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2014 Fiscal Year Final Research Report

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

Research Project

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Project/Area Number 24540124
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKobe 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

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Published: 2016-06-03  

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