American Institute of Mathematical Sciences

Advanced
June  2005, 4(2): 311-339. doi: 10.3934/cpaa.2005.4.311

Method of the distance function to the Bence-Merriman-Osher algorithm for motion by mean curvature

Y. Goto 1, , K. Ishii 2, and  T. Ogawa 3,

1. 

NTT Comware, Minato, Tokyo, 108-8019, Japan
2. 

Faculty of Maritime Sciences, Kobe University, Higashinada, Kobe 658-0022, Japan
3. 

Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan

Received  March 2003 Revised  May 2004 Published  March 2005

A new proof of the convergence of the Bence-Merriman-Osher algorithm for the motion of mean curvature is given. The idea is making use of the approximate distance function to the interface and analogous argument in the singular limiting problem for the Allen-Cahn equation via an auxiliary function given by the primitive function of the heat kernel.
Keywords: signed distance function, Motionby mean curvature, viscosity solutions., Bence-Merrina-Osher algorithm.
Mathematics Subject Classification: 35K65,35K55,65M99.
Citation: Y. Goto, K. Ishii, T. Ogawa. Method of the distance function to the Bence-Merriman-Osher algorithm for motion by mean curvature. Communications on Pure and Applied Analysis, 2005, 4 (2) : 311-339. doi: 10.3934/cpaa.2005.4.311
[1]

Kateřina Škardová, Tomáš Oberhuber, Jaroslav Tintěra, Radomír Chabiniok. Signed-distance function based non-rigid registration of image series with varying image intensity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1145-1160. doi: 10.3934/dcdss.2020386
[2]

Changfeng Gui, Huaiyu Jian, Hongjie Ju. Properties of translating solutions to mean curvature flow. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 441-453. doi: 10.3934/dcds.2010.28.441
[3]

Kin Ming Hui. Existence of self-similar solutions of the inverse mean curvature flow. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 863-880. doi: 10.3934/dcds.2019036
[4]

Chiara Corsato, Colette De Coster, Pierpaolo Omari. Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape. Conference Publications, 2015, 2015 (special) : 297-303. doi: 10.3934/proc.2015.0297
[5]

Hongjie Ju, Jian Lu, Huaiyu Jian. Translating solutions to mean curvature flow with a forcing term in Minkowski space. Communications on Pure and Applied Analysis, 2010, 9 (4) : 963-973. doi: 10.3934/cpaa.2010.9.963
[6]

John D. Towers. An explicit finite volume algorithm for vanishing viscosity solutions on a network. Networks and Heterogeneous Media, 2022, 17 (1) : 1-13. doi: 10.3934/nhm.2021021
[7]

Michel C. Delfour. Hadamard Semidifferential, Oriented Distance Function, and some Applications. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1917-1951. doi: 10.3934/cpaa.2021076
[8]

Rhudaina Z. Mohammad, Karel Švadlenka. Multiphase volume-preserving interface motions via localized signed distance vector scheme. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 969-988. doi: 10.3934/dcdss.2015.8.969
[9]

Jun Wang, Wei Wei, Jinju Xu. Translating solutions of non-parametric mean curvature flows with capillary-type boundary value problems. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3243-3265. doi: 10.3934/cpaa.2019146
[10]

Alberto Farina, Miguel Angel Navarro. Some Liouville-type results for stable solutions involving the mean curvature operator: The radial case. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 1233-1256. doi: 10.3934/dcds.2020076
[11]

Ruyun Ma, Man Xu. Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2701-2718. doi: 10.3934/dcdsb.2018271
[12]

Guowei Dai, Alfonso Romero, Pedro J. Torres. Global bifurcation of solutions of the mean curvature spacelike equation in certain standard static spacetimes. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3047-3071. doi: 10.3934/dcdss.2020118
[13]

Alessio Pomponio. Oscillating solutions for prescribed mean curvature equations: euclidean and lorentz-minkowski cases. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3899-3911. doi: 10.3934/dcds.2018169
[14]

Diego Castellaneta, Alberto Farina, Enrico Valdinoci. A pointwise gradient estimate for solutions of singular and degenerate pde's in possibly unbounded domains with nonnegative mean curvature. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1983-2003. doi: 10.3934/cpaa.2012.11.1983
[15]

Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial and Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117
[16]

Peter Frolkovič, Karol Mikula, Jozef Urbán. Distance function and extension in normal direction for implicitly defined interfaces. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 871-880. doi: 10.3934/dcdss.2015.8.871
[17]

G. Kamberov. Prescribing mean curvature: existence and uniqueness problems. Electronic Research Announcements, 1998, 4: 4-11.

[18]

Giulio Colombo, Luciano Mari, Marco Rigoli. Remarks on mean curvature flow solitons in warped products. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1957-1991. doi: 10.3934/dcdss.2020153
[19]

Zhengchao Ji. Cylindrical estimates for mean curvature flow in hyperbolic spaces. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1199-1211. doi: 10.3934/cpaa.2021016
[20]

Georgi I. Kamberov. Recovering the shape of a surface from the mean curvature. Conference Publications, 1998, 1998 (Special) : 353-359. doi: 10.3934/proc.1998.1998.353

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (97)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy