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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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An elastic flow for nonlinear spline interpolations in $\mathbb {R}^n$
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by Chun-Chi Lin, Hartmut R. Schwetlick and Dung The Tran PDF
Trans. Amer. Math. Soc. 375 (2022), 4893-4942 Request permission

Abstract:

In this paper we use the method of geometric flow on the problem of nonlinear spline interpolations for non-closed curves in $n$-dimensional Euclidean spaces. The method applies theory of fourth-order parabolic PDEs to each piece of the curve between two successive knot points at which certain dynamic boundary conditions are imposed. We show the existence of global solutions of the elastic flow in suitable Hölder spaces. In the asymptotic limit, as time approaches infinity, solutions subconverge to a stationary solution of the problem. The method of geometric flows provides a new approach for the problem of nonlinear spline interpolations.
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Additional Information
  • Chun-Chi Lin
  • Affiliation: Department of Mathematics, National Taiwan Normal University, Taipei 116, Taiwan
  • MR Author ID: 632143
  • ORCID: 0000-0001-5682-0698
  • Email: chunlin@math.ntnu.edu.tw
  • Hartmut R. Schwetlick
  • Affiliation: Department of Mathematical Sciences, University of Bath, United Kingdom
  • MR Author ID: 668083
  • Email: h.schwetlick@bath.ac.uk
  • Dung The Tran
  • Affiliation: Department of Mathematics, National Taiwan Normal University, Taipei 116, Taiwan
  • Email: tranthedung56@gmail.com
  • Received by editor(s): September 9, 2012
  • Received by editor(s) in revised form: December 18, 2021
  • Published electronically: May 4, 2022
  • Additional Notes: This work was partially supported by the research grant of the National Science Council of Taiwan (NSC-100-2115-M-003-003), the National Center for Theoretical Sciences at Taipei, and the Max-Planck-Institut für Mathematik in den Naturwissenschaften in Leipzig. The third author received financial support from Taiwan MoST 108-2115-M-003-003-MY2.
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 4893-4942
  • MSC (2020): Primary 35K55; Secondary 41A15
  • DOI: https://doi.org/10.1090/tran/8639
  • MathSciNet review: 4439495