Highly degenerate harmonic mean curvature flow |
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Authors: | M C Caputo P Daskalopoulos |
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Institution: | (1) Department of Mathematics, University of Texas at Austin, Austin, TX, USA;(2) Department of Mathematics, Columbia University, New York, NY, USA |
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Abstract: | We study the evolution of a weakly convex surface in with flat sides by the Harmonic Mean Curvature flow. We establish the short time existence as well as the optimal regularity
of the surface and we show that the boundaries of the flat sides evolve by the curve shortening flow. It follows from our
results that a weakly convex surface with flat sides of class C
k,γ
, for some and 0 < γ ≤ 1, remains in the same class under the flow. This distinguishes this flow from other, previously studied, degenerate parabolic
equations, including the porous medium equation and the Gauss curvature flow with flat sides, where the regularity of the
solution for t > 0 does not depend on the regularity of the initial data.
M. C. Caputo partially supported by the NSF grant DMS-03-54639. P. Daskalopoulos partially supported by the NSF grants DMS-01-02252,
DMS-03-54639 and the EPSRC in the UK. |
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Keywords: | Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 35XX 53XX |
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