Limits of Solutions to a Parabolic Monge-Ampere Equation

Details Limits of Solutions to a Parabolic Monge-Ampere Equation Limits of Solutions to a Parabolic Monge-Ampere Equation

2008-02-01

arxiv.org/abs/0802.0208v1

Mathematics

Analysis of PDEs

Scientific paper

We present the results from our earlier paper (arXiv:math/0602484) on the affine normal flow on noncompact convex hypersurfaces in affine space from a more PDE point of view, emphasizing the estimates involved. Our results concern the limits of solutions to a parabolic Monge-Ampere equation on $S^n$, where a sequence of smooth strictly convex initial value functions increase monotonically to a limiting initial value function which is infinite on at least a hemisphere of $S^n$. We prove long-time existence and instantaneous smoothing for quite general initial data, and we characterize ancient solutions as ellipsoids or paraboloids. We make essential use of estimates of Andrews and Gutierrez-Huang, and barriers due to Calabi.

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