American Institute of Mathematical Sciences

March  2021, 14(3): 1093-1102. doi: 10.3934/dcdss.2020385

An application of interpolation inequalities between the deviation of curvature and the isoperimetric ratio to the length-preserving flow

 Graduate School of Science and Engineering, Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama City, Saitama 338-8570, Japan

Received  January 2019 Revised  February 2020 Published  March 2021 Early access  June 2020

In recent work of Nagasawa and the author, new interpolation inequalities between the deviation of curvature and the isoperimetric ratio were proved. In this paper, we apply such estimates to investigate the large-time behavior of the length-preserving flow of closed plane curves without a convexity assumption.

Citation: Kohei Nakamura. An application of interpolation inequalities between the deviation of curvature and the isoperimetric ratio to the length-preserving flow. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1093-1102. doi: 10.3934/dcdss.2020385
References:
 [1] G. Dziuk, E. Kuwert and R. Schätzle, Evolution of elastic curves in $\Bbb R^n$: Existence and computation, SIAM J. Math. Anal., 33 (2002), 1228-1245.  doi: 10.1137/S0036141001383709. [2] J. Escher and G. Simonett, The volume preserving mean curvature flow near spheres, Proc. Amer. Math. Soc., 126 (1998), 2789-2796.  doi: 10.1090/S0002-9939-98-04727-3. [3] M. Gage, On an area-preserving evolution equation for plane curves, Nonlinear Problems in Geometry, Contemp. Math., Amer. Math. Soc., Providence, RI, 51, (1986), 51–62. doi: 10.1090/conm/051/848933. [4] L. S. Jiang and S. L. Pan, On a non-local curve evolution problem in the plane, Comm. Anal. Geom., 16 (2008), 1-26.  doi: 10.4310/CAG.2008.v16.n1.a1. [5] L. Ma and A. Q. Zhu, On a length preserving curve flow, Monatsh. Math., 165 (2012), 57-78.  doi: 10.1007/s00605-011-0302-8. [6] U. F. Mayer, A singular example for the averaged mean curvature flow, Experiment. Math., 10 (2001), 103-107.  doi: 10.1080/10586458.2001.10504432. [7] T. Nagasawa and K. Nakamura, Interpolation inequalities between the deviation of curvature and the isoperimetric ratio with applications to geometric flows, Adv. Differential Equations, 24 (2019), 581-608. [8] D. Ševčovič and S. Yazaki, Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity, Math. Methods Appl. Sci., 35 (2012), 1784-1798.  doi: 10.1002/mma.2554.

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References:
 [1] G. Dziuk, E. Kuwert and R. Schätzle, Evolution of elastic curves in $\Bbb R^n$: Existence and computation, SIAM J. Math. Anal., 33 (2002), 1228-1245.  doi: 10.1137/S0036141001383709. [2] J. Escher and G. Simonett, The volume preserving mean curvature flow near spheres, Proc. Amer. Math. Soc., 126 (1998), 2789-2796.  doi: 10.1090/S0002-9939-98-04727-3. [3] M. Gage, On an area-preserving evolution equation for plane curves, Nonlinear Problems in Geometry, Contemp. Math., Amer. Math. Soc., Providence, RI, 51, (1986), 51–62. doi: 10.1090/conm/051/848933. [4] L. S. Jiang and S. L. Pan, On a non-local curve evolution problem in the plane, Comm. Anal. Geom., 16 (2008), 1-26.  doi: 10.4310/CAG.2008.v16.n1.a1. [5] L. Ma and A. Q. Zhu, On a length preserving curve flow, Monatsh. Math., 165 (2012), 57-78.  doi: 10.1007/s00605-011-0302-8. [6] U. F. Mayer, A singular example for the averaged mean curvature flow, Experiment. Math., 10 (2001), 103-107.  doi: 10.1080/10586458.2001.10504432. [7] T. Nagasawa and K. Nakamura, Interpolation inequalities between the deviation of curvature and the isoperimetric ratio with applications to geometric flows, Adv. Differential Equations, 24 (2019), 581-608. [8] D. Ševčovič and S. Yazaki, Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity, Math. Methods Appl. Sci., 35 (2012), 1784-1798.  doi: 10.1002/mma.2554.
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