper, the function ξ was not necessarily Lipschitz continuous in x, thus the statements on uniqueness and regularity of the solution to the ODE below were wrong. Under the new assumptions above we have , thus the equation (1)has a unique solution for some T small enough by [1, Theorem 9.2]. More precisely, we define , where and is small enough such that there exists a -extension (here we use the notation of [1, p. 91]). Applying [1, Theorem 9.2] we find a unique solution to 1, where the domain is relatively open by [1, Theorem 8.3] and contains . From the compactness of we find some small enough such that . Since ξ vanishes at the boundary of I we find that is indeed a diffeomorphism on for t small enough by continuity of . Moreover, using [1, Theorem 9.2] again we find that exists and is continuous, which shows that and and its spatial derivative vanish on the boundary. Finally, from the Cauchy–Kovalevskaya theorem [2, Theorems 1.40 and 1.41] we find the analyticity of Φ and whence of in the interior of their domains.The general case with follows from [1, Theorem 9.5 and Remark 9.6]. From here onwards we can follow the proof from 3.For the other direction, we let be given as in [3, Beginning of the Proof of Theorem 1.1 on p. 2054] with the regularity under the new assumptions and consider the function As stated below [3, (2.1)], the function is well defined for t and small enough. Moreover, from the regularity assumption on we find that , whence by [1, Theorem 9.5 and Remark 9.6] there exists a unique solution to the problem Similarly to the case above we find that is a diffeomorphism for t small enough and , where the local analyticity again follows from the Cauchy–Kovalevskaya theorem [2, Theorems 1.40 and 1.41]. Continuing as in [3, Proof of Theorem 1.1] we have the same regularity for , which solves [3, (1.3)]. ACKNOWLEDGEMENTS The author would like to thank Prof. Dr. Oliver Schnürer for drawing his attention to the mistake in the statement and the proof of [3, Theorem 1.1]. REFERENCES 1H. Amann, Ordinary differential equations, de Gruyter Stud. Math., vol. 13, Walter de Gruyter & Co., Berlin, 1990. 2G. B. Folland, Introduction to partial differential equations, second edition, Princeton University Press, Princeton, NJ, 1995. 3A. Spener, Short time existence for the elastic flow of clamped curves, Math. Nachr. 290 (2017), no. 13, 2052– 2077. Citing Literature Volume291, Issue13September 2018Pages 2115-2116 ReferencesRelatedInformation