ROLLING STONES WITH NONCONVEX SIDES II: ALL TIME REGULARITY OF INTERFACE AND SURFACE

Title & Authors
ROLLING STONES WITH NONCONVEX SIDES II: ALL TIME REGULARITY OF INTERFACE AND SURFACE
Lee, Ki-Ahm; Rhee, Eun-Jai;

Abstract
In this paper we consider the evolution of the rolling stone with a rotationally symmetric nonconvex compact initial surface $\small{{\Sigma}_0}$ under the Gauss curvature flow. Let $\small{X:S^n{\times}[0,\;{\infty}){\rightarrow}\mathbb{R}^{n+1}}$ be the embeddings of the sphere in $\small{\mathbb{R}^{n+1}}$ such that $\Sigma(t) Keywords free boundary problems;degenerate fully nonlinear equations;Gauss curvature flow; Language English Cited by References 1. B. Andrews, Gauss curvature ow: The fate of the rolling stones, Invent. Math. 138 (1999), no. 1, 151-161. 2. D. Chopp, L. C. Evans, and H. Ishii, Waiting time effects for Gauss curvature flow, Indiana Univ. Math. J. 48 (1999), no. 1, 311-334. 3. B. Chow, Deforming convex hypersurfaces by the nth root of the Gaussian curvature, J. Differential Geom. 22 (1985), no. 1, 117-138. 4. B. Chow, On Harnack's inequality and entropy for the Gaussian curvature flow, Comm. Pure Appl. Math. 44 (1991), no. 4, 469-483. 5. P. Daskalopoulos and R. Hamilton, The free boundary in the Gauss curvature flow with flat sides, J. Reine Angew. Math. 510 (1999), 187-227. 6. P. Daskalopoulos and R. Hamilton, The free boundary for the n-dimensional porous medium equation, Internat. Math. Res. Notices 1997 (1997), no. 17, 817-831. 7. P. Daskalopoulos and R. Hamilton, Regularity of the free boundary for the porous medium equation, J. Amer. Math. Soc. 11 (1998), no. 4, 899-965. 8. P. Daskalopoulos and R. Hamilton,$C^{{\infty}}$-regularity of the interface of the evolution pp-Laplacian equation, Math. Res. Lett. 5 (1998), no. 5, 685-701. 9. P. Daskalopoulos, R. Hamilton, and K. Lee, All time$C^{{\infty}}\$-regularity of the interface in degenerate diffusion: a geometric approach, Duke Math. J. 108 (2001), no. 2, 295-327.

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