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

DOI:
https://doi.org/10.1090/S0002-9939-98-04727-3

MathSciNet review:
1485470

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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.

**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**

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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

DOI:
https://doi.org/10.1090/S0002-9939-98-04727-3

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