2014 Fiscal Year Final Research Report

Studies on approximate problems, regularity and singularity for mean curvature flow

Research Project

PDF

Project/Area Number	24540124
Research Category	Grant-in-Aid for Scientific Research (C)
Allocation Type	Multi-year Fund
Section	一般
Research Field	General mathematics (including Probability theory/Statistical mathematics)
Research Institution	Kobe University
Principal Investigator	ISHII Katsuyuki 神戸大学, 海事科学研究科(研究院), 教授 (40232227)
Co-Investigator(Kenkyū-buntansha)	NAITO Yuki 愛媛大学, 大学院理工学研究科, 教授 (10231458) UEDA Yoshihiro 神戸大学, 大学院海事科学研究科, 准教授 (50534856)
Co-Investigator(Renkei-kenkyūsha)	TAKAHASHI Futoshi 大阪市立大学, 大学院理学研究科, 教授 (10374901) KUWAMURA Masataka 神戸大学, 大学院人間発達環境学研究科, 教授 (30270333) OHNUMA Masaki 徳島大学, ソシオ・アーツ・サイエンス研究部, 准教授 (90304500) AKAGI Goro 神戸大学, 大学院システム情報学研究科, 准教授 (60360202) ISHIWATA Tetsuya 芝浦工業大学, システム理工学部, 准教授 (50334917)
Project Period (FY)	2012-04-01 – 2015-03-31
Keywords	平均曲率流 / 近似アルゴリズム / 等高面の方法 / 粘性解 / 正則性 / 特異性
Outline of Final Research Achievements	In this research project we study some approximate problems, regularity and singularity for the motion of a curve or a surface by its mean curvature, called mean curvature flow. We study some threshold-type algorithms for mean curvature flow, which is proposed by Chambolle in 2004. We prove the convergence of his algorithm by using the mathematical morphology in image processing, the level-set method, the signed distance function and the theory of viscosity solutions. As an application, we prove the convergence to the planar motion of a curve by non-smooth interface energies. As for the generalized mean curvature flow constructed by the above algorithms we show that if it does not fatten and the gradient of the auxiliary fucntion of the generalized flow does not vanish, then the generalized motion becomes smooth near such portions. We obtain some idea to prove the convergence of an algorithm for mean curvature flow in higher codimension and can expect the future study.
Free Research Field	非線形解析