𝐶^{{}2,𝛼} estimate of a parabolic Monge-Ampère equation on 𝑆^{{}𝑛}

Tsai, Dong-Ho

doi:10.1090/S0002-9939-03-06848-5

$C^\{2,\alpha \}$ estimate of a parabolic Monge-Ampère equation on $S^\{n\}$
HTML articles powered by AMS MathViewer

by Dong-Ho Tsai PDF

Proc. Amer. Math. Soc. 131 (2003), 3067-3074 Request permission

Abstract:

We consider a special type of parabolic Monge-Ampère equation on $S^{n}$ arising from convex hypersurfaces expansion in Euclidean spaces. We obtained a $C^{2,\alpha }$ parabolic estimate of the support functions for the convex hypersurfaces assuming that we have already had a $C^{2}$ parabolic estimate.

References

Ben Andrews, Motion of hypersurfaces by Gauss curvature, Pacific J. Math. 195 (2000), no. 1, 1–34. MR 1781612, DOI 10.2140/pjm.2000.195.1
Luis A. Caffarelli, Interior $W^{2,p}$ estimates for solutions of the Monge-Ampère equation, Ann. of Math. (2) 131 (1990), no. 1, 135–150. MR 1038360, DOI 10.2307/1971510
Bennett Chow, Geometric aspects of Aleksandrov reflection and gradient estimates for parabolic equations, Comm. Anal. Geom. 5 (1997), no. 2, 389–409. MR 1483984, DOI 10.4310/CAG.1997.v5.n2.a5
Bennett Chow and Robert Gulliver, Aleksandrov reflection and nonlinear evolution equations. I. The $n$-sphere and $n$-ball, Calc. Var. Partial Differential Equations 4 (1996), no. 3, 249–264. MR 1386736, DOI 10.1007/BF01254346
Bennett Chow and Dong-Ho Tsai, Nonhomogeneous Gauss curvature flows, Indiana Univ. Math. J. 47 (1998), no. 3, 965–994. MR 1665729, DOI 10.1512/iumj.1998.47.1546
N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 1, 161–175, 239 (Russian). MR 563790
O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Lineĭ nye i kvazilineĭ nye uravneniya parabolicheskogo tipa, Izdat. “Nauka”, Moscow, 1967 (Russian). MR 0241821
Kaising Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm. Pure Appl. Math. 38 (1985), no. 6, 867–882. MR 812353, DOI 10.1002/cpa.3160380615
John I. E. Urbas, Correction to: “An expansion of convex hypersurfaces” [J. Differential Geom. 33 (1991), no. 1, 91–125; MR1085136 (91j:58155)], J. Differential Geom. 35 (1992), no. 3, 763–765. MR 1163459