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On Planar Curves with Position-Dependent Curvature

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Abstract

Motivated by homothetic solutions in curvature-driven flows of planar curves, as well as their many physical applications, this article carries out a systematic study of oriented smooth curves whose curvature \(\kappa \) is a given function of position or direction. The analysis is informed by a dynamical systems point of view. Though focussed on situations where the prescribed curvature depends only on the distance r from one distinguished point, the basic dynamical concepts are seen to be applicable in other situations as well. As an application, a complete classification of all closed solutions of \(\kappa = ar^{b}\), with arbitrary real constants \(a,b\), is established.

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References

  1. Abresch, U., Langer, J.: The normalized curve shortening flow and homothetic solutions. J. Differ. Geom. 23, 175–196 (1986)

    Article  MathSciNet  Google Scholar 

  2. Alefeld, G., Herzberger, J.: Introduction to Interval Computations. Academic Press, Cambridge (1983)

    MATH  Google Scholar 

  3. Amann, H.: Ordinary Differential Equations: An Introduction to Non-linear Analysis. De Gruyter, Berlin (1990)

    Book  Google Scholar 

  4. Andrews, B.: Evolving convex curves. Calc. Var. Partial Differ. Equ. 7, 315–371 (1998)

    Article  MathSciNet  Google Scholar 

  5. Andrews, B.: Classification of limiting shapes for isotropic curve flows. J. Am. Math. Soc. 16, 443–459 (2002)

    Article  MathSciNet  Google Scholar 

  6. Arai, Z., Kokubu, H., Pilarczyk, P.: Recent development in rigorous computational methods in dynamical systems. Jpn. J. Ind. Appl. Math. 26, 393–417 (2009)

    Article  MathSciNet  Google Scholar 

  7. Arnold, V.I.: Ordinary Differential Equations, 3rd edn. Springer, Berlin (1992)

    Google Scholar 

  8. Ayala, J.: On the topology of the spaces of curvature constrained plane curves. Adv. Geom. 17, 283–292 (2017)

    Article  MathSciNet  Google Scholar 

  9. Benguria, R.D., Depassier, C., Loss, M.: Monotonicity of the period of a non linear oscillator. Nonlinear Anal. 140, 61–68 (2016)

    Article  MathSciNet  Google Scholar 

  10. Berger, M., Gostiaux, B.: Differential Geometry: Manifolds, Curves, and Surfaces. Graduate Texts in Mathematics, vol. 115. Springer, Berlin (1988)

    Book  Google Scholar 

  11. Chicone, C.: The monotonicity of the period function for planar Hamiltonian vector fields. J. Differ. Equ. 69, 310–321 (1987)

    Article  MathSciNet  Google Scholar 

  12. Chicone, C., Dumortier, F.: A quadratic system with a nonmonotonic period function. Proc. Am. Math. Soc. 102, 706–710 (1988)

    Article  MathSciNet  Google Scholar 

  13. Chow, S.-N., Wang, D.: On the monotonicity of the period function of some second order equations. Časopis Pěst. Mat. 111, 14–25 (1986)

    Article  MathSciNet  Google Scholar 

  14. Cima, A., Gasull, A., Mañosas, F.: Period function for a class of Hamiltonian systems. J. Differ. Equ. 168, 180–199 (2000)

    Article  MathSciNet  Google Scholar 

  15. Cima, A., Mañosas, F., Villadelprat, J.: Isochronicity for several classes of Hamiltonian systems. J. Differ. Equ. 157, 373–413 (1999)

    Article  MathSciNet  Google Scholar 

  16. Curtiss, D.R.: Recent extensions of Descartes’ rule of signs. Ann. Math. 19, 251–278 (1918)

    Article  MathSciNet  Google Scholar 

  17. de Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Hoboken (1976)

    MATH  Google Scholar 

  18. Domokos, G., Gibbons, G.W.: The evolution of pebble size and shape in space and time. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468, 3059–3079 (2012)

    MathSciNet  MATH  Google Scholar 

  19. Domokos, G., Sipos, A.A., Várkonyi, P.L.: Continuous and discrete models for abrasion processes. Period. Pol. Arch. 40, 3–8 (2009)

    Article  Google Scholar 

  20. Dubins, L.E.: On curves of minimal length with constraint on average curvature and prescribed initial and terminal positions and tangents. Am. J. Math. 79, 497–516 (1957)

    Article  MathSciNet  Google Scholar 

  21. Elezović, N., Giordano, C., Pečarić, J.: The best bounds in Gautschi’s inequality. Math. Inequal. Appl. 3, 239–252 (2000)

    MathSciNet  MATH  Google Scholar 

  22. Fehér, E., Domokos, G., Krauskopf, B.: Computing planar shape and critical point evolution under curvature-driven flows. arXiv:2010.11169 [v1] (2020)

  23. Firey, W.J.: Shapes of worn stones. Mathematika 21, 1–11 (1974)

    Article  MathSciNet  Google Scholar 

  24. Hill, T.P.: On the oval shapes of beach stones. Appl. Math. 2, 16–38 (2022)

    Google Scholar 

  25. Irving, M.C.: Smooth Dynamical Systems. Advanced Series in Nonlinear Dynamics, vol. 17. World Scientific, Singapore (2001)

    Google Scholar 

  26. Kim, H.-S., Cheong, O.: The cost of bounded curvature. Comput. Geom. 46, 648–672 (2013)

    Article  MathSciNet  Google Scholar 

  27. Klingenberg, W.: A Course in Differential Geometry. Graduate Texts in Mathematics, vol. 51. Springer (1978)

  28. Kühnel, W.: Differential Geometry: Curves-Surfaces-Manifolds. Student Mathematical Library, vol. 16, 2nd edn. American Mathematical Society, Providence (2006)

    MATH  Google Scholar 

  29. Miyamoto, Y., Yagasaki, K.: Monotonicity of the first eigenvalue and the global bifurcation diagram for the branch of interior peak solutions. J. Differ. Equ. 254, 342–367 (2013)

    Article  MathSciNet  Google Scholar 

  30. Perko, L.: Differential Equations and Dynamical Systems. Texts in Applied Mathematics, vol. 7, 3rd edn. Springer, Berlin (2001)

    Book  Google Scholar 

  31. Rothe, F.: Remarks on periods of planar Hamiltonian systems. SIAM J. Math. Anal. 24, 129–154 (1993)

    Article  MathSciNet  Google Scholar 

  32. Rudin, W.: Real and Complex Analysis. McGraw Hill, New York (1973)

    MATH  Google Scholar 

  33. Tucker, W.: Validated Numerics. A Short Introduction to Rigorous Computations. Princeton University Press, Princeton (2011)

    Book  Google Scholar 

  34. Urbas, J.: Convex curves moving homothetically by negative powers of their curvature. Asian J. Math. 3, 635–656 (1999)

    Article  MathSciNet  Google Scholar 

  35. Villadelprat, J., Zhang, X.: The period function of Hamiltonian systems with separable variables. J. Dyn. Differ. Equ. 32, 741–767 (2020)

    Article  MathSciNet  Google Scholar 

  36. Walter, W.: Ordinary Differential Equations. Graduate Texts in Mathematics, vol. 182. Springer, Berlin (1998)

    Book  Google Scholar 

  37. Yagasaki, K.: Monotonicity of the period function for \(u^{\prime \prime }-u+u^p=0\) with \(p\in {\mathbb{R}}\) and \(p>1\). J. Differ. Equ. 255, 1988–2001 (2013)

  38. Zevin, A.A., Pinsky, M.A.: Monotonicity criteria for an energy-period function in planar Hamiltonian systems. Nonlinearity 14, 1425–1432 (2001)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author was partially supported by an Nserc Discovery Grant. He owes deep gratitude to T.P. Hill and K.E. Morrison who in 2005 conjectured (correctly, as it turned out) that the only Jordan solution of \(\kappa = r\) is the (counter-clockwise oriented) unit circle, and who greatly helped this work come to fruition through continued interest and advice. Insightful comments by an anonymous referee led to a much improved presentation. Thanks also to J. Muldowney, M. Niksirat, T. Schmah, and C. Xu for several enlightening conversations over the years. Parts of this work were completed while the author was a visitor at the Universität Wien. He is much indebted to R. Zweimüller for many kind acts of hospitality.

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  1. Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada

    Arno Berger

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Correspondence to Arno Berger.

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Berger, A. On Planar Curves with Position-Dependent Curvature. J Dyn Diff Equat (2022). https://doi.org/10.1007/s10884-022-10168-9

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  • DOI: https://doi.org/10.1007/s10884-022-10168-9

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