Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The volume preserving mean curvature flow
near spheres

Authors: Joachim Escher and Gieri Simonett
Journal: Proc. Amer. Math. Soc. 126 (1998), 2789-2796
MSC (1991): Primary 53C42, 58G11, 58F39; Secondary 35K99
MathSciNet review: 1485470
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: By means of a center manifold analysis we investigate the averaged mean curvature flow near spheres. In particular, we show that there exist global solutions to this flow starting from non-convex initial hypersurfaces.

References [Enhancements On Off] (What's this?)

  • 1. A. D. ALEXANDROV, Uniqueness theorems for surfaces in the large I, Vestnik Leningrad Univ. 11, 5-7 (1956); English transl., Amer. Math. Soc. Transl. (2) 21 (1962), 341-353. MR 19:167c; MR 27:698a
  • 2. H. AMANN, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in H. J. Schmeisser, H. Triebel, editors, Function Spaces, Differential Operators and Nonlinear Analysis, Teubner, Stuttgart, Leipzig (1993), 9-126. MR 94m:35153
  • 3. G. DA PRATO & P. GRISVARD, Equations d'évolution abstraites non linéaires de type parabolique, Ann. Mat. Pura Appl., (4) 120, 329-396 (1979). MR 81d:34052
  • 4. G. DA PRATO & A. LUNARDI, Stability, instability and center manifold theorem for fully nonlinear autonomous parabolic equations in Banach space, Arch. Rational Mech. Anal. 101, 115-144 (1988). MR 89e:35019
  • 5. J. ESCHER & G. SIMONETT, On Hele-Shaw models with surface tension, Math. Res. Lett. 3, 467-474 (1996). MR 97i:35145
  • 6. J. ESCHER & G. SIMONETT, Classical solutions for Hele-Shaw models with surface tension, Adv. Differential Equations 2, 619-642 (1997). CMP 97:10
  • 7. J. ESCHER & G. SIMONETT, A center manifold analysis for the Mullins-Sekerka model, J. Differential Equations, to appear.
  • 8. M. GAGE, On an area-preserving evolution equation for plane curves, Nonlinear Problems in Geometry, D.M. DeTurck, editor, Contemp. Math. 51, AMS, Providence, 51-62, (1986). MR 87g:53003
  • 9. M. GAGE & R. HAMILTON, The heat equation shrinking convex plane curves, J. Differential Geom. 23, 69-96 (1986). MR 87m:53003
  • 10. G. HUISKEN, The volume preserving mean curvature flow, J. Reine Angew. Math. 382, 35-48 (1987). MR 89d:53015
  • 11. H. B. LAWSON, Lectures on Minimal Submanifolds, Publish or Perish, Berkeley, 1980. MR 82d:53035b
  • 12. A. LUNARDI, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995. MR 96e:47039
  • 13. G. SIMONETT, Center manifolds for quasilinear reaction-diffusion systems, Differential Integral Equations 8, 753-796 (1995). MR 96a:35091

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 53C42, 58G11, 58F39, 35K99

Retrieve articles in all journals with MSC (1991): 53C42, 58G11, 58F39, 35K99

Additional Information

Joachim Escher
Affiliation: Mathematical Institute, University of Basel, CH-4051 Basel, Switzerland
Address at time of publication: FB 17 Mathematics, University of Kassel, D-34132 Kassel, Germany

Gieri Simonett
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240

Keywords: Generalized motion by mean curvature, center manifolds
Received by editor(s): December 14, 1996
Received by editor(s) in revised form: February 7, 1997
Communicated by: Peter Li
Article copyright: © Copyright 1998 American Mathematical Society