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Networks and Heterogeneous Media

2013, Volume 8, Issue 1: 9-22. doi: 10.3934/nhm.2013.8.9
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A nonlinear partial differential equation for the volume preserving mean curvature flow

  • 1.
    Department of Applied Mathematics, University of Crete, 714 09 Heraklion
  • Received: 01 September 2011

  • Primary: 35; Secondary: 37.

  • 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.

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

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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.


    [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
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    [10] E. Kreyszig, "Differential Geometry," Dover Publications, New York, 1991.
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