ROLLING STONES WITH NONCONVEX SIDES I: REGULARITY THEORY

Title & Authors
ROLLING STONES WITH NONCONVEX SIDES I: REGULARITY THEORY
Lee, Ki-Ahm; Rhee, Eun-Jai;

Abstract
In this paper, we consider the regularity theory and the existence of smooth solution of a degenerate fully nonlinear equation describing the evolution of the rolling stones with nonconvex sides: $\small{\{M(h)=h_t-F(t,z,z^{\alpha}h_{zz})\;in\;\{0}$<$\small{z{\leq}1\}{\times}[0,T] \\ h_t(z,t)=H(h_z(z,t),h)\;{on}\;\{z=0\}}$. We establish the Schauder theory for $\small{C^{2,{\alpha}}}$-regularity of h.
Keywords
rolling stone;degenerate fully nonlinear equations;free boundary problem;
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 ow, Comm. Pure Appl. Math. 44 (1991), no. 4, 469-483.

5.
P. Daskalopoulos and R. Hamilton, The free boundary in the Gauss curvature ow with at 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.

10.
P. Daskalopoulos and K. Lee Free-Boundary Regularity on the Focusing Problem for the Gauss Curvature Flow with Flat sides, Math. Z. 237 (2001), no. 4, 847-874.

11.
P. Daskalopoulos and K. Lee, Worn stones with at sides all time regularity of the interface, Invent. Math. 156 (2004), no. 3, 445-493.

12.
P. Daskalopoulos and K. Lee, Holder regularity of solutions of degenerate elliptic and parabolic equations, J. Funct. Anal. 201 (2003), no. 2, 341-379.

13.
P. Daskalopoulos and E. Rhee Free-boundary regularity for generalized porous medium equations, Commun. Pure Appl. Anal. 2 (2003), no. 4, 481-494.

14.
W. Firey, Shapes of worn stones, Mathematica 21 (1974), 1-11.

15.
R. Hamilton, Worn stones with at sides, A tribute to Ilya Bakelman (College Station, TX, 1993), 69-78, Discourses Math. Appl., 3, Texas A & M Univ., College Station, TX, 1994.

16.
H. Ishii and T. Mikami, A mathematical model of the wearing process of a nonconvex stone, SIAM J. Math. Anal. 33 (2001), no. 4, 860-876.

17.
H. Ishii and T. Mikami, A level set approach to the wearing process of a nonconvex stone, Calc. Var. Partial Differential Equations 19 (2004), no. 1, 53-93.

18.
N. V. Krylov and N. 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.

19.
G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientic Publishing Co., Inc., River Edge, NJ, 1996.

20.
K.-A. Lee and E. Rhee, Rolling Stones with nonconvex sides II: All time regularity of Interface and surface, preprint.

21.
K. Tso, Deforming a hypersurface by its gauss-Kronecker curvature, Comm. Pure Appl. Math. 38 (1985), no. 6, 867-882.

22.
L. Wang, On the regularity theory of fully nonlinear parabolic equations I, Comm. Pure Appl. Math. 45 (1992), no. 1, 27-76.

23.
L. Wang, On the regularity theory of fully nonlinear parabolic equations II, Comm. Pure Appl. Math. 45 (1992), no. 2, 141-178.