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

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 & 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. Google Scholar
[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. Google Scholar
[3]	S. Altschuler, Singularities of the curve shrinking flow for space curves,, J. Differential Geometry, 34 (1991), 491. Google Scholar
[4]	B. Andrews, Contraction of convex hypersurfaces by their affine normal,, J. Differential Geometry, 43 (1996), 207. Google Scholar
[5]	B. Andrews, Evolving convex curves,, Calc. Var., 7 (1998), 315. doi: 10.1007/s005260050111. Google Scholar
[6]	S. Angenent, On the formation of singularities in the curve shortening flow,, J. Differential Geometry, 33 (1991), 601. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[14]	I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications,, Oxford Science Publications, (1995). Google Scholar
[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. Google Scholar
[16]	M. E. Gage and R. Hamilton, The heat equation shrinking convex plane curves,, J. Differential Geometry, 23 (1986), 69. Google Scholar
[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. Google Scholar
[18]	M. Grayson, The heat equation shrinking embedded curves to round points,, J. Differential Geometry, 26 (1987), 285. Google Scholar
[19]	M. E. Gurtin, Thermodynamics of Evolving Phase Boundaries in the Plane,, Clarendon Press, (1993). Google Scholar
[20]	M.-Y. Jiang, Remarks on the 2-dimensional $L_p$-Minkowski problem,, Advanced Nonlinear Studies, 10 (2010), 297. Google Scholar
[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. Google Scholar
[22]	H. Matano and J. Wei, On anisotropic curvature flow equations,, preprint., (). Google Scholar
[23]	G. Sapiro and A. Tannenbaum, On affine plane curve evolution,, J. Funct. Anal., 119 (1994), 79. doi: 10.1006/jfan.1994.1004. Google Scholar

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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. Google Scholar
[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. Google Scholar
[3]	S. Altschuler, Singularities of the curve shrinking flow for space curves,, J. Differential Geometry, 34 (1991), 491. Google Scholar
[4]	B. Andrews, Contraction of convex hypersurfaces by their affine normal,, J. Differential Geometry, 43 (1996), 207. Google Scholar
[5]	B. Andrews, Evolving convex curves,, Calc. Var., 7 (1998), 315. doi: 10.1007/s005260050111. Google Scholar
[6]	S. Angenent, On the formation of singularities in the curve shortening flow,, J. Differential Geometry, 33 (1991), 601. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[14]	I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications,, Oxford Science Publications, (1995). Google Scholar
[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. Google Scholar
[16]	M. E. Gage and R. Hamilton, The heat equation shrinking convex plane curves,, J. Differential Geometry, 23 (1986), 69. Google Scholar
[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. Google Scholar
[18]	M. Grayson, The heat equation shrinking embedded curves to round points,, J. Differential Geometry, 26 (1987), 285. Google Scholar
[19]	M. E. Gurtin, Thermodynamics of Evolving Phase Boundaries in the Plane,, Clarendon Press, (1993). Google Scholar
[20]	M.-Y. Jiang, Remarks on the 2-dimensional $L_p$-Minkowski problem,, Advanced Nonlinear Studies, 10 (2010), 297. Google Scholar
[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. Google Scholar
[22]	H. Matano and J. Wei, On anisotropic curvature flow equations,, preprint., (). Google Scholar
[23]	G. Sapiro and A. Tannenbaum, On affine plane curve evolution,, J. Funct. Anal., 119 (1994), 79. doi: 10.1006/jfan.1994.1004. Google Scholar

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