Self-similar solutions for the anisotropic affine curve shortening problem

Calculus of Variations and Partial Differential Equations

Volumn 13, Issue 3, 2001, Pages 311-337

  Ai, Jun ^a,b   Chou, Kai Seng ^a   Wei, Juncheng ^a  

^a CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

^b SUN YAT SEN UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0346491847     PISSN: 09442669     EISSN: None     Source Type: Journal    
DOI: 10.1007/s005260000075     Document Type: Article

References (29)

1. U. Abresh, J. Langer, The normalized curve shortening flow and homothetic solutions, J. Diff. Geom. 23 (1986), 175-196
2. J. Ai, A variational theory for some group invariant solutions, Thesis in preparation
3. S. Altschuler, Singularities of the curve shrinking flow for space curves, J. Diff. Geom. 34 (1991), 491-514
4. B. Andrews, Contraction of convex hypersurfaces by their affine normal, J. Diff. Geom. 43 (1996), 207-230
5. B. Andrews, Evolving convex curves, Calc. Var. 7 (1998), 315-371
6. S. Angenent, On the formation of singularities in the curve shortening flow, J. Diff. Geom. 33 (1991), 601-634
7. S. Angenent, M.E. Gurtin, Multiphase thermodynamics with interfacial structure 2. evolution of an isothermal interface, Arch. Rational Mech. Anal. 108 (1989), 323-391
8. 2, Trans. Amer. Math. Soc. 303 (1987), 369-382
9. W.X. Chen, C.M. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math. 145 (1997), 547-564
10. K.C. Chang, J.Q. Liu, On Nirenberg's problem, International J. Math. 4(1993), 35-58
11. 2, Acta Math. 159 (1987), 215-259
12. n, Duke Math. J. 64 (1991), 27-69
13. S.Y.A. Chang, M.J. Gursky, P.C Yang, The scalar curvature equation on 2- and 3-spheres, Calc. Var. 1 (1993), 205-229
14. K.S. Chou, L. Zhang, On the uniqueness of stable ultimate shapes for the anisotropic curve shortening problem
15. K.S. Chou, X.P. Zhu, Variational identities and conservation laws, Preprint, 1996
16. K.S. Chou, X.P. Zhu, Anisotropic flows for convex plane curves, Duke Math. J. 97 (1999), 579-619
17. C. Dohmen, Y. Giga, Selfsimilar shrinking curves for anisotropic curvature flow equations, Proc. Japan Acad. Ser. A 70 (1994), 252-255
18. C. Dohmen, Y. Giga, N. Mizoguchi, Existence of selfsimilar shrinking curves for anisotropic curvature flow equations, Calc. Var. 4 (1996), 103-119
19. M.E. Gage, Evolving plane curve by curvature in relative geometries , Duke Math. J. 72 (1993), 441-466
20. M.E. Gage, R. Hamilton, The heat equation shrinking convex plane curves, J. Diff. Geom. 23 (1996), 69-96
21. M.E. Gage, Y. Li, Evolving plane curve by curvature in relative geometries II, Duke Math. 75 (1994), 79-98
22. M. Grayson, The heat equation shrinking embedded curves to round points, J. Diff. Geom. 26 (1987), 285-314
23. M.E. Gurtin, Thermodynamics of evolving phase boundaries in the plane, Clarendon press, Oxford 1993
24. 2, Duke Math. J. 61 (1990), 679-703. N.H. Ibragimov, Lie Group Analysis of Differential Equations Vol. 3, CRC Press, Boca Raton 1996
25. 2, Duke Math. J. 61 (1990), 679-703. N.H. Ibragimov, Lie Group Analysis of Differential Equations Vol. 3, CRC Press, Boca Raton 1996
26. Y. Li, On Nirenberg's problem and related topics, Topological Math. Nonlinear Anal. 6(1994), 309-329
27. G. Sapiro, A. Tannenbaum, On affine plane curve evolution, J. Funct. Anal. 119 (1994), 79-120
28. n, J. Funct. Anal. 157 (1998), 292-325
29. X.W. Xu, P.C. Yang, Remarks on prescribing Gauss curvature, Trans. Amer. Math. Soc. 336 (1993), 831-840