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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Gage and R. Hamilton, The Heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69-96.  Google Scholar

[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.  Google Scholar

[8]

M. A. Grayson, The Heat Equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285-314.  Google Scholar

[9]

G. Huisken, The volume preserving mean curvature flow, J. Reine Angew. Math., 382 (1987), 35-48. doi: 10.1515/crll.1987.382.35.  Google Scholar

[10]

E. Kreyszig, "Differential Geometry," Dover Publications, New York, 1991.  Google Scholar

[11]

N. Shimakura, "Partial Differential Operators of Elliptic Type," Translations of Mathematical Monographs, 99, 1992.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Gage and R. Hamilton, The Heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69-96.  Google Scholar

[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.  Google Scholar

[8]

M. A. Grayson, The Heat Equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285-314.  Google Scholar

[9]

G. Huisken, The volume preserving mean curvature flow, J. Reine Angew. Math., 382 (1987), 35-48. doi: 10.1515/crll.1987.382.35.  Google Scholar

[10]

E. Kreyszig, "Differential Geometry," Dover Publications, New York, 1991.  Google Scholar

[11]

N. Shimakura, "Partial Differential Operators of Elliptic Type," Translations of Mathematical Monographs, 99, 1992.  Google Scholar

[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.  Google Scholar

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