Convexified Gauss curvature flow of sets: A stochastic approximation

SIAM Journal on Mathematical Analysis

Volumn 36, Issue 2, 2005, Pages 552-579

  Ishii, Hitoshi ^a   Mikami, Toshio ^b  

^a WASEDA UNIVERSITY   (Japan)

^b HOKKAIDO UNIVERSITY   (Japan)

Author keywords

Convexified Gauss curvature flow; Stochastic approximation; Viscosity solution; Weak solution

Indexed keywords

ANISOTROPY; APPROXIMATION THEORY; BOUNDARY LAYER FLOW; CONTINUUM MECHANICS; EIGENVALUES AND EIGENFUNCTIONS; FUNCTIONS; GRAPH THEORY; MAPPING; MATHEMATICAL MODELS; STOCHASTIC PROGRAMMING; THEOREM PROVING; VECTORS; VISCOSITY;

CONVEXIFIED GAUSS CURVATURE FLOW; STOCHASTIC APPROXIMATION; VISCOSITY SOLUTIONS; WEAK SOLUTION;

SET THEORY;

EID: 14944371769     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141002420509     Document Type: Article

References (23)

1. D. ADALSTEINSSON, L. C. EVANS, AND H. ISHII, The level set method for etching and deposition, Math. Models Methods Appl. Sci., 7 (1997), pp. 1153-1186.
2. B. ANDREWS, Gauss curvature flow: The fate of the rolling stones, Invent. Math., 138 (1999), pp. 151-161.
3. B. ANDREWS, Motion of hypersurfaces by Gauss curvature, Pacific J. Math., 195 (2000), pp. 1-34.
4. I. J. BAKELMAN, Convex Analysis and Nonlinear Geometric Elliptic Equations, Springer-Verlag, Berlin, Heidelberg, New York, 1994.
5. Y.-G. CHEN, Y. GIGA, AND S. GOTO, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), pp. 749-786.
6. D. CHOPP, L. C. EVANS, AND H. ISHII, Waiting time effects for Gauss curvature flows, Indiana Univ. Math. J., 48 (1999), pp. 311-334.
7. B. CHOW, Deforming convex hypersurfaces by the nth root of the Gaussian curvature, J. Differential Geom., 22 (1985), pp. 117-138.
8. M. G. CRANDALL, H. ISHII, AND P.-L. LIONS, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), pp. 1-67.
9. S. N. ETHIER AND T. G. KURTZ, Markov Processes: Characterization and Convergence, John Wiley and Sons, New York, 1986.
10. L. C. EVANS AND J. SPRUCK, Motion of level sets by mean curvature I, J. Differential Geom., 33 (1991), pp. 635-681.
11. W. J. FIREY, Shapes of worn stones, Mathematika, 21 (1974), pp. 1-11.
12. M.-H. GIGA AND Y. GIGA, Crystalline and level set flow-convergence of a crystalline algorithm for a general anisotropic curvature flow in the plain, in Free Boundary Problems: Theory and Applications, I (Chiba, 1999), GAKUTO Internat. Ser. Math. Sci. Appl. 13, Gakkōosho, Tokyo, 2000, pp. 64-79.
13. P. M. GIRÃO, Convergence of a crystalline algorithm for the motion of a simple closed convex curve by weighted curvature, SIAM J. Numer. Anal., 32 (1995), pp. 886-899.
14. R. HAMILTON, Worn stones with flat sides, in A Tribute to Ilya Bakelman (College Station, TX, 1993), Discourses Math. Appl. 3, Texas A & M University, College Station, TX, 1994, pp. 69-78.
15. N. IKEDA AND S. WATANABE, Stochastic Differential Equations and Diffusion Processes, North-Holland/Kodansha, Tokyo, 1981.
16. H. ISHII, Gauss curvature flow and its approximation, in Free Boundary Problems: Theory and Applications, II (Chiba, 1999), GAKUTO Internat. Ser. Math. Sci. Appl. 14, Gakkõtosho, Tokyo, 2000, pp. 198-206.
17. H. ISHII AND T. MIKAMI, A mathematical model of the wearing process of a nonconvex stone, SIAM J. Math. Anal., 33 (2001), pp. 860-876.
18. H. ISHII AND T. MIKAMI, A two dimensional random crystalline algorithm for Gauss curvature flow, Adv. in Appl. Probab., 34 (2002), pp. 491-504.
19. H. ISHII AND T. MIKAMI, A level set approach to the wearing process of a nonconvex stone, Calc. Var. Partial Differential Equations, 19 (2003), pp. 53-93.
20. H. ISHII AND T. MIKAMI, Motion of a graph by R-curvature, Arch. Ration. Mech. Anal., 171 (2004), pp. 1-23.
21. H. ISHII AND P. E. SOUGANIDIS, Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor, Tohoku Math. J. (2), 47 (1995), pp. 227-250.
22. S. OSHER AND J. SETHIAN, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), pp. 12-49.
23. K. TSO, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm. Pure Appl. Math., 38 (1985), pp. 867-882.