The self-similar expanding curve for the curvature flow equation
HTML articles powered by AMS MathViewer
- by Hua-Huai Chern, Jong-Shenq Guo and Chu-Pin Lo
- Proc. Amer. Math. Soc. 131 (2003), 3191-3201
- DOI: https://doi.org/10.1090/S0002-9939-03-07055-2
- Published electronically: April 30, 2003
- PDF | Request permission
Abstract:
We study a two-point free boundary problem for the curvature flow equation. By studying the corresponding nonlinear initial value problem, we obtain the existence and uniqueness of the forward self-similar solution of this problem. The corresponding curve is called the self-similar expanding curve. We also derive the asymptotic stability of this curve.References
- Steven J. Altschuler and Lang-Fang Wu, Convergence to translating solutions for a class of quasilinear parabolic boundary problems, Math. Ann. 295 (1993), no. 4, 761–765. MR 1214961, DOI 10.1007/BF01444916
- Steven J. Altschuler and Lang F. Wu, Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle, Calc. Var. Partial Differential Equations 2 (1994), no. 1, 101–111. MR 1384396, DOI 10.1007/BF01234317
- J. D. Buckmaster and G. S. S. Ludford, Theory of laminar flames, Electronic & Electrical Engineering Research Studies: Pattern Recognition & Image Processing Series, vol. 2, Cambridge University Press, Cambridge-New York, 1982. MR 666866
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
- Luis A. Caffarelli and Juan L. Vázquez, A free-boundary problem for the heat equation arising in flame propagation, Trans. Amer. Math. Soc. 347 (1995), no. 2, 411–441. MR 1260199, DOI 10.1090/S0002-9947-1995-1260199-7
- Yun Gang Chen, Yoshikazu Giga, and Shun’ichi Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom. 33 (1991), no. 3, 749–786. MR 1100211
- Bennett Chow and Dong-Ho Tsai, Geometric expansion of convex plane curves, J. Differential Geom. 44 (1996), no. 2, 312–330. MR 1425578
- K. Deckelnick, C. M. Elliott, and G. Richardson, Long time asymptotics for forced curvature flow with applications to the motion of a superconducting vortex, Nonlinearity 10 (1997), no. 3, 655–678. MR 1448581, DOI 10.1088/0951-7715/10/3/005
- L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom. 33 (1991), no. 3, 635–681. MR 1100206
- Victor A. Galaktionov, Josephus Hulshof, and Juan L. Vazquez, Extinction and focusing behaviour of spherical and annular flames described by a free boundary problem, J. Math. Pures Appl. (9) 76 (1997), no. 7, 563–608. MR 1472115, DOI 10.1016/S0021-7824(97)89963-1
- Y. Giga, N. Ishimura, and Y. Kohsaka, Spiral solutions for a weakly anisotropic curvature flow equation, Hokkaido University Preprint Series in Mathematics, Series #529, June 2001.
- J.-S. Guo and Y. Kohsaka, Two-point free boundary problem for heat equation, preprint.
- Morton E. Gurtin, Thermomechanics of evolving phase boundaries in the plane, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. MR 1402243
- Danielle Hilhorst and Josephus Hulshof, A free boundary focusing problem, Proc. Amer. Math. Soc. 121 (1994), no. 4, 1193–1202. MR 1233975, DOI 10.1090/S0002-9939-1994-1233975-9
- Gerhard Huisken, Nonparametric mean curvature evolution with boundary conditions, J. Differential Equations 77 (1989), no. 2, 369–378. MR 983300, DOI 10.1016/0022-0396(89)90149-6
- Hitoshi Imai, Naoyuki Ishimura, and TaKeo Ushijima, A crystalline motion of spiral-shaped curves with symmetry, J. Math. Anal. Appl. 240 (1999), no. 1, 115–127. MR 1728200, DOI 10.1006/jmaa.1999.6599
- James Keener and James Sneyd, Mathematical physiology, Interdisciplinary Applied Mathematics, vol. 8, Springer-Verlag, New York, 1998. MR 1673204
- Yoshihito Kohsaka, Free boundary problem for quasilinear parabolic equation with fixed angle of contact to a boundary, Nonlinear Anal. 45 (2001), no. 7, Ser. A: Theory Methods, 865–894. MR 1845031, DOI 10.1016/S0362-546X(99)00422-8
- Karol Mikula and Daniel Ševčovič, Evolution of plane curves driven by a nonlinear function of curvature and anisotropy, SIAM J. Appl. Math. 61 (2001), no. 5, 1473–1501. MR 1824511, DOI 10.1137/S0036139999359288
- Hirokazu Ninomiya and Masaharu Taniguchi, Traveling curved fronts of a mean curvature flow with constant driving force, Free boundary problems: theory and applications, I (Chiba, 1999) GAKUTO Internat. Ser. Math. Sci. Appl., vol. 13, Gakk\B{o}tosho, Tokyo, 2000, pp. 206–221. MR 1793036
- Hirokazu Ninomiya and Masaharu Taniguchi, Stability of traveling curved fronts in a curvature flow with driving force, Methods Appl. Anal. 8 (2001), no. 3, 429–449. MR 1904754, DOI 10.4310/MAA.2001.v8.n3.a4
- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Springer-Verlag, New York, 1984. Corrected reprint of the 1967 original. MR 762825, DOI 10.1007/978-1-4612-5282-5
- J. A. Sethian, Level set methods and fast marching methods, 2nd ed., Cambridge Monographs on Applied and Computational Mathematics, vol. 3, Cambridge University Press, Cambridge, 1999. Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science. MR 1700751
- J. L. Vazquez, The free boundary problem for the heat equation with fixed gradient condition, Free boundary problems, theory and applications (Zakopane, 1995) Pitman Res. Notes Math. Ser., vol. 363, Longman, Harlow, 1996, pp. 277–302. MR 1462990
Bibliographic Information
- Hua-Huai Chern
- Affiliation: Department of Computer and Information Sciences, National Taiwan Ocean University, 2, Pei-Ning Road, Keelung, Taiwan
- Email: felix@cs.ntou.edu.tw
- Jong-Shenq Guo
- Affiliation: Department of Mathematics, National Taiwan Normal University, 88, S-4 Ting Chou Road, Taipei 117, Taiwan
- Email: jsguo@math.ntnu.edu.tw
- Chu-Pin Lo
- Affiliation: Department of Applied Mathematics, Providence University, 200, Chung-Chi Road, Shalu, Taichung County 433, Taiwan
- Email: cplo@pu.edu.tw
- Received by editor(s): May 16, 2002
- Published electronically: April 30, 2003
- Communicated by: David S. Tartakoff
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 3191-3201
- MSC (2000): Primary 35B60, 34A12, 35B35
- DOI: https://doi.org/10.1090/S0002-9939-03-07055-2
- MathSciNet review: 1992860