Barriers for a class of geometric evolution problems

Giovanni Bellettini; Matteo Novaga

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1997)

  • Volume: 8, Issue: 2, page 119-128
  • ISSN: 1120-6330

Abstract

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We present some general results on minimal barriers in the sense of De Giorgi for geometric evolution problems. We also compare minimal barriers with viscosity solutions for fully nonlinear geometric problems of the form u t + F ⁢ t , x , ∇ ⁡ u , ∇ 2 ⁡ u = 0 . If F is not degenerate elliptic, it turns out that we obtain the same minimal barriers if we replace F with F + , which is defined as the smallest degenerate elliptic function above F .

How to cite

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Bellettini, Giovanni, and Novaga, Matteo. "Barriers for a class of geometric evolution problems." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 8.2 (1997): 119-128. <http://eudml.org/doc/244151>.

@article{Bellettini1997,
abstract = {We present some general results on minimal barriers in the sense of De Giorgi for geometric evolution problems. We also compare minimal barriers with viscosity solutions for fully nonlinear geometric problems of the form \( u\_\{t\} + F(t, x, \nabla u, \nabla^\{2\} u) = 0 \). If \( F \) is not degenerate elliptic, it turns out that we obtain the same minimal barriers if we replace \( F \) with \( F^\{+\} \), which is defined as the smallest degenerate elliptic function above \( F \).},
author = {Bellettini, Giovanni, Novaga, Matteo},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Barriers; Nonlinear partial differential equations of parabolic type; Mean curvature flow; Viscosity solutions; minimal barriers; geometric evolution equations; viscosity solutions},
language = {eng},
month = {7},
number = {2},
pages = {119-128},
publisher = {Accademia Nazionale dei Lincei},
title = {Barriers for a class of geometric evolution problems},
url = {http://eudml.org/doc/244151},
volume = {8},
year = {1997},
}

TY - JOUR
AU - Bellettini, Giovanni
AU - Novaga, Matteo
TI - Barriers for a class of geometric evolution problems
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1997/7//
PB - Accademia Nazionale dei Lincei
VL - 8
IS - 2
SP - 119
EP - 128
AB - We present some general results on minimal barriers in the sense of De Giorgi for geometric evolution problems. We also compare minimal barriers with viscosity solutions for fully nonlinear geometric problems of the form \( u_{t} + F(t, x, \nabla u, \nabla^{2} u) = 0 \). If \( F \) is not degenerate elliptic, it turns out that we obtain the same minimal barriers if we replace \( F \) with \( F^{+} \), which is defined as the smallest degenerate elliptic function above \( F \).
LA - eng
KW - Barriers; Nonlinear partial differential equations of parabolic type; Mean curvature flow; Viscosity solutions; minimal barriers; geometric evolution equations; viscosity solutions
UR - http://eudml.org/doc/244151
ER -

References

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  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. BELLETTINI, G. - NOVAGA, M., Minimal barriers for geometric evolutions. Preprint Univ. Pisa, gennaio 1997; J. Diff. Eq., to appear. Zbl0882.35028MR1467354DOI10.1006/jdeq.1997.3288
  3. BELLETTINI, G. - NOVAGA, M., Comparison results between minimal barriers and viscosity solutions for geometric evolutions. Preprint Univ. Pisa n. 2.252.998, ottobre 1996. Zbl0904.35041MR1632984
  4. BELLETTINI, G. - PAOLINI, M., Some results on minimal barriers in the sense of De Giorgi applied to driven motion by mean curvature. Rend. Acc. Naz. Sci., XLMem. Mat. (5), 19, 1995, 43-67. Zbl0944.53039MR1387549
  5. 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
  6. 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
  7. DE GIORGI, E., New problems on minimizing movements. In: J.-L. LIONS - C. BAIOCCHI (eds.), Boundary Value Problems for Partial Differential Equations and Applications. Vol. 29, Masson, Paris1993. Zbl0851.35052MR1260440
  8. DE GIORGI, E., Barriers, boundaries, motion of manifolds. Conference held at Dipartimento di Matematica of Pavia, March 18, 1994. 
  9. DE GIORGI, E., Congetture riguardanti alcuni problemi di evoluzione. Duke Math. J., 81, 1996, 255-268. Zbl0874.35027MR1395405DOI10.1215/S0012-7094-96-08114-4
  10. EVANS, L. C. - SPRUCK, J., Motion of level sets by mean curvature. I. J. Differential Geom., 33, 1991, 635-681. Zbl0726.53029MR1100206
  11. 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
  12. GOTO, S., Generalized motions of hypersurfaces whose growth speed depends superlinearly on the curvature tensor. Differential Integral Equations, 7, 1994, 323-343. Zbl0808.35007MR1255892
  13. ILMANEN, T., The level-set flow on a manifold. In: R. GREENE - S. T. YAU (eds.), Proc. of the 1990 Summer Inst. in Diff. Geom.Amer. Math. Soc., 1992. Zbl0827.53014MR1216585
  14. ILMANEN, T., Elliptic Regularization and Partial Regularity for Motion by Mean Curvature. Memoirs of the Amer. Math. Soc., 250, 1994, 1-90. Zbl0798.35066MR1196160
  15. ISHII, H. - SOUGANIDIS, P. E., Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor. Tohoku Math. J., 47, 1995, 227-250. Zbl0837.35066MR1329522DOI10.2748/tmj/1178225593
  16. SONER, H.-M., Motion of a set by the curvature of its boudnary. J. Differential Equations, 101, 1993, 313-372. Zbl0769.35070MR1204331DOI10.1006/jdeq.1993.1015

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