# American Institute of Mathematical Sciences

February  2016, 36(2): 785-803. doi: 10.3934/dcds.2016.36.785

## $2\pi$-Periodic self-similar solutions for the anisotropic affine curve shortening problem II

 1 LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, China 2 Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

Received  April 2014 Published  August 2015

The existence of $2\pi$-periodic positive solutions of the equation $$u'' + u = \displaystyle{\frac{a(x)}{u^3}}$$ is studied, where $a$ is a positive smooth $2\pi$-periodic function. Under some non-degenerate conditions on $a$, the existence of $2\pi$-periodic solutions to the equation is established.
Citation: Meiyue Jiang, Juncheng Wei. $2\pi$-Periodic self-similar solutions for the anisotropic affine curve shortening problem II. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 785-803. doi: 10.3934/dcds.2016.36.785
##### References:
 [1] U. Abresch and J. Langer, The normalized curved shortening flow and homothetic solutions,, J. Differential Geometry, 23 (1986), 175. [2] J. Ai, K. S. Chou and J. Wei, Self-similar solutions for the anisotropic affine curve shortening problem,, Calc. Var., 13 (2001), 311. doi: 10.1007/s005260000075. [3] S. Altschuler, Singularities of the curve shrinking flow for space curves,, J. Differential Geometry, 34 (1991), 491. [4] B. Andrews, Contraction of convex hypersurfaces by their affine normal,, J. Differential Geometry, 43 (1996), 207. [5] B. Andrews, Evolving convex curves,, Calc. Var., 7 (1998), 315. doi: 10.1007/s005260050111. [6] S. Angenent, On the formation of singularities in the curve shortening flow,, J. Differential Geometry, 33 (1991), 601. [7] S. Angenent and M. E. Gurtin, Multiphase thermodynamics with interfacial structure evolution of an isothermal interface,, Arch. Rational Mech. Anal., 108 (1989), 323. doi: 10.1007/BF01041068. [8] W. X. Chen, $L_p$-Minkowski problem with not necessarily positive data,, Adv. in Math., 201 (2006), 77. doi: 10.1016/j.aim.2004.11.007. [9] K. S. Chou and L. Zhang, On the uniqueness of stable ultimate shapes for the anisotropic curve-shorting problem,, Manuscripta Math., 102 (2000), 101. doi: 10.1007/s002291020101. [10] K. S. Chou and X. P. Zhu, Anisotropic flows for convex plane curves,, Duke Math. J., 97 (1999), 579. doi: 10.1215/S0012-7094-99-09722-3. [11] M. del Pino, R. Manásevich and A. Montero, $T$-periodic solutions for some second order differential equation with singularities,, Proc. Roy. Soc. Edinburgh, 120 (1992), 231. doi: 10.1017/S030821050003211X. [12] C. Dohmen and Y. Giga, Self-similar shrinking curves for anisotropic curvature flow equations,, Proc. Japan Acad., 70 (1994), 252. doi: 10.3792/pjaa.70.252. [13] C. Dohmen, Y. Giga and N. Mizoguchi, Existence of self-similar shrinking curves for anisotropic curvature flow equations,, Calc. Var., 4 (1996), 103. doi: 10.1007/BF01189949. [14] I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications,, Oxford Science Publications, (1995). [15] M. E. Gage, Evolving plane curve by curvature in relative geometries,, Duke Math. J., 72 (1993), 441. doi: 10.1215/S0012-7094-93-07216-X. [16] M. E. Gage and R. Hamilton, The heat equation shrinking convex plane curves,, J. Differential Geometry, 23 (1986), 69. [17] M. E. Gage and Y. Li, Evolving plane curve by curvature in relative geometries II,, Duke Math. J., 75 (1994), 79. doi: 10.1215/S0012-7094-94-07503-0. [18] M. Grayson, The heat equation shrinking embedded curves to round points,, J. Differential Geometry, 26 (1987), 285. [19] M. E. Gurtin, Thermodynamics of Evolving Phase Boundaries in the Plane,, Clarendon Press, (1993). [20] M.-Y. Jiang, Remarks on the 2-dimensional $L_p$-Minkowski problem,, Advanced Nonlinear Studies, 10 (2010), 297. [21] M.-Y. Jiang, L. Wang and J. Wei, $2\pi$-periodic self-similar solutions for the anisotropic affine curve shortening problem,, Calc. Var., 41 (2011), 535. doi: 10.1007/s00526-010-0375-6. [22] H. Matano and J. Wei, On anisotropic curvature flow equations,, preprint., (). [23] G. Sapiro and A. Tannenbaum, On affine plane curve evolution,, J. Funct. Anal., 119 (1994), 79. doi: 10.1006/jfan.1994.1004.

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##### References:
 [1] U. Abresch and J. Langer, The normalized curved shortening flow and homothetic solutions,, J. Differential Geometry, 23 (1986), 175. [2] J. Ai, K. S. Chou and J. Wei, Self-similar solutions for the anisotropic affine curve shortening problem,, Calc. Var., 13 (2001), 311. doi: 10.1007/s005260000075. [3] S. Altschuler, Singularities of the curve shrinking flow for space curves,, J. Differential Geometry, 34 (1991), 491. [4] B. Andrews, Contraction of convex hypersurfaces by their affine normal,, J. Differential Geometry, 43 (1996), 207. [5] B. Andrews, Evolving convex curves,, Calc. Var., 7 (1998), 315. doi: 10.1007/s005260050111. [6] S. Angenent, On the formation of singularities in the curve shortening flow,, J. Differential Geometry, 33 (1991), 601. [7] S. Angenent and M. E. Gurtin, Multiphase thermodynamics with interfacial structure evolution of an isothermal interface,, Arch. Rational Mech. Anal., 108 (1989), 323. doi: 10.1007/BF01041068. [8] W. X. Chen, $L_p$-Minkowski problem with not necessarily positive data,, Adv. in Math., 201 (2006), 77. doi: 10.1016/j.aim.2004.11.007. [9] K. S. Chou and L. Zhang, On the uniqueness of stable ultimate shapes for the anisotropic curve-shorting problem,, Manuscripta Math., 102 (2000), 101. doi: 10.1007/s002291020101. [10] K. S. Chou and X. P. Zhu, Anisotropic flows for convex plane curves,, Duke Math. J., 97 (1999), 579. doi: 10.1215/S0012-7094-99-09722-3. [11] M. del Pino, R. Manásevich and A. Montero, $T$-periodic solutions for some second order differential equation with singularities,, Proc. Roy. Soc. Edinburgh, 120 (1992), 231. doi: 10.1017/S030821050003211X. [12] C. Dohmen and Y. Giga, Self-similar shrinking curves for anisotropic curvature flow equations,, Proc. Japan Acad., 70 (1994), 252. doi: 10.3792/pjaa.70.252. [13] C. Dohmen, Y. Giga and N. Mizoguchi, Existence of self-similar shrinking curves for anisotropic curvature flow equations,, Calc. Var., 4 (1996), 103. doi: 10.1007/BF01189949. [14] I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications,, Oxford Science Publications, (1995). [15] M. E. Gage, Evolving plane curve by curvature in relative geometries,, Duke Math. J., 72 (1993), 441. doi: 10.1215/S0012-7094-93-07216-X. [16] M. E. Gage and R. Hamilton, The heat equation shrinking convex plane curves,, J. Differential Geometry, 23 (1986), 69. [17] M. E. Gage and Y. Li, Evolving plane curve by curvature in relative geometries II,, Duke Math. J., 75 (1994), 79. doi: 10.1215/S0012-7094-94-07503-0. [18] M. Grayson, The heat equation shrinking embedded curves to round points,, J. Differential Geometry, 26 (1987), 285. [19] M. E. Gurtin, Thermodynamics of Evolving Phase Boundaries in the Plane,, Clarendon Press, (1993). [20] M.-Y. Jiang, Remarks on the 2-dimensional $L_p$-Minkowski problem,, Advanced Nonlinear Studies, 10 (2010), 297. [21] M.-Y. Jiang, L. Wang and J. Wei, $2\pi$-periodic self-similar solutions for the anisotropic affine curve shortening problem,, Calc. Var., 41 (2011), 535. doi: 10.1007/s00526-010-0375-6. [22] H. Matano and J. Wei, On anisotropic curvature flow equations,, preprint., (). [23] G. Sapiro and A. Tannenbaum, On affine plane curve evolution,, J. Funct. Anal., 119 (1994), 79. doi: 10.1006/jfan.1994.1004.
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