Two examples of fattening for the curvature flow with a driving force

Giovanni Bellettini; Maurizio Paolini

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1994)

  • Volume: 5, Issue: 3, page 229-236
  • ISSN: 1120-6330

Abstract

top
We provide two examples of a regular curve evolving by curvature with a forcing term, which degenerates in a set having an interior part after a finite time.

How to cite

top

Bellettini, Giovanni, and Paolini, Maurizio. "Two examples of fattening for the curvature flow with a driving force." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 5.3 (1994): 229-236. <http://eudml.org/doc/244251>.

@article{Bellettini1994,
abstract = {We provide two examples of a regular curve evolving by curvature with a forcing term, which degenerates in a set having an interior part after a finite time.},
author = {Bellettini, Giovanni, Paolini, Maurizio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Nonlinear partial differential equations of parabolic type; Mean curvature flow; Viscosity solutions; mean curvature flow; viscosity solutions; fattening},
language = {eng},
month = {9},
number = {3},
pages = {229-236},
publisher = {Accademia Nazionale dei Lincei},
title = {Two examples of fattening for the curvature flow with a driving force},
url = {http://eudml.org/doc/244251},
volume = {5},
year = {1994},
}

TY - JOUR
AU - Bellettini, Giovanni
AU - Paolini, Maurizio
TI - Two examples of fattening for the curvature flow with a driving force
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1994/9//
PB - Accademia Nazionale dei Lincei
VL - 5
IS - 3
SP - 229
EP - 236
AB - We provide two examples of a regular curve evolving by curvature with a forcing term, which degenerates in a set having an interior part after a finite time.
LA - eng
KW - Nonlinear partial differential equations of parabolic type; Mean curvature flow; Viscosity solutions; mean curvature flow; viscosity solutions; fattening
UR - http://eudml.org/doc/244251
ER -

References

top
  1. ALMGREN, F. - TAYLOR, J. E. - WANG, L., Curvature-driven flows: a variational approach. SIAM J. Control Optim., 31, 1993, 387-437. Zbl0783.35002MR1205983DOI10.1137/0331020
  2. ANGENENT, S., On the formation of singularities in the curve shortening flow. J. Differential Geom., 33, 3, 1991, 601-633. Zbl0731.53002MR1100205
  3. BARLES, G. - SONER, H. M. - SOUGANIDIS, P. E., Front propagation and phase field theory. SIAM J. Control Optim., 31, 1993, 439-469. Zbl0785.35049MR1205984DOI10.1137/0331021
  4. BRAKKE, K. A., The Motion of a Surface by its Mean Curvature. Princeton University Press, Princeton1978. Zbl0386.53047MR485012
  5. BRONSARD, L. - KOHN, R. V., Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics. J. Differential Equations, 90, 1991, 211-237. Zbl0735.35072MR1101239DOI10.1016/0022-0396(91)90147-2
  6. CHEN, Y. G. - GIGA, Y. - GOTO, S., Uniqueness and existence of viscosity solutions of generalized mean curvature flow equation. J. Differential Geom., 33, 1991, 749-786. Zbl0696.35087MR1100211
  7. CRANDALL, M. G. - ISHII, H. - LIONS, P. L., Users guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc.(N.S.), 27, 1992, 1-67. Zbl0755.35015MR1118699DOI10.1090/S0273-0979-1992-00266-5
  8. . Trans. Amer. Math. Soc., 227, 1983, 1-42. Zbl0599.35024MR690039DOI10.2307/1999343
  9. DE GIORGI, E., Some conjectures on flow by mean curvature. In: M. L. BENEVENTO - T. BRUNO - C. SBORDONE (eds.), Methods of real analysis and partial differential equations. Liguori, Napoli1990. Zbl0840.35042
  10. DE GIORGI, E., Conjectures on limits of some quasi linear parabolic equations and flow by mean curvature. Lecture delivered at the meeting on Partial Differential Equations and Related Topics in honour of L. Niremberg, Trento, September 3-7, 1990. Zbl0802.35063
  11. DE GIORGI, E., Congetture sui limiti delle soluzioni di alcune equazioni paraboliche quasi lineari. In: Nonlinear Analysis. A Tribute in Honour of G. Prodi. S.N.S. Quaderni, Pisa1991, 173-187. Zbl0840.35012
  12. EVANS, L. C. - SONER, H.-M. - SOUGANIDIS, P. E., Phase transitions and generalized motion by mean curvature. Comm. Pure Appl. Math., 45, 1992, 1097-1123. Zbl0801.35045MR1177477DOI10.1002/cpa.3160450903
  13. EVANS, L. C. - SPRUCK, J., Motion of level sets by mean curvature. I. J. Differential Geom., 33, 1991, 635-681. Zbl0726.53029MR1100206
  14. EVANS, L. C. - SPRUCK, J., Motion of level sets by mean curvature. II. Trans. Amer. Math. Soc., 330, 1992, 321-332. Zbl0776.53005MR1068927DOI10.2307/2154167
  15. EVANS, L. C. - SPRUCK, J., Motion of level sets by mean curvature. III. J. Geom. An., 2, 1992, 121-150. Zbl0768.53003MR1151756DOI10.1007/BF02921385
  16. GIGA, Y. - GOTO, S. - ISHII, H. - SATO, M. H., Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Indiana Univ. Math. J., 40, 1991, 443-470. Zbl0836.35009MR1119185DOI10.1512/iumj.1991.40.40023
  17. GRAYSON, M., The heat equation shrinks embedded plane curves to round points. J. Differential Geom., 26, 1987, 285-314. Zbl0667.53001MR906392
  18. HUISKEN, G., Asymptotic behavior for singularities of the mean curvature flow. J. Differential Geom., 31, 1990, 285-299. Zbl0694.53005MR1030675
  19. ILMANEN, T., Generalized flow of sets by mean curvature on a manifold. Indiana Univ. Math. J., 41, 1992, 671-705. Zbl0759.53035MR1189906DOI10.1512/iumj.1992.41.41036
  20. ILMANEN, T., Convergence of the Allen-Cahn equation to Brakkes motion by mean curvature. J. Differential Geom., 38, 1993, 417-461. Zbl0784.53035MR1237490
  21. JENSEN, R., The maximum principle for viscosity solutions of second-order fully nonlinear partial differential equations. Arch. Rational Mech. Anal., 101, 1988, 1-27. Zbl0708.35019MR920674DOI10.1007/BF00281780
  22. LIONS, P.-L., Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, I. Comm. Partial Differential Equations, 8, 1983, 1101-1134. Zbl0716.49022MR709164DOI10.1080/03605308308820297
  23. PAOLINI, M. - VERDI, C., Asymptotic and numerical analyses of the mean curvature flow with a space-dependent relaxation parameter. Asymptotic Anal., 5, 1992, 553-574. Zbl0757.65078MR1169358
  24. SONER, H. M., Motion of a set by the curvature of its boundary. J. Differential Equations, 101, 1993, 313-372. Zbl0769.35070MR1204331DOI10.1006/jdeq.1993.1015
  25. SONER, H. M., Ginzburg-Landau equation and motion by mean curvature, I: convergence. Research report n. 93-NA-026, August 1993, Carnegie Mellon University. Zbl0935.35060
  26. SONER, H.-M. - SOUGANIDIS, P. E., Singularities and uniqueness of cylindrically symmetric surfaces moving by mean curvature. Comm. Partial Differential Equations, 18, 1993, 859-894. Zbl0804.53006MR1218522DOI10.1080/03605309308820954

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.