A nonlinear partial differential equation for the volume preserving mean curvature flow

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March 2013, 8(1): 9-22. doi: 10.3934/nhm.2013.8.9

A nonlinear partial differential equation for the volume preserving mean curvature flow

Dimitra Antonopoulou ^1, and Georgia Karali ^1,

1.	Department of Applied Mathematics, University of Crete, 714 09 Heraklion

Received September 2011 Published April 2013

Abstract
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We analyze the evolution of multi-dimensional normal graphs over the unit sphere under volume preserving mean curvature flow and derive a non-linear partial differential equation in polar coordinates. Furthermore, we construct finite difference numerical schemes and present numerical results for the evolution of non-convex closed plane curves under this flow, to observe that they become convex very fast.

Keywords: Nonlinear parabolic equations, geometric evolution equations, normal graphs, volume preserving mean curvature flow, numerics..

Mathematics Subject Classification: Primary: 35; Secondary: 3.

Citation: Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks and Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9

References:

[1]	N. D. Alikakos and A. Freire, The normalized mean curvature flow for a small bubble in a Riemannian manifold, J. Differential Geom., 64 (2003), 247-303.
[2]	D. C. Antonopoulou, G. D. Karali and I. M. Sigal, Stability of spheres under volume preserving mean curvature flow, Dynamics of PDE, 7 (2010), 327-344.
[3]	J. Escher and G. Simonett, A center manifold analysis for the mullins-sekerka model, J. Differential Eq., 143 (1998), 267-292. doi: 10.1006/jdeq.1997.3373.
[4]	J. Escher and G. Simonett, The volume preserving mean curvature flow near spheres, Proc. Amer. Math. Soc., 126 (1998), 2789-2796. doi: 10.1090/S0002-9939-98-04727-3.
[5]	M. Gage, On an area-preserving evolution equation for plane curves, Nonlinear Problems in Geometry, D. M. DeTurck, editor, Contemp. Math., 51 (1986), AMS, Providence, 51-62. doi: 10.1090/conm/051/848933.
[6]	M. Gage and R. Hamilton, The Heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69-96.
[7]	Z. Gang and I. M. Sigal, Neck pinching dynamics under mean curvature flow, J. Geom. Anal., 19 (2009), 36-80. doi: 10.1007/s12220-008-9050-y.
[8]	M. A. Grayson, The Heat Equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285-314.
[9]	G. Huisken, The volume preserving mean curvature flow, J. Reine Angew. Math., 382 (1987), 35-48. doi: 10.1515/crll.1987.382.35.
[10]	E. Kreyszig, "Differential Geometry," Dover Publications, New York, 1991.
[11]	N. Shimakura, "Partial Differential Operators of Elliptic Type," Translations of Mathematical Monographs, 99, 1992.
[12]	M. Struwe, Geometric evolution problems. Nonlinear partial differential equations in differential geometry, IAS/Park City Math. Ser., 2, Amer. Math. Soc., Providence, RI, Park City, UT, (1992), 257-339.

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References:

[1]	N. D. Alikakos and A. Freire, The normalized mean curvature flow for a small bubble in a Riemannian manifold, J. Differential Geom., 64 (2003), 247-303.
[2]	D. C. Antonopoulou, G. D. Karali and I. M. Sigal, Stability of spheres under volume preserving mean curvature flow, Dynamics of PDE, 7 (2010), 327-344.
[3]	J. Escher and G. Simonett, A center manifold analysis for the mullins-sekerka model, J. Differential Eq., 143 (1998), 267-292. doi: 10.1006/jdeq.1997.3373.
[4]	J. Escher and G. Simonett, The volume preserving mean curvature flow near spheres, Proc. Amer. Math. Soc., 126 (1998), 2789-2796. doi: 10.1090/S0002-9939-98-04727-3.
[5]	M. Gage, On an area-preserving evolution equation for plane curves, Nonlinear Problems in Geometry, D. M. DeTurck, editor, Contemp. Math., 51 (1986), AMS, Providence, 51-62. doi: 10.1090/conm/051/848933.
[6]	M. Gage and R. Hamilton, The Heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69-96.
[7]	Z. Gang and I. M. Sigal, Neck pinching dynamics under mean curvature flow, J. Geom. Anal., 19 (2009), 36-80. doi: 10.1007/s12220-008-9050-y.
[8]	M. A. Grayson, The Heat Equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285-314.
[9]	G. Huisken, The volume preserving mean curvature flow, J. Reine Angew. Math., 382 (1987), 35-48. doi: 10.1515/crll.1987.382.35.
[10]	E. Kreyszig, "Differential Geometry," Dover Publications, New York, 1991.
[11]	N. Shimakura, "Partial Differential Operators of Elliptic Type," Translations of Mathematical Monographs, 99, 1992.
[12]	M. Struwe, Geometric evolution problems. Nonlinear partial differential equations in differential geometry, IAS/Park City Math. Ser., 2, Amer. Math. Soc., Providence, RI, Park City, UT, (1992), 257-339.

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