| Project/Area Number |
16K17622
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Mathematical analysis
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| Research Institution | Kyoto University (2017-2019)
The University of Tokyo (2016) |
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Principal Investigator |
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| Project Period (FY) |
2016-04-01 – 2020-03-31
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| Project Status |
Completed (Fiscal Year 2019)
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| Budget Amount *help |
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
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| Keywords | 平均曲率流 / 幾何学的測度論 / フェイズフィールド法 / 特異極限問題 / 変分問題 / バリフォールド / 弱解 / Allen-Cahn方程式 / 動的境界条件 / 極小曲面 |
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| Outline of Final Research Achievements |
The mean curvature flow equation is an equation to describe the motion of metal grain boundaries in the annealing process. The existence of a time-global existence of the weak solution for the mean curvature flow equation is known, but when there are boundary condition or forcing term, it is difficult to prove the existence of the solution.
In this work, we showed that the solutions for Allen-Cahn equation with dynamic boundary condition converges to a time-global weak solution for the mean curvature flow equation with dynamic boundary condition, under suitable assumptions. We also proved the existence of time-global weak solutions for the mean curvature flow equation with forcing term belonging to the Sobolev class, using the Allen-Cahn equation with forcing term.
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| Academic Significance and Societal Importance of the Research Achievements |
本研究では、境界条件や外力項を課した場合における、Allen-Cahn方程式と平均曲率流方程式との関係性を明らかにした。従来の外力項付きAllen-Cahn方程式では証明に必要な評価を得ることが出来なかったが、適切な補正項を加えることにより、それを解決した。この補正項を加える方法は他の方程式への応用も期待できる。
また、弱解の構成で用いたフェイズフィールド法は数値計算にも用いられる手法であり、本研究で得られた結果は実学への応用も期待できる。
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