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

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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

Meiyue Jiang ^1, and Juncheng Wei ^2,

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

Abstract
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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.

Keywords: anisotropic affine curve shortening problem., Self-similar solutions.

Mathematics Subject Classification: Primary: 34B15, 34B16; Secondary: 44J9.

Citation: Meiyue Jiang, Juncheng Wei. $2\pi$-Periodic self-similar solutions for the anisotropic affine curve shortening problem II. Discrete and Continuous Dynamical Systems, 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), 175-196.
[2]	J. Ai, K. S. Chou and J. Wei, Self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var., 13 (2001), 311-337. doi: 10.1007/s005260000075.
[3]	S. Altschuler, Singularities of the curve shrinking flow for space curves, J. Differential Geometry, 34 (1991), 491-514.
[4]	B. Andrews, Contraction of convex hypersurfaces by their affine normal, J. Differential Geometry, 43 (1996), 207-230.
[5]	B. Andrews, Evolving convex curves, Calc. Var., 7 (1998), 315-371. doi: 10.1007/s005260050111.
[6]	S. Angenent, On the formation of singularities in the curve shortening flow, J. Differential Geometry, 33 (1991), 601-633.
[7]	S. Angenent and M. E. Gurtin, Multiphase thermodynamics with interfacial structure evolution of an isothermal interface, Arch. Rational Mech. Anal., 108 (1989), 323-391. doi: 10.1007/BF01041068.
[8]	W. X. Chen, $L_p$-Minkowski problem with not necessarily positive data, Adv. in Math., 201 (2006), 77-89. 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), 101-110. doi: 10.1007/s002291020101.
[10]	K. S. Chou and X. P. Zhu, Anisotropic flows for convex plane curves, Duke Math. J., 97 (1999), 579-619. 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, Sect. A, 120 (1992), 231-243. doi: 10.1017/S030821050003211X.
[12]	C. Dohmen and Y. Giga, Self-similar shrinking curves for anisotropic curvature flow equations, Proc. Japan Acad., Ser. A, 70 (1994), 252-255. 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), 103-119. 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), 441-466. 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), 69-96.
[17]	M. E. Gage and Y. Li, Evolving plane curve by curvature in relative geometries II, Duke Math. J., 75 (1994), 79-98. 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), 285-314.
[19]	M. E. Gurtin, Thermodynamics of Evolving Phase Boundaries in the Plane, Clarendon Press, Oxford 1993.
[20]	M.-Y. Jiang, Remarks on the 2-dimensional $L_p$-Minkowski problem, Advanced Nonlinear Studies, 10 (2010), 297-313.
[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), 535-565. 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), 79-120. doi: 10.1006/jfan.1994.1004.

show all references

References:

[1]	U. Abresch and J. Langer, The normalized curved shortening flow and homothetic solutions, J. Differential Geometry, 23 (1986), 175-196.
[2]	J. Ai, K. S. Chou and J. Wei, Self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var., 13 (2001), 311-337. doi: 10.1007/s005260000075.
[3]	S. Altschuler, Singularities of the curve shrinking flow for space curves, J. Differential Geometry, 34 (1991), 491-514.
[4]	B. Andrews, Contraction of convex hypersurfaces by their affine normal, J. Differential Geometry, 43 (1996), 207-230.
[5]	B. Andrews, Evolving convex curves, Calc. Var., 7 (1998), 315-371. doi: 10.1007/s005260050111.
[6]	S. Angenent, On the formation of singularities in the curve shortening flow, J. Differential Geometry, 33 (1991), 601-633.
[7]	S. Angenent and M. E. Gurtin, Multiphase thermodynamics with interfacial structure evolution of an isothermal interface, Arch. Rational Mech. Anal., 108 (1989), 323-391. doi: 10.1007/BF01041068.
[8]	W. X. Chen, $L_p$-Minkowski problem with not necessarily positive data, Adv. in Math., 201 (2006), 77-89. 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), 101-110. doi: 10.1007/s002291020101.
[10]	K. S. Chou and X. P. Zhu, Anisotropic flows for convex plane curves, Duke Math. J., 97 (1999), 579-619. 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, Sect. A, 120 (1992), 231-243. doi: 10.1017/S030821050003211X.
[12]	C. Dohmen and Y. Giga, Self-similar shrinking curves for anisotropic curvature flow equations, Proc. Japan Acad., Ser. A, 70 (1994), 252-255. 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), 103-119. 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), 441-466. 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), 69-96.
[17]	M. E. Gage and Y. Li, Evolving plane curve by curvature in relative geometries II, Duke Math. J., 75 (1994), 79-98. 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), 285-314.
[19]	M. E. Gurtin, Thermodynamics of Evolving Phase Boundaries in the Plane, Clarendon Press, Oxford 1993.
[20]	M.-Y. Jiang, Remarks on the 2-dimensional $L_p$-Minkowski problem, Advanced Nonlinear Studies, 10 (2010), 297-313.
[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), 535-565. 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), 79-120. doi: 10.1006/jfan.1994.1004.

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