## The volume preserving mean curvature flow near spheres

HTML articles powered by AMS MathViewer

- by Joachim Escher and Gieri Simonett PDF
- Proc. Amer. Math. Soc.
**126**(1998), 2789-2796 Request permission

## 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

- Saunders MacLane and O. F. G. Schilling,
*Infinite number fields with Noether ideal theories*, Amer. J. Math.**61**(1939), 771–782. MR**19**, DOI 10.2307/2371335 - Herbert Amann,
*Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems*, Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992) Teubner-Texte Math., vol. 133, Teubner, Stuttgart, 1993, pp. 9–126. MR**1242579**, DOI 10.1007/978-3-663-11336-2_{1} - Giuseppe Da Prato and Pierre Grisvard,
*Equations d’évolution abstraites non linéaires de type parabolique*, Ann. Mat. Pura Appl. (4)**120**(1979), 329–396 (French, with English summary). MR**551075**, DOI 10.1007/BF02411952 - G. Da Prato and A. Lunardi,
*Stability, instability and center manifold theorem for fully nonlinear autonomous parabolic equations in Banach space*, Arch. Rational Mech. Anal.**101**(1988), no. 2, 115–141. MR**921935**, DOI 10.1007/BF00251457 - Joachim Escher and Gieri Simonett,
*On Hele-Shaw models with surface tension*, Math. Res. Lett.**3**(1996), no. 4, 467–474. MR**1406012**, DOI 10.4310/MRL.1996.v3.n4.a5 - J. ESCHER & G. SIMONETT, Classical solutions for Hele-Shaw models with surface tension,
*Adv. Differential Equations***2**, 619–642 (1997). - J. ESCHER & G. SIMONETT, A center manifold analysis for the Mullins-Sekerka model,
*J. Differential Equations*, to appear. - Michael Gage,
*On an area-preserving evolution equation for plane curves*, Nonlinear problems in geometry (Mobile, Ala., 1985) Contemp. Math., vol. 51, Amer. Math. Soc., Providence, RI, 1986, pp. 51–62. MR**848933**, DOI 10.1090/conm/051/848933 - M. Gage and R. S. Hamilton,
*The heat equation shrinking convex plane curves*, J. Differential Geom.**23**(1986), no. 1, 69–96. MR**840401**, DOI 10.4310/jdg/1214439902 - Gerhard Huisken,
*The volume preserving mean curvature flow*, J. Reine Angew. Math.**382**(1987), 35–48. MR**921165**, DOI 10.1515/crll.1987.382.35 - H. Blaine Lawson Jr.,
*Lectures on minimal submanifolds. Vol. I*, Monografías de Matemática [Mathematical Monographs], vol. 14, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1977. MR**527121** - Alessandra Lunardi,
*Analytic semigroups and optimal regularity in parabolic problems*, Progress in Nonlinear Differential Equations and their Applications, vol. 16, Birkhäuser Verlag, Basel, 1995. MR**1329547**, DOI 10.1007/978-3-0348-9234-6 - Gieri Simonett,
*Center manifolds for quasilinear reaction-diffusion systems*, Differential Integral Equations**8**(1995), no. 4, 753–796. MR**1306591**

## 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
- Email: escher@mathematik.uni-kassel.de
**Gieri Simonett**- Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
- Email: simonett@math.vanderbilt.edu
- Received by editor(s): December 14, 1996
- Received by editor(s) in revised form: February 7, 1997
- Communicated by: Peter Li
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**126**(1998), 2789-2796 - MSC (1991): Primary 53C42, 58G11, 58F39; Secondary 35K99
- DOI: https://doi.org/10.1090/S0002-9939-98-04727-3
- MathSciNet review: 1485470