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

Dimitra Antonopoulou; Georgia Karali; Dimitra Antonopoulou; Georgia Karali

doi:10.3934/nhm.2013.8.9

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

Dimitra Antonopoulou ^{1
,},
Georgia Karali ^{1
,}

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.
- Nonlinear parabolic equations,
- geometric evolution equations,
- normal graphs,
- volume preserving mean curvature flow,
- numerics
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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Abstract

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.

References

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