Article Contents
Article Contents

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

• 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.

Mathematics Subject Classification: Primary: 53A04, 53E10, 35B40, 35K55.

 Citation:

•  [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.