Abstract
In this article we show that for initial data close to local minimizers of the Möbius energy the gradient flow exists for all time and converges smoothly to a local minimizer after suitable reparametrizations. To prove this, we show that the heat flow of the Möbius energy possesses a quasilinear structure which allows us to derive new short-time existence results for this evolution equation and a Łojasiewicz-Simon gradient inequality for the Möbius energy.
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Amann, H.: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), vol. 133 of Teubner-Texte Math, pp. 9–126. Teubner, Stuttgart (1993)
Angenent S.B.: Nonlinear analytic semiflows. Proc. R. Soc. Edinburgh Sect. A 115(1–2), 91–107 (1990)
Blatt S.: Loss of convexity and embeddedness for geometric evolution equations of higher order. J. Evol. Equ. 10, 21–27 (2010)
Bourdaud G.: Une algèbre maximale d’opérateurs pseudo-différentiels. Comm. Partial Differ. Equ. 13(9), 1059–1083 (1988)
Chill R.: On the łojasiewicz-simon gradient inequality. J. Funct. Anal. 201, 572–601 (2003)
Freedman M.H., He Z.-X., Wang Z.: Möbius energy of knots and unknots. Ann. Math. (2) 139(1), 1–50 (1994)
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. In: Classics in Mathematics. Springer-Verlag, Berlin (2001). Reprint of the 1998 edition
He Z.: A formula for the non-integer powers of the Laplacian. Acta Math. Sin. (Engl. Ser.) 15(1), 21–24 (1999)
He Z.-X.: The Euler-Lagrange equation and heat flow for the M öbius energy. Comm. Pure Appl. Math. 53(4), 399–431 (2000)
Hille, E., Phillips, R.S.: Functional Analysis and Semi-Groups. American Mathematical Society (1957)
Lunardi, A.: Analytic semigroups and optimal regularity in parabolic problems. In: Progress in Nonlinear Differential Equations and their Applications, vol. 16. Birkhäuser Verlag, Basel (1995)
O’Hara J.: Energy of a knot. Topology 30(2), 241–247 (1991)
Reiter, P.: Repulsive knot energies and pseudodifferential calculus: rigorous analysis and regularity theory for O’Hara’s knot energy family \({E^{(\alpha)},\alpha\in[2,3)}\). PhD thesis, RWTH Aachen (2009)
von der Mosel H.: Minimizing the elastic energy of knots. Asymptot. Anal. 18(1–2), 49–65 (1998)
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Communicated by J. Jost.
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Blatt, S. The gradient flow of the Möbius energy near local minimizers. Calc. Var. 43, 403–439 (2012). https://doi.org/10.1007/s00526-011-0416-9
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DOI: https://doi.org/10.1007/s00526-011-0416-9