Skip to main content
SpringerLink
Log in

Motion by curvature of planar curves with end points moving freely on a line

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

This paper deals with the motion by curvature of planar curves having end points moving freely along a line with fixed contact angles to this line. We first prove the existence and uniqueness of self-similar shrinking solution. Then we show that the curve shrinks to a point in a self-similar manner, if initially the curve is a graph.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Abresch U., Langer J.: The normalized curve shortening flow and homothetic solutions. J. Differ. Geom. 23, 175–196 (1986)

    MathSciNet  MATH  Google Scholar 

  2. Adams B.L., Ta’asan S., Kinderlehrer D., Livshits I., Mason D.E., Wu C.-T., Mullins W.W., Rother G.S., Rollett A.D., Saylor D.M.: Extracting grain boundary and surface energy measurement of triple junction geometry. Interface Sci. 7, 321–338 (1999)

    Article  Google Scholar 

  3. Adams B.L., Kinderlehrer D., Mullins W.W., Rollett A.D., Ta’asan S.: Extracting the relative grain boundary free energy and mobility function from the geometry of microstructures. Scripta Materialia 38, 531–536 (1998)

    Article  Google Scholar 

  4. Allen S., Cahn J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta. Metall. 27, 1084–1095 (1979)

    Google Scholar 

  5. Altschuler S.J.: Singularities of the curve shrinking flow for space curves. J. Differ. Geom. 34, 491–514 (1991)

    MathSciNet  MATH  Google Scholar 

  6. Andrews B.: Classification of limiting shapes for isotropic curve flows. J. Am. Math. Soc. 16, 443–459 (2003)

    Article  MATH  Google Scholar 

  7. Angenent S.: On the formation of sinularities in the curve shortening flow. J. Differ. Geom. 33, 601–633 (1991)

    MathSciNet  MATH  Google Scholar 

  8. Angenent S., Gurtin M.E.: Multiphase thermomechanics with interfacial structure. 2. Evolution of an isothermal interface. Arch. Ration. Mech. Anal. 108, 323–391 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bellettini, G., Novaga, M.: Curvature evolution of nonconvex lens-shaped domains. J. Reine Angew. Math. (to appear)

  10. Brakke K.A.: The Motion of a Surface by its Mean Curvature. Princeton University Press, Princeton (1978)

    MATH  Google Scholar 

  11. Bronsard L., Kohn R.: Motion by mean curvature as the singular limit of Ginzburgh-Landau dynamics. J. Differ. Equ. 90, 211–237 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bronsard L., Reitich F.: On three-phase boundary motion and the singular limit of a vector valued Ginzburg-Landau equation. Arch. Ration. Mech. Anal. 124, 355–379 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chang Y.-L., Guo J.-S., Kohsaka Y.: On a two-point free boundary problem for a quasilinear parabolic equation. Asymptot. Anal. 34, 333–358 (2003)

    MathSciNet  MATH  Google Scholar 

  14. Chen X.: Generation and propagation of interface in reaction–diffusion equations. J. Differ. Equ. 96, 116–141 (1992)

    Article  MATH  Google Scholar 

  15. Chen X., Guo J.-S.: Self-similar solutions of a 2-D multiple phase curvature flow. Physica D 229, 22–34 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chen Y.G., Giga Y., Goto S.: Uniqueness and existence of viscosity solution of generalized mean curvature flow equations. J. Differ. Geom. 33, 749–786 (1991)

    MathSciNet  MATH  Google Scholar 

  17. Daskalopoulos P., Hamilton R.: Regularity of the free boundary for the porous medium equation. J. Am. Math. Soc. 11, 899–965 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. De Mottoni P., Schatzman M.: Development of interfaces in R N. Proc. Roy. Soc. Edinb. Sect. A 116, 207–220 (1990)

    MathSciNet  MATH  Google Scholar 

  19. DiBenedetto E.: Degenerate Parabolic Equations. Springer, New York (1993)

    MATH  Google Scholar 

  20. Evans L.C., Spruck J.: Motion of level sets by mean curvature I. J. Differ. Geom. 33, 635–681 (1991)

    MathSciNet  MATH  Google Scholar 

  21. Evans L.C., Spruck J.: Motion of level sets by mean curvature II. Trans. Am. Math. Soc. 330, 321–332 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  22. Evans L.C., Spruck J.: Motion of level sets by mean curvature III. J. Geom. Anal. 2, 121–150 (1992)

    MathSciNet  MATH  Google Scholar 

  23. Evans L.C., Spruck J.: Motion of level sets by mean curvature IV. J. Geom. Anal. 5, 77–114 (1995)

    MathSciNet  Google Scholar 

  24. Evans L.C., Soner H.M., Souganidis P.E.: The Allen–Cahn equation and the generalized motion by mean curvature. Commun. Pure Appl. Math. XLV, 1097–1123 (1992)

    Article  MathSciNet  Google Scholar 

  25. Fife, P.C.: Dynamics of Internal Layers and Diffusive Interfaces. In: CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1988)

  26. Giga, Y.: Surface evolution equations. A level set approach, Monographs in Mathematics, 99, Birkhäuser, Basel (2006)

  27. Grayson M.A.: The heat equation shrinks embedded plane curves to round points. J. Differ. Geom. 26, 285–314 (1987)

    MathSciNet  MATH  Google Scholar 

  28. Guo J.-S., Hu B.: On a two-point free boundary problem. Quart. Appl. Math. 64, 413–431 (2006)

    MathSciNet  MATH  Google Scholar 

  29. Gurtin M.: Thermomechanics of Evolving Phase Boundaries in the Plane. Oxford Science Publication, London (1993)

    MATH  Google Scholar 

  30. Herring C.: Surface tension as motivation for sintering. In: Kingston, W. (ed) The Physica of Powder Metallury, McGraw-Hill, New York (1951)

    Google Scholar 

  31. Herring C.: The use of classical macroscopic concepts in surface-energy problems. In: Gomer, R., Smith, S. (eds) Structure and Properties of Solid Surfaces, University of Chicago Press, Chicago (1952)

    Google Scholar 

  32. Ilmanen, T.: Elliptic Regularization and Partial Regularity for Motion by Mean Curvature. Mem. Am. Math. Soc. 108(520) (1994)

  33. Kinderlehrer D., Liu C.: Evolution of grain boundaries. Math. Models Methods Appl. Sci. 11, 713–729 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kinderlehrer, D., Livshits, I., Manolache, F., Rollett, A.D., Ta’asan, S.: An approach to the mesoscale simulation of grain growth. In: Aindow, M. et al. (ed.) Influences of interface and dislocation behavior on microstructure evolution. Mat. Res. Soc. Symp. Proc., vol. 652 (2001)

  35. Ladyzenskaya, O.A., Solonnikov, V.A., Uralt́zeva, N.N.: Linear and quasilinear equations of parabolic type. Transl. Math. Mono., vol. 23. American Mathematical Society, Providence (1968)

  36. Kobayashi R.: Modelling and numerical simulation of dendritic crystal growth. Physica D 63, 410–423 (1993)

    Article  MATH  Google Scholar 

  37. Kobayashi R., Warren J.A., Carter W.C.: Vector valued phase field model for crystallization and grain boundary formation. Physica D 119, 415–423 (1998)

    Article  Google Scholar 

  38. Kobayashi R., Warren J.A., Carter W.C.: A continuum model of grain boundaries. Physica D 140, 141–150 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  39. Mantegazza C., Novaga M., Tortorelli V.M.: Motion by curvature of planar networks. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3, 235–324 (2004)

    MathSciNet  MATH  Google Scholar 

  40. Mullins, W.: Solid surface morphologies governed by capillarity. In: Metal Surface: Structure, Energetics and Kinetics, pp. 17–66. Am Soc. Metals, Metals Park (1963)

  41. Mullins W.: Two dimensional motion of idealized grain boundaries. J. Appl. Phys. 27, 900–904 (1956)

    Article  MathSciNet  Google Scholar 

  42. Mullins W.: On idealized two-dimensional grain growth. Scripta Metall. 22, 1441–1444 (1988)

    Article  Google Scholar 

  43. Rubinstein J., Sternberg P., Keller J.B.: Fast reaction, slow diffusion, and curve shortening. SIAM J. Appl. Math. 111, 116–133 (1989)

    Article  MathSciNet  Google Scholar 

  44. Rubinstein J., Sternberg P., Keller J.B.: Front interaction and nonhomogeneous equilibria for tristable reaction–diffusion equations. SIAM J. Appl. Math. 53, 1669–1685 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  45. Schnürer, O.C., Azouani, A., Georgi, M., Hell, J., Jangle, N., Koeller, A., Marxen, T., Ritthaler, S., Sáez, M., Schulze, F., Smith, B.: Evolution of convex lens-shaped networks under curve shortening flow, Mathematics, arXiv:0711.1108 in http://www.au.arxiv.org/

  46. Soner H.M.: Motion of a set by the mean curvature of its boundary. J. Differ. Equ. 101, 313–372 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  47. Sutton A., Baluffi R.: Interface in Crystalline Materials. Oxford Science Publication, London (1995)

    Google Scholar 

  48. Woodruff D.: The Solid-Liquid Interface. Cambridge University Press, London (1973)

    Google Scholar 

  49. Zelenjak T.I.: Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable. Differ. Equ. 4, 17–22 (1968)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Xinfu Chen

  2. Department of Mathematics, Tamkang University, Tamsui, Taipei County, 25137, Taiwan

    Jong-Shenq Guo

Authors
  1. Xinfu Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jong-Shenq Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jong-Shenq Guo.

Additional information

This work is partially supported by the National Science Foundation grant DMS-0504691 and the National Science Council of the Republic of China grant NSC 95-2115-M-003-001. We would like to thank the referee for careful reading and valuable comments.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, X., Guo, JS. Motion by curvature of planar curves with end points moving freely on a line. Math. Ann. 350, 277–311 (2011). https://doi.org/10.1007/s00208-010-0558-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-010-0558-7

Keywords

Search

Navigation