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Mean curvature flow through singularities for surfaces of rotation. (English) Zbl 0847.58072
The authors consider generalized “viscosity” solutions of the mean curvature evolution equation. Much of the paper is devoted to the case of initial values given by a compact, smooth, rotationally symmetric hypersurface obtained by rotating a graph around an axis. The authors prove the regularity of the solutions except at isolated points and give an estimate on the number of such points.
Reviewer: M.Biroli (Monza)

MSC:
58J35 Heat and other parabolic equation methods for PDEs on manifolds
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
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[1] Altschuler, S. J. Singularities of the curve shrinking flow for space curves.Journal of Differential Geometry 34(2), 491–514 (1991). · Zbl 0754.53006
[2] Altschuler, S. J., Angenent, S. B., and Giga, Y. Generalized motion by mean curvature for surfaces of rotation.Advanced Studies in Pure Mathematics, to appear; also Hokkaido University Preprint #118 (1991).
[3] Altschuler, S. J., and Grayson, M. Shortening space curves and flow through singularities.Journal of Differential Geometry 35, 283–298 (1992). · Zbl 0782.53001
[4] Angenent, S. B. The zeroset a a solution of a parabolic equation.Journal für die reine und angewandte Mathematik 390, 79–96 (1988). · Zbl 0644.35050
[5] Angenent, S.B. On the formation of singularities in the curve shortening problem.Journal of Differential Geometry 33, 601–633 (1991). · Zbl 0731.53002
[6] Angenent, S. B. Parabolic equations for curves on surfaces-part 2.Annals of Mathematics 133, 171–215 (1991). · Zbl 0749.58054 · doi:10.2307/2944327
[7] Angenent, S. B. Solutions of the 1-D porous medium equation are determined by their free boundary.Journal of the London Math. Soc. 42, 339–353 (1990). · Zbl 0679.35040 · doi:10.1112/jlms/s2-42.2.339
[8] Angenent, S. B. Shrinking doughnuts. To appear in theProceedings of the Conference on Nonlinear Parabolic PDE, Gregynog-Wales, August 1989.
[9] Brakke, K. A.The Motion of a Surface by Its Mean Curvature. Math. Notes. Princeton University Press, Princeton, NJ, 1978. · Zbl 0386.53047
[10] Chen, X.-Y. Private communication, 1990.
[11] Chen, X.-Y., and Matano, H. Convergence, asymptotic periodicity, and finite point blow-up in one dimensional semilinear heat equations.Journal of Differential Equations 78, 160–190 (1989). · Zbl 0692.35013 · doi:10.1016/0022-0396(89)90081-8
[12] Chen, Y.-G., Giga, Y., and Goto, S. Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations.Journal of Differential Geometry 33, 749–786 (1991); announcement in:Proc. Japan Acad. Ser. A 65 207–210 (1989). · Zbl 0696.35087
[13] Chen, Y.-G., Giga, Y., and Goto, S. Analysis toward snow crystal growth.Proceedings of International Symposium on Functional Analysis and Related Topics, ed. S. Koshi, pp. 43–57. World Scientific, Singapore, 1991. · Zbl 0817.76090
[14] Crandall, M. G., Ishii, H., and Lions, P. L. User’s guide to viscosity solutions of second order partial differential equations. Preprint. · Zbl 0755.35015
[15] DeGiorgi, E. Some conjectures on flow by mean curvature. Preprint.
[16] Dziuk, G., and Kawohl, B. On rotationally symmetric mean curvature flow.J. of Diff. Equations 93, 142–149 (1991). · Zbl 0749.53001 · doi:10.1016/0022-0396(91)90024-4
[17] Ecker, K., and Huisken, G. Mean curvature for entire graphs.Annals of Mathematics 130, 453–471 (1989). · Zbl 0696.53036 · doi:10.2307/1971452
[18] Ecker, K., and Huisken, G. Interior estimates for hypersurfaces moving by mean curvature.Inventiones Mathematicae 105, 547–569 (1991). · Zbl 0725.53009 · doi:10.1007/BF01232278
[19] Evans, L. C., Soner, H. M., and Souganidis, P. E. Phase transitions and generalized mean curvature flow equations. To appear inComm. Pure Appl. Math. · Zbl 0801.35045
[20] Evans, L. C., and Spruck, J. Motion of level sets by mean curvature I.Journal of Differential Geometry 33, 635–681 (1991). · Zbl 0726.53029
[21] Evans, L. C., and Spruck, J. Motion of level sets by mean curvature II.Trans. Amer. Math. Soc. 330, 321–332 (1992). · Zbl 0776.53005 · doi:10.1090/S0002-9947-1992-1068927-8
[22] Evans, L. C., and Spruck, J. Motion of level sets by mean curvature III.J. Geom. Anal. 2, 121–150 (1992). · Zbl 0768.53003 · doi:10.1007/BF02921385
[23] Gage, M., and Hamilton, R. S. The heat equation shrinking convex plane curves.Journal of Differential Geometry 23, 69–96 (1986). · Zbl 0621.53001
[24] Giga, Y., and Goto, S. Motion of hypersurfaces and geometric equations.J. Math. Soc. Japan 44, 99–111 (1992). · Zbl 0739.53005 · doi:10.2969/jmsj/04410099
[25] Giga, Y., and Goto, S.On the Evolution of Phase Boundaries, eds. M. Gurtin and G. McFadden, IMA Volumes in Mathematics and Its Applications, Vol. 43, pp. 51–66. Springer-Verlag, New York, 1992. · Zbl 0771.35027
[26] Giga, Y., Goto, S., and Ishii, H. Global existence of weak solutions for interface equations coupled with diffusion equations.SIAM J. Math. Anal. 23, 821–835 (1992). · Zbl 0754.35061 · doi:10.1137/0523043
[27] Giga, Y., Goto, S., Ishii, H., and Sato, M.-H. Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains.Indiana University Math. Journal 40, 443–470 (1990). · Zbl 0836.35009 · doi:10.1512/iumj.1991.40.40023
[28] Giga, Y., and Kohn, R. Asymptotically self-similar blow-up of semilinear heat equations.Comm. Pure and Appl. Math. 38, 297–319 (1985). · Zbl 0585.35051 · doi:10.1002/cpa.3160380304
[29] Galaktionov, Victor A., and Posashkov, Sergei A. On some monotonicity in time properties for a quasilinear parabolic equation with source. InDegenerate Diffusions, eds. W.-M. Ni, L. A. Peletier, and J. L. Vazquez, IMA Volumes in Mathematics and Its Applications, Vol. 47, pp. 77–93. Springer-Verlag, New York, 1993. · Zbl 0827.35054
[30] Grayson, M. The heat equation shrinks embedded plane curves to round points.Journal of Differential Geometry 26, 285–314 (1987). · Zbl 0667.53001
[31] Grayson, M. A short note on the evolution of a surface by its mean curvature.Duke Math. Journal 58, 555–558 (1989). · Zbl 0677.53059 · doi:10.1215/S0012-7094-89-05825-0
[32] Huisken, G. Flow by mean curvature of convex surfaces into spheres.Journal of Differential Geometry 20, 237–266 (1984). · Zbl 0556.53001
[33] Huisken, G. Asymptotic behaviour for singularities of the mean curvature flow.Journal of Differential Geometry 31, 285–299 (1991). · Zbl 0694.53005
[34] Huisken, G. Local and global behaviour of hypersurfaces moving by mean curvature. C. M. A. Australian National University Preprint CMA-R34-90, 1990.
[35] Ilmanen, T. Generalized flow of sets by mean curvature on a manifold. Preprint. · Zbl 0759.53035
[36] Ilmanen, T. The level-set flow on a manifold. Preprint. · Zbl 0827.53014
[37] Korevaar, N. An easy proof of the interior gradient bound for solutions to the prescribed mean curvature equation.Nonlinear functional analysis and its applications, Proc. Symp. Pure Math. 45, part 2, 81–89 (1986).
[38] Ladyzhenskaya, O. A., Solonnikov, V., and Ural’ceva, N. Linear and quasilinear equations of parabolic type.Translations of Mathematical Monographs,23 A. M. S. (1968).
[39] Matano, H. Nonincrease of the lapnumber for a solution of a one–dimensional semi–linear parabolic equation.J. Fac. Sci. Univ. Tokyo, Sec. IA,29, 401–441 (1982). · Zbl 0496.35011
[40] Ohta, T., Jasnow, D., and Kawasaki, K. Universal scaling in the motion of random interfaces.Physics Review Letters 49, 1223–1226 (1982). · doi:10.1103/PhysRevLett.49.1223
[41] Osher, S., and Sethian, J. A. Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations.J. Comp. Phys. 79, 12–49 (1988). · Zbl 0659.65132 · doi:10.1016/0021-9991(88)90002-2
[42] Sethian, J. A. Curvature and evolution of fronts.Comm. Math. Physics 101, 12–49 (1985). · Zbl 0619.76087 · doi:10.1007/BF01210742
[43] Sethian, J. A. Numerical for propagating interfaces: Hamilton-Jacobi equations and conservation laws.Journal of Differential Geometry 31, 131–161 (1990). · Zbl 0691.65082
[44] Soner, H. M. Motion of a set by the curvature of its boundary. To appear inJournal of Differential Equations. · Zbl 0769.35070
[45] Soner, H. M., and Souganidis, P. E. Uniqueness and singularities of cylindrically symmetric surfaces moving by mean curvature. Preprint. · Zbl 0804.53006
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