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$2\pi$-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 |
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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