Dynamical aspects of geometric evolution equations defined on curves or surfaces

2015 Fiscal Year Final Research Report

• Project/Area Number: 24740097
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Global analysis
• Research Institution: Tohoku University
• Principal Investigator: Okabe Shinya 東北大学, 理学(系)研究科(研究院), 准教授 (70435973)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Keywords: 幾何学的発展方程式 / 変分法

Outline of Final Research Achievements

We consider the following problems: (A) Dynamical aspects of geometric evolution equations defined on planar curves; (B) Obstacle problem for parabolic biharmonic equation. Regarding the theme (A), we proved that (A1) a planar open curve with infinite length converges to the borderline elastica as time goes to infinity; (A2) if a solution of geometric evolution equation converges to an equilibrium along a sequence of time, then the curve converges to an equilibrium as time goes to infinity, provided that the set of the equilibria has a discrete structure. Concerning the theme (B), we considered (B1) obstacle problem for the parabolic biharmonic equation, and (B2) two-obstacle problem for the parabolic biharmonic equation. For each problems (B1) and (B2), we proved the long time existence of solutions and investigated the regularity of the solutions.

Free Research Field

非線形解析学